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. 00:00:23.370 --> 00:00:24.403 PROFESSOR: Apparently. 00:00:28.190 --> 00:00:29.210 We'll start slow. 00:00:29.210 --> 00:00:31.088 So last time we were talking, we just 00:00:31.088 --> 00:00:32.630 started talking about effective field 00:00:32.630 --> 00:00:34.310 theory with a fine tuning. 00:00:34.310 --> 00:00:36.620 And what actually that means takes 00:00:36.620 --> 00:00:38.060 a little bit of discussion. 00:00:41.840 --> 00:00:44.380 So what you could mean by a fine tuning 00:00:44.380 --> 00:00:47.350 is that you have something that's irrelevant. 00:00:47.350 --> 00:00:49.610 You look at the operator, you think it's irrelevant, 00:00:49.610 --> 00:00:50.530 but it's not. 00:00:50.530 --> 00:00:51.340 It's relevant. 00:00:51.340 --> 00:00:53.800 Something that you should include even at lowest order 00:00:53.800 --> 00:00:55.300 in your power accounting. 00:00:55.300 --> 00:00:57.815 But saying something is irrelevant 00:00:57.815 --> 00:01:00.190 means that you have a power counting, that you understand 00:01:00.190 --> 00:01:02.650 the power counting for the theory, what the correct power 00:01:02.650 --> 00:01:04.010 counting is. 00:01:04.010 --> 00:01:05.710 So in this example that I'll give, 00:01:05.710 --> 00:01:07.960 what "irrelevant" will mean is that you basically 00:01:07.960 --> 00:01:10.480 do a dimensional power counting, which 00:01:10.480 --> 00:01:11.980 is how we usually think of defining 00:01:11.980 --> 00:01:13.450 irrelevant and relevant. 00:01:13.450 --> 00:01:15.520 Do a dimensional power counting and you end up 00:01:15.520 --> 00:01:17.047 finding an operator that-- 00:01:17.047 --> 00:01:18.880 you find that the operator looks irrelevant, 00:01:18.880 --> 00:01:20.920 but when you do calculations you can 00:01:20.920 --> 00:01:22.618 see that should be relevant. 00:01:22.618 --> 00:01:24.160 And that means, really, what it means 00:01:24.160 --> 00:01:26.433 is that this natural power counting of dimensions 00:01:26.433 --> 00:01:27.850 is not the right one, and you have 00:01:27.850 --> 00:01:29.692 to do something more complicated. 00:01:29.692 --> 00:01:31.150 But it is still is a sense in which 00:01:31.150 --> 00:01:37.110 it can be thought of as a fine tuning, because as you'll see, 00:01:37.110 --> 00:01:39.330 the changing of the power counting from the naive one 00:01:39.330 --> 00:01:41.700 to the more complicated power counting 00:01:41.700 --> 00:01:46.330 involves some kind of tuning, if you like. 00:01:46.330 --> 00:01:47.850 And it really, in this case, we'll 00:01:47.850 --> 00:01:49.730 see actually that it corresponds not 00:01:49.730 --> 00:01:52.230 to expanding around the trivial fixed point, where you would 00:01:52.230 --> 00:01:54.060 have a free theory, but expanding 00:01:54.060 --> 00:01:55.560 around an interactive fixed point. 00:01:58.455 --> 00:01:59.830 So it'll be a little non-trivial, 00:01:59.830 --> 00:02:02.080 but we'll be doing this in the context of two 00:02:02.080 --> 00:02:03.490 nucleon effective field theory. 00:02:03.490 --> 00:02:05.710 And the advantage of this is that the nucleons are 00:02:05.710 --> 00:02:07.390 going to be non-relativistic. 00:02:07.390 --> 00:02:09.857 So P is going to be much less than M pi. 00:02:09.857 --> 00:02:11.440 It's going to be a very simple theory. 00:02:11.440 --> 00:02:14.620 Everything that's an exchange particle gets integrated out. 00:02:14.620 --> 00:02:16.690 It's just a theory of contact interactions, 00:02:16.690 --> 00:02:19.210 and derivatives of contact interactions. 00:02:19.210 --> 00:02:20.800 And because it's non-relativistic, 00:02:20.800 --> 00:02:22.510 we can actually calculate all the loops 00:02:22.510 --> 00:02:24.093 to all orders and perturbation theory, 00:02:24.093 --> 00:02:25.490 and we'll do that in a minute. 00:02:25.490 --> 00:02:28.820 So this theory, we can calculate a lot of stuff. 00:02:28.820 --> 00:02:32.980 And so we'll actually be able to see how this non-trivial fine 00:02:32.980 --> 00:02:35.713 tuning works, and explore it from multiple directions, 00:02:35.713 --> 00:02:37.380 and we'll be sure that what we're saying 00:02:37.380 --> 00:02:38.500 is actually correct. 00:02:41.340 --> 00:02:43.720 So it can be a lesson for understanding 00:02:43.720 --> 00:02:46.330 some of the concepts and other effective field theories 00:02:46.330 --> 00:02:49.360 with the fine tuning, which you might want to design, 00:02:49.360 --> 00:02:53.407 where you don't have as much ability to calculate. 00:02:53.407 --> 00:02:55.240 All right, so let's start off with something 00:02:55.240 --> 00:02:59.650 simple, which is elastic scattering. 00:02:59.650 --> 00:03:01.930 And that's mostly what we're going to talk about. 00:03:07.350 --> 00:03:12.300 Two particles in, two particles out. 00:03:12.300 --> 00:03:15.430 Center of mass frame. 00:03:15.430 --> 00:03:21.183 They scatter to some P's coming in, P primes going out. 00:03:21.183 --> 00:03:23.100 And this is basically a problem that you could 00:03:23.100 --> 00:03:24.390 treat with quantum mechanics. 00:03:27.342 --> 00:03:31.810 It's like non-relativistic scattering. 00:03:31.810 --> 00:03:41.650 So if you have a single partial wave, 00:03:41.650 --> 00:03:43.290 then this scattering is described 00:03:43.290 --> 00:03:45.930 by a phase shift, delta. 00:03:45.930 --> 00:03:51.870 And the relation of the phase shift to the amplitude, 00:03:51.870 --> 00:03:57.600 with our normalization for the amplitude, is this. 00:03:57.600 --> 00:03:59.700 So this is the S-matrix, it's just a phase, 00:03:59.700 --> 00:04:02.670 and that's the relation of the S-matrix to the amplitude. 00:04:02.670 --> 00:04:05.922 And this thing is the amplitude. 00:04:05.922 --> 00:04:07.380 And I guess the other thing we know 00:04:07.380 --> 00:04:12.960 is that, by energy conservation, the magnitude of P 00:04:12.960 --> 00:04:16.529 is equal to the magnitude of P prime. 00:04:16.529 --> 00:04:19.860 All right, so if we rearrange this equation, 00:04:19.860 --> 00:04:23.760 and we write it as A and put the phase, solve for a. 00:04:29.736 --> 00:04:30.810 Could do that. 00:04:34.550 --> 00:04:37.363 So that gives that equation, which we 00:04:37.363 --> 00:04:38.655 can rearrange a little further. 00:04:47.954 --> 00:05:00.690 Now, there's kind of a conventional way of writing it, 00:05:00.690 --> 00:05:02.420 which is that the amplitude should 00:05:02.420 --> 00:05:06.440 be given by 1 over something that's P cotangent delta. 00:05:06.440 --> 00:05:10.370 Scattering angle, or that S-matrix angle, 00:05:10.370 --> 00:05:13.440 and then minus this IP. 00:05:13.440 --> 00:05:14.000 OK? 00:05:14.000 --> 00:05:15.860 So the I of-- 00:05:15.860 --> 00:05:19.250 the I's just-- shows up here in this part, 00:05:19.250 --> 00:05:21.660 and that's the complex part of the amplitude. 00:05:21.660 --> 00:05:23.810 That's basically going to be related to unitarity, 00:05:23.810 --> 00:05:26.960 that that IP is there. 00:05:26.960 --> 00:05:30.326 This S-matrix is obviously unitary. 00:05:30.326 --> 00:05:33.830 This digress is 1. 00:05:33.830 --> 00:05:37.760 All right, so let me tell you something 00:05:37.760 --> 00:05:44.000 about non-relativistic scattering, 00:05:44.000 --> 00:05:46.700 which on the face of it, looks kind of non-trivial. 00:06:10.990 --> 00:06:13.690 So this thing, P cotangent delta. 00:06:13.690 --> 00:06:16.450 So here, I was doing a single partial wave. 00:06:16.450 --> 00:06:18.460 Yeah, OK, no, it's fine. 00:06:18.460 --> 00:06:20.590 So what is this L? 00:06:20.590 --> 00:06:22.360 This L is the partial I am considering. 00:06:22.360 --> 00:06:23.690 S wave, P wave. 00:06:23.690 --> 00:06:27.310 So L is 0 for the S wave, L is 1 for the P wave. 00:06:27.310 --> 00:06:30.100 And the statement is that if you have a short range potential, 00:06:30.100 --> 00:06:35.860 and you pick a wave, then this P cotangent delta 00:06:35.860 --> 00:06:38.590 with the appropriate power of P can be a Taylor series 00:06:38.590 --> 00:06:41.398 expansion of P. OK? 00:06:41.398 --> 00:06:42.940 And this is actually something that's 00:06:42.940 --> 00:06:46.000 quite difficult to prove in quantum mechanics, 00:06:46.000 --> 00:06:49.043 this particular fact. 00:06:49.043 --> 00:06:50.960 And it's called the effective range expansion. 00:07:00.807 --> 00:07:03.390 It's difficult to prove because when you do quantum mechanics, 00:07:03.390 --> 00:07:05.250 you pick a potential. 00:07:05.250 --> 00:07:08.140 And I'm saying that this is true for any potential. 00:07:08.140 --> 00:07:11.140 So if you start doing quantum mechanics with some potential, 00:07:11.140 --> 00:07:13.380 you've got to prove that you can put it in this form 00:07:13.380 --> 00:07:16.510 irrespective of what the choice of that potential would be. 00:07:16.510 --> 00:07:18.360 And that makes it a little bit tricky 00:07:18.360 --> 00:07:22.002 to do in a quantum mechanical setup. 00:07:22.002 --> 00:07:23.460 But we'll see actually that this is 00:07:23.460 --> 00:07:25.950 very easy to prove from an effective field theory setup. 00:07:34.104 --> 00:07:36.280 So, as a way of getting into this effective field 00:07:36.280 --> 00:07:40.450 theory of two nucleons, let's prove this fact. 00:07:40.450 --> 00:07:42.320 So what is the Lagrangian for this theory? 00:07:46.930 --> 00:07:49.980 There's no-- it's not a gauge theory, 00:07:49.980 --> 00:07:53.920 so we just have ordinary derivatives. 00:07:53.920 --> 00:07:56.170 So if you like, you can think of what I'm writing here 00:07:56.170 --> 00:07:58.780 as kind of like our IV.D term, except I think 00:07:58.780 --> 00:08:00.790 the center of mass frame, and this 00:08:00.790 --> 00:08:02.560 would be like the kinetic energy term, 00:08:02.560 --> 00:08:06.040 but now it's just partial squared with no D, et cetera. 00:08:20.370 --> 00:08:23.028 And then there's a bunch of contact interactions. 00:08:23.028 --> 00:08:24.570 So there's a whole bunch of operators 00:08:24.570 --> 00:08:27.540 that are involved in nucleon and fields 00:08:27.540 --> 00:08:30.450 with some Wilson coefficients. 00:08:30.450 --> 00:08:34.350 The notation here is this S is kind 00:08:34.350 --> 00:08:36.850 of a pseudonym for the channel. 00:08:36.850 --> 00:08:38.909 So this S here-- 00:08:38.909 --> 00:08:41.970 maybe it should be a script S or something, 00:08:41.970 --> 00:08:43.679 is telling me what channel I'm in, 00:08:43.679 --> 00:08:46.980 and if in a spectroscopic notation, 00:08:46.980 --> 00:08:52.717 you'd say you're in the 2S plus 1 LJ channel. 00:08:52.717 --> 00:08:54.300 So this would be the angular momentum, 00:08:54.300 --> 00:08:55.470 total momentum in the spin. 00:08:58.970 --> 00:09:04.010 And these operators here, for our purposes, 00:09:04.010 --> 00:09:07.892 are four nuclear fields with 2M derivatives. 00:09:14.300 --> 00:09:17.650 Now, this is not really the complete theory 00:09:17.650 --> 00:09:18.650 for a couple of reasons. 00:09:18.650 --> 00:09:20.210 Well, there's higher order relativistic corrections 00:09:20.210 --> 00:09:21.560 indicated by the dots. 00:09:21.560 --> 00:09:24.230 There would also be dots over here 00:09:24.230 --> 00:09:26.660 That could have to do with having more nucleon fields. 00:09:26.660 --> 00:09:28.340 For example, I could have-- 00:09:28.340 --> 00:09:30.612 instead of just four, I could have six. 00:09:30.612 --> 00:09:32.570 But I don't need to worry about those operators 00:09:32.570 --> 00:09:37.113 for two-to-two scattering, so I'm leaving them out. 00:09:37.113 --> 00:09:38.780 So this is actually the complete theory, 00:09:38.780 --> 00:09:45.060 if we include these dots here, for two-to-two scattering. 00:09:45.060 --> 00:09:55.100 And this nucleon field, [? its ?] spin [? at ?] half, 00:09:55.100 --> 00:09:56.630 and it's isospin been at half too, 00:09:56.630 --> 00:09:58.610 so it includes both the proton and the neutron. 00:10:04.680 --> 00:10:13.260 Nucleons are fermions, and that implies, actually, a relation, 00:10:13.260 --> 00:10:15.850 because the wave function has to be anti-symmetric. 00:10:15.850 --> 00:10:23.040 And so actually, you know that you can associate isospin 00:10:23.040 --> 00:10:25.410 and the angular momentum in the following 00:10:25.410 --> 00:10:27.370 way because of this fact. 00:10:27.370 --> 00:10:31.770 So all the isotriplets have -1 to the S plus L even, 00:10:31.770 --> 00:10:37.770 and the isosinglets have -1 to the S plus L odd. 00:10:37.770 --> 00:10:40.230 So that cuts down by a factor of two 00:10:40.230 --> 00:10:42.990 the number of combinations you have to consider. 00:10:42.990 --> 00:10:45.000 And basically, what this theory has 00:10:45.000 --> 00:10:49.320 is for some given channel in and some given channel 00:10:49.320 --> 00:11:01.700 out, which I could denote in general different, 00:11:01.700 --> 00:11:04.220 we get operators that just will have some power 00:11:04.220 --> 00:11:05.810 of the center of mass momentum. 00:11:05.810 --> 00:11:06.750 P to the 2M. 00:11:09.500 --> 00:11:13.670 And actually, just by angular momentum conservation, 00:11:13.670 --> 00:11:16.460 that has to be the same as that. 00:11:16.460 --> 00:11:19.010 And so all that can change is the L's. 00:11:19.010 --> 00:11:24.680 And so if S is 0, S is either 0 or 1, 00:11:24.680 --> 00:11:27.500 because we have 2 spin half particles. 00:11:27.500 --> 00:11:31.010 If S is zero, L will be equal to L prime 00:11:31.010 --> 00:11:34.130 because J is equal to J prime, and there's 00:11:34.130 --> 00:11:36.110 no shift of the spin. 00:11:36.110 --> 00:11:39.920 So that's one possibility, and if S is 1, then-- 00:11:42.630 --> 00:11:45.450 so S here is the same, if S is 1, 00:11:45.450 --> 00:11:55.480 then you can have L minus L prime which is 2, so, or 0. 00:11:55.480 --> 00:11:56.220 OK? 00:11:56.220 --> 00:11:57.720 You're shifting by one unit, and you 00:11:57.720 --> 00:12:03.390 can compensate either by having minus L prime be 0 or 2. 00:12:08.880 --> 00:12:14.930 OK, so we conserve J. So we can enumerate 00:12:14.930 --> 00:12:16.340 all the possible partial waves. 00:12:16.340 --> 00:12:19.530 We'll mostly focus on the S wave. 00:12:19.530 --> 00:12:26.360 So, let me write out some of these operators for you. 00:12:33.540 --> 00:12:36.290 So the first operator has no derivatives, 00:12:36.290 --> 00:12:42.220 and I can write it in a way that makes the partial wave explicit 00:12:42.220 --> 00:12:43.786 if I do the following. 00:12:50.940 --> 00:12:54.135 And then there would be some derivative operator. 00:12:54.135 --> 00:12:55.760 And I'm going to pick the normalization 00:12:55.760 --> 00:12:59.930 to make our lives as easy as possible, 00:12:59.930 --> 00:13:02.440 as we usually do when we're setting up the operator basis. 00:13:22.660 --> 00:13:29.242 There's the first two guys where this derivative operator 00:13:29.242 --> 00:13:33.736 is like, [? grad squared ?] to the left. 00:13:33.736 --> 00:13:36.857 [? Grad ?] left dot [? grad ?] right. 00:13:36.857 --> 00:13:38.190 That [? grad ?] go to the right. 00:13:41.980 --> 00:13:44.720 And these P's, if you look at them, 00:13:44.720 --> 00:13:48.080 they're just matrices in the spin and isospin space. 00:13:48.080 --> 00:13:53.260 So the two we're focusing on are the S waves. 00:13:53.260 --> 00:13:55.570 And in the S wave you either have 1S0 or 3S1. 00:14:16.310 --> 00:14:18.350 And so we've encoded all the, sort of, 00:14:18.350 --> 00:14:21.230 complexity in just these matrices, which kind of just 00:14:21.230 --> 00:14:26.460 go along for the ride, and I'll tell you what they are. 00:14:26.460 --> 00:14:33.980 So I sigma 2 projects you onto a spin singlet, and I tau 2 tau 00:14:33.980 --> 00:14:37.290 I projects you onto an isotriplet. 00:14:37.290 --> 00:14:43.700 And then likewise, 3S1, which is a spin triplet, 00:14:43.700 --> 00:14:50.540 you put I sigma 2 sigma I. I tau 2. 00:14:50.540 --> 00:14:52.815 I tau 2 and I sigma 2 are just because of the way 00:14:52.815 --> 00:14:53.690 I wrote the operator. 00:14:53.690 --> 00:14:56.060 I wrote it, instead of writing N dagger N, 00:14:56.060 --> 00:14:58.958 I wrote N transpose N, all dagger. 00:14:58.958 --> 00:15:00.500 And that means, basically, you should 00:15:00.500 --> 00:15:02.167 think about the way this operator works, 00:15:02.167 --> 00:15:05.960 is it annihilates two nucleons in a particular spin wave, 00:15:05.960 --> 00:15:09.680 or a particular spin, isospin channel, 00:15:09.680 --> 00:15:13.180 and then creates them again. 00:15:13.180 --> 00:15:18.340 So annihilate, create. 00:15:18.340 --> 00:15:20.080 So I just put the two fields that 00:15:20.080 --> 00:15:21.587 are doing the annihilating together, 00:15:21.587 --> 00:15:23.920 and the two fields that are doing the creating together. 00:15:23.920 --> 00:15:26.560 And that's nice because you're annihilating them 00:15:26.560 --> 00:15:27.570 in a particular channel. 00:15:32.110 --> 00:15:36.190 So with those conventions, our Feynman Rules 00:15:36.190 --> 00:15:39.500 are particularly simple. 00:15:39.500 --> 00:15:43.600 If we just have a C0 in some channel, 00:15:43.600 --> 00:15:47.020 then the Feynman Rule is just minus IC0, 00:15:47.020 --> 00:15:52.870 and if we have one of these higher C2 operators 00:15:52.870 --> 00:15:56.320 in the center of mass frame, it's just minus IC2P squared. 00:16:02.172 --> 00:16:03.630 In the center of mass frame, that's 00:16:03.630 --> 00:16:04.620 the center of mass momentum. 00:16:04.620 --> 00:16:06.370 And remember, in the center of mass frame, 00:16:06.370 --> 00:16:08.670 P squared was equal to P prime squared. 00:16:08.670 --> 00:16:12.120 And so we can actually just write down the Feynman Rule 00:16:12.120 --> 00:16:13.890 for the complete set of operators 00:16:13.890 --> 00:16:16.129 there if we adopt this convention. 00:16:20.350 --> 00:16:22.693 So if you insert a guy with 2M derivatives-- 00:16:22.693 --> 00:16:24.360 derivatives always have to come in pairs 00:16:24.360 --> 00:16:27.140 because of angular momentum. 00:16:35.077 --> 00:16:36.660 You just have that Feynman Rule, sum's 00:16:36.660 --> 00:16:43.300 over the number of derivatives. 00:16:43.300 --> 00:16:45.080 So this is the complete, in this theory, 00:16:45.080 --> 00:16:53.682 this is the complete tree-level amplitude 00:16:53.682 --> 00:16:55.140 from those interactions over there. 00:16:57.970 --> 00:17:01.285 It's very nice theory. 00:17:01.285 --> 00:17:01.785 Simple. 00:17:04.660 --> 00:17:05.847 What about loops? 00:17:05.847 --> 00:17:07.180 We are going to need some loops. 00:17:10.950 --> 00:17:14.609 So let's look at the following loop, 00:17:14.609 --> 00:17:17.251 and I'll start by looking at just E equals 0. 00:17:22.069 --> 00:17:24.050 Let's just take it, take the nuclear arms 00:17:24.050 --> 00:17:26.599 and then scatter them again. 00:17:26.599 --> 00:17:28.460 So, in terms of the momentum flow 00:17:28.460 --> 00:17:31.070 I have some Q going this way, and then I 00:17:31.070 --> 00:17:36.800 have minus Q going that way, that's my loop momenta. 00:17:36.800 --> 00:17:39.453 So I get 2 coupling C0. 00:17:39.453 --> 00:17:40.800 Want these to be 0's. 00:17:43.680 --> 00:17:44.180 Ergo, DDQ. 00:17:48.061 --> 00:17:56.580 And if I just kept the HQET type term in my kinetic term, 00:17:56.580 --> 00:17:59.220 then it would look like that, OK? 00:17:59.220 --> 00:18:00.060 So this is just-- 00:18:03.090 --> 00:18:06.840 for the moment, if we just keep partial DT 00:18:06.840 --> 00:18:11.970 in the kinetic term, which is what we were doing in HQET, 00:18:11.970 --> 00:18:14.200 then we would get that. 00:18:14.200 --> 00:18:15.750 And that's a bad integral. 00:18:15.750 --> 00:18:17.912 It's got a pinch singularity. 00:18:17.912 --> 00:18:19.120 It's an ill-defined interval. 00:18:26.400 --> 00:18:28.620 So just from that little algebra, 00:18:28.620 --> 00:18:30.855 we see that actually keeping just the partial D 00:18:30.855 --> 00:18:32.940 by DT in the kinetic term is not the right thing 00:18:32.940 --> 00:18:34.920 to do for this theory. 00:18:34.920 --> 00:18:39.330 And that's because the kinetic energy is a relevant operator 00:18:39.330 --> 00:18:40.743 in quantum mechanics. 00:18:45.090 --> 00:18:47.160 Whenever you write down the Schrodinger equation, 00:18:47.160 --> 00:18:47.700 you kept it. 00:18:54.417 --> 00:18:56.000 So the right power counting and should 00:18:56.000 --> 00:19:00.100 have E, which is of order P squared over 2M. 00:19:00.100 --> 00:19:02.675 So the partial T term and the [? grad ?] scored over M term 00:19:02.675 --> 00:19:03.675 should be the same size. 00:19:13.580 --> 00:19:16.720 So this is generically true of two heavy particles. 00:19:25.030 --> 00:19:32.350 They have a different power counting for the kinetic term, 00:19:32.350 --> 00:19:33.120 than HQET. 00:19:39.452 --> 00:19:41.410 Any two heavy particles, whether it's too heavy 00:19:41.410 --> 00:19:44.350 quarks, two heavy nucleons, two heavy anything. 00:19:44.350 --> 00:19:46.580 All right, P should be [? order ?] P squared over 2M. 00:19:46.580 --> 00:19:49.157 AUDIENCE: E being the kinetic energy? 00:19:49.157 --> 00:19:50.740 PROFESSOR: So really, what I mean here 00:19:50.740 --> 00:19:54.400 is just the partial T term in the action 00:19:54.400 --> 00:19:56.990 over there should be the same size as the [? grad ?] squared 00:19:56.990 --> 00:19:57.490 over 2M. 00:20:04.140 --> 00:20:05.620 Yes, I have pulled out the mass. 00:20:05.620 --> 00:20:06.870 Yeah. 00:20:06.870 --> 00:20:10.260 So just like in HQET, to get this partial T 00:20:10.260 --> 00:20:12.750 we pulled out the mass, and we have 00:20:12.750 --> 00:20:15.192 a kind of non-relativistic type expansion. 00:20:15.192 --> 00:20:17.400 And the difference here is that we need the partial T 00:20:17.400 --> 00:20:18.775 to be of [? order grad squared ?] 00:20:18.775 --> 00:20:22.410 over 2M, and that leads to effectively counting velocities 00:20:22.410 --> 00:20:25.260 rather than-- 00:20:25.260 --> 00:20:29.190 because you have to count energies different than P's. 00:20:29.190 --> 00:20:31.410 We won't spend too long talking about 00:20:31.410 --> 00:20:34.080 that because we have other things to discuss here, 00:20:34.080 --> 00:20:36.870 but this is a whole interesting subject in itself. 00:20:36.870 --> 00:20:39.172 The power counting and what it means. 00:20:39.172 --> 00:20:41.130 One thing that's kind of interesting here which 00:20:41.130 --> 00:20:44.370 we won't cover, which I can't help but mention, 00:20:44.370 --> 00:20:47.050 is that say you did quarks, which was a gauge theory. 00:20:47.050 --> 00:20:48.720 This is not a gauge theory that we're talking about, 00:20:48.720 --> 00:20:51.303 but let's-- but you could do a gauge theory that has this type 00:20:51.303 --> 00:20:54.090 of power counting and it has exactly the same problem. 00:20:54.090 --> 00:20:58.140 Just replace these heavy nucleons by quarks, 00:20:58.140 --> 00:21:01.410 and replace this dot here by cooling potential. 00:21:01.410 --> 00:21:05.130 Exactly the same problem if you try to use HQET. 00:21:05.130 --> 00:21:07.170 And in that theory, too, you need 00:21:07.170 --> 00:21:08.670 E to be of order P squared over 2, 00:21:08.670 --> 00:21:11.120 and that's called non-relativistic QCD. 00:21:11.120 --> 00:21:12.870 Or you could do heavy electrons, where you 00:21:12.870 --> 00:21:15.990 have QED as the gauge theory. 00:21:15.990 --> 00:21:19.200 Non-relativistic QED, same issue. 00:21:19.200 --> 00:21:22.650 E has to be of order P squared over M. And in gauge theory, 00:21:22.650 --> 00:21:24.780 there's even a further complication, which 00:21:24.780 --> 00:21:27.330 is basically that there's gauge particles that 00:21:27.330 --> 00:21:29.940 want to talk to E, and there's gauge particles that 00:21:29.940 --> 00:21:33.010 want to talk to P, and those are different sizes. 00:21:33.010 --> 00:21:35.820 So you have something called ultra-soft photons 00:21:35.820 --> 00:21:38.700 and soft photons that are the gauge particles for E, 00:21:38.700 --> 00:21:42.660 and the gauge particles for P. Kind of an interesting theory. 00:21:42.660 --> 00:21:44.320 We want to have time to talk about it. 00:21:44.320 --> 00:21:47.240 Somebody wants to talk about it for their project, 00:21:47.240 --> 00:21:48.090 it's kind of fun. 00:21:52.980 --> 00:21:56.790 OK, so we have to keep P squared over 2M. 00:21:56.790 --> 00:21:59.610 At least knowing a little quantum mechanics, 00:21:59.610 --> 00:22:01.650 we know that that's true. 00:22:01.650 --> 00:22:03.450 Or running into this pinch singularity, 00:22:03.450 --> 00:22:08.810 we see that trying to do something different than that 00:22:08.810 --> 00:22:09.560 leads to problems. 00:22:21.080 --> 00:22:23.380 So let's redo our calculation here, but now keeping 00:22:23.380 --> 00:22:27.380 that term, and see what we get. 00:22:27.380 --> 00:22:29.740 Same bubble diagram. 00:22:29.740 --> 00:22:34.060 Let me send in on each of these legs E over 2, 00:22:34.060 --> 00:22:36.070 so E is the total energy that I'm sending in. 00:22:40.528 --> 00:22:42.228 Convenient normalization. 00:22:47.710 --> 00:22:51.400 Let's see how having the kinetic energy fixes this pinch pole. 00:23:16.390 --> 00:23:18.610 OK, so if you look at the poles in the complex plane 00:23:18.610 --> 00:23:22.010 here, what's happened is you've split them like this. 00:23:22.010 --> 00:23:25.160 So that you've moved them along the real axis. 00:23:25.160 --> 00:23:27.280 And so now you can just think of a contour, 00:23:27.280 --> 00:23:32.860 for example, if you want to think in the complex Q0 plane, 00:23:32.860 --> 00:23:37.200 you can think of doing a contour integral like that. 00:23:37.200 --> 00:23:40.640 Everything is well defined, convergence at infinity, 00:23:40.640 --> 00:23:42.370 everybody's happy. 00:23:42.370 --> 00:23:46.200 So we can close above, pick the polar. 00:23:46.200 --> 00:23:55.227 Q0 is E over 2, minus Q squared over 2M, plus I0. 00:23:55.227 --> 00:23:56.560 Plug it back into the other one. 00:24:03.760 --> 00:24:08.110 My notation is that N is going to be D minus 1. 00:24:08.110 --> 00:24:10.150 So when I do one of the integrals by contour, 00:24:10.150 --> 00:24:11.140 I go down a dimension. 00:24:11.140 --> 00:24:13.480 I'll call that N. So it's N here. 00:24:24.630 --> 00:24:25.740 This integral we can do. 00:24:33.905 --> 00:24:35.655 You have to be careful about the epsilons. 00:24:45.930 --> 00:24:48.240 Because they tell us tell us whether it's minus IP 00:24:48.240 --> 00:24:50.805 or plus IP. 00:24:50.805 --> 00:24:53.870 ME is set up in my convention. 00:24:53.870 --> 00:24:56.450 ME is P squared. 00:24:56.450 --> 00:24:59.948 So this is giving a P, but it's giving either a minus 00:24:59.948 --> 00:25:01.990 IP or a plus IP, depending on the sign of the I0, 00:25:01.990 --> 00:25:05.800 but I know it's a minus IP. 00:25:05.800 --> 00:25:07.240 So I used dimreg here. 00:25:11.878 --> 00:25:13.420 Because if you look at this integral, 00:25:13.420 --> 00:25:16.240 there's three pairs of Q upstairs, two downstairs. 00:25:16.240 --> 00:25:18.458 So it's power law divergent. 00:25:18.458 --> 00:25:20.500 But we don't see power law divergences in dimreg, 00:25:20.500 --> 00:25:23.155 we just get this finite answer. 00:25:29.120 --> 00:25:32.030 And actually, that finite answer is exactly the imaginary part 00:25:32.030 --> 00:25:34.670 that you need if you want to cut to graph, 00:25:34.670 --> 00:25:37.970 and say that the cut of the forward scattering 00:25:37.970 --> 00:25:42.360 is the same as this amplitude squared. 00:25:42.360 --> 00:25:44.390 So it's exactly, in essence, this 00:25:44.390 --> 00:25:47.090 is the piece that you need to be there by unitary. 00:25:50.210 --> 00:25:51.867 If you were trying to keep the E, 00:25:51.867 --> 00:25:53.450 you could think, well, maybe if I just 00:25:53.450 --> 00:25:54.950 kept the E in this calculation, it 00:25:54.950 --> 00:25:56.610 would solve this pinch problem. 00:25:56.610 --> 00:26:00.050 But it doesn't really do it because if you really 00:26:00.050 --> 00:26:03.290 stick with the partial D by DT as your leading order action, 00:26:03.290 --> 00:26:06.830 then the equations of motion are equal 0, so. 00:26:06.830 --> 00:26:10.145 You have to take equal 0 to go on [INAUDIBLE].. 00:26:10.145 --> 00:26:12.020 So you can't really avoid the pinch that way, 00:26:12.020 --> 00:26:15.980 you really have to take this kinetic term. 00:26:15.980 --> 00:26:18.470 you have to take the kinetic term to have both the partial 00:26:18.470 --> 00:26:21.440 DT and the [? grad ?] [? squared ?] over 2M. 00:26:21.440 --> 00:26:23.510 OK, any questions so far? 00:26:28.417 --> 00:26:30.000 All right, well there's something here 00:26:30.000 --> 00:26:31.680 that might bother you. 00:26:31.680 --> 00:26:34.350 We've got an M upstairs. 00:26:34.350 --> 00:26:35.730 M is big. 00:26:35.730 --> 00:26:39.970 M is the mass the nucleon, and it's appearing upstairs. 00:26:39.970 --> 00:26:41.250 That's always a bad sign. 00:26:43.920 --> 00:26:45.450 Usually a bad sign. 00:26:45.450 --> 00:26:48.720 Well, at least that's something we should worry about. 00:26:48.720 --> 00:26:50.490 So let's figure out where all the M's are. 00:26:53.910 --> 00:26:57.540 Let's count all the M's in the theory, 00:26:57.540 --> 00:26:59.025 holding the momentum fixed. 00:27:04.960 --> 00:27:07.410 So if we hold the momentum fixed, 00:27:07.410 --> 00:27:11.520 then [? grad ?] has no M's. 00:27:11.520 --> 00:27:17.340 But partial T scales like 1 over M. 00:27:17.340 --> 00:27:25.360 And correspondingly, T scales like M. OK. 00:27:25.360 --> 00:27:30.050 X and [? grad ?] don't scale but partial T and T do. 00:27:30.050 --> 00:27:32.950 And that's exactly what makes these two terms the same size. 00:27:38.300 --> 00:27:42.640 So we can ask, what is the scaling of our nucleon field? 00:27:42.640 --> 00:27:47.860 And if we want to do that, we go back to our action 00:27:47.860 --> 00:27:52.410 and we just say the action shouldn't have any scaling. 00:27:52.410 --> 00:27:55.900 T has a scaling, so there's an M in here. 00:27:55.900 --> 00:27:58.690 This guy here, the whole thing scales nicely, 1 over M. 00:27:58.690 --> 00:28:00.770 From what we just said over there. 00:28:00.770 --> 00:28:04.630 This M cancels that 1 over M, so this guy 00:28:04.630 --> 00:28:08.390 is therefore M to the 0. 00:28:08.390 --> 00:28:13.000 The nucleon field has no scaling with them. 00:28:13.000 --> 00:28:15.370 So then, once you've done the kinetic term to figure out 00:28:15.370 --> 00:28:17.260 the scaling of the nucleon field, you can do, 00:28:17.260 --> 00:28:20.875 then, any other interaction. 00:28:20.875 --> 00:28:22.750 So that the power counting, the kinetic term, 00:28:22.750 --> 00:28:26.080 is always which determines the power counting of the field. 00:28:26.080 --> 00:28:28.443 That's true in any theory, any effective theory, 00:28:28.443 --> 00:28:29.110 because you're-- 00:28:32.080 --> 00:28:34.180 that's the basis of the fluctuations you're 00:28:34.180 --> 00:28:37.578 describing by that field. 00:28:37.578 --> 00:28:39.370 And once you've fixed that counting, you've 00:28:39.370 --> 00:28:41.120 got all the counting you need, and now you 00:28:41.120 --> 00:28:44.590 can go count other operators like the one with 2M 00:28:44.590 --> 00:28:48.100 derivatives. 00:28:48.100 --> 00:28:50.290 And this guy here just has-- 00:28:50.290 --> 00:28:53.410 you can have just vector derivatives, no time 00:28:53.410 --> 00:28:55.840 derivatives, just vector derivatives, 2M of them, 00:28:55.840 --> 00:28:56.830 and nucleon fields. 00:28:56.830 --> 00:29:00.640 So there's no M's in there, that's M to the 0. 00:29:00.640 --> 00:29:05.260 And this guy here is an M. So this guy here 00:29:05.260 --> 00:29:17.380 is C2M, must be a 1 over M. So therefore, any C2M scales like 00:29:17.380 --> 00:29:24.490 1 over M. And now you see it's not 00:29:24.490 --> 00:29:26.680 such a problem, because here I have 2 C0's, 00:29:26.680 --> 00:29:28.270 each one of them scales like 1 over M, 00:29:28.270 --> 00:29:30.730 I pick up one more M in the numerator, 00:29:30.730 --> 00:29:33.790 and that just makes the whole thing feel like one over M. 00:29:33.790 --> 00:29:36.220 Which is the same order that the tree level was scaling, 00:29:36.220 --> 00:29:39.550 the tree level with scaling like 1 over M. 00:29:39.550 --> 00:29:44.230 So loop diagrams and the tree level both scale like 1 over M, 00:29:44.230 --> 00:29:46.030 and that's actually generically true 00:29:46.030 --> 00:29:48.560 because every time I add a loop I add a coupling, 00:29:48.560 --> 00:29:51.350 and so those two M's can compensate each other. 00:29:51.350 --> 00:29:52.690 So there's no issue with M's. 00:30:05.090 --> 00:30:10.700 So the one that graph is the same size as tree level, 00:30:10.700 --> 00:30:16.190 and they both go like 1 over M. If you now 00:30:16.190 --> 00:30:22.730 do dimension counting, you can say given that we've identified 00:30:22.730 --> 00:30:26.990 that the C has an M in it, if you 00:30:26.990 --> 00:30:31.880 ask about the dimensions of C, -2 minus 2M. 00:30:31.880 --> 00:30:36.390 Because the dimensions of the nucleon field are 3 halves. 00:30:36.390 --> 00:30:39.080 And we're doing some expansion in P much less than lambda, 00:30:39.080 --> 00:30:40.850 so just by dimensional power counting 00:30:40.850 --> 00:30:47.710 we expect that the coefficients would be of the following size. 00:30:47.710 --> 00:30:51.200 We know that there's an M and that it's the only M, 00:30:51.200 --> 00:30:52.910 and all the rest of the dimensions 00:30:52.910 --> 00:30:54.980 you should think of as being made up 00:30:54.980 --> 00:30:56.910 by the stuff you integrated it out. 00:30:56.910 --> 00:30:58.700 So it could be the pion, for example, 00:30:58.700 --> 00:31:00.540 setting the scale lambda. 00:31:00.540 --> 00:31:02.300 So what you'd expect for the C2M's is 00:31:02.300 --> 00:31:05.060 that this is how big they are, and that the derivatives are 00:31:05.060 --> 00:31:07.998 then being suppressed because of these lambdas 00:31:07.998 --> 00:31:10.040 and you're expanding for P much less than lambda. 00:31:13.540 --> 00:31:14.590 OK. 00:31:14.590 --> 00:31:15.690 Are we happy so far? 00:31:20.470 --> 00:31:22.120 All right, and we'll see when you 00:31:22.120 --> 00:31:24.740 do matching calculations, this M [? law's ?] always there. 00:31:24.740 --> 00:31:29.770 And this is just a nice, elegant way of figuring that out. 00:31:29.770 --> 00:31:34.445 Because this is a very simple argument. 00:31:34.445 --> 00:31:34.945 All right. 00:31:34.945 --> 00:31:36.903 AUDIENCE: I don't know if I totally understand. 00:31:36.903 --> 00:31:39.820 How do you know that C2M-- 00:31:39.820 --> 00:31:41.380 PROFESSOR: Goes like 1 over M? 00:31:41.380 --> 00:31:42.160 AUDIENCE: Yeah-- 00:31:42.160 --> 00:31:43.860 PROFESSOR: So the kinetic term-- 00:31:43.860 --> 00:31:47.350 so at first you know this, right? 00:31:47.350 --> 00:31:48.550 AUDIENCE: Yeah. 00:31:48.550 --> 00:31:49.690 PROFESSOR: Yeah, so I see. 00:31:49.690 --> 00:31:50.860 You're worried about whether there could be 00:31:50.860 --> 00:31:52.080 1 over M squared corrections? 00:31:52.080 --> 00:31:52.705 AUDIENCE: Yeah. 00:31:52.705 --> 00:31:53.020 PROFESSOR: Yeah. 00:31:53.020 --> 00:31:55.240 In principle, there could be 1 over M squared corrections 00:31:55.240 --> 00:31:57.282 from relativistic corrections, so that this would 00:31:57.282 --> 00:31:58.810 be the leading order term. 00:31:58.810 --> 00:31:59.500 Yeah. 00:31:59.500 --> 00:32:01.750 Not worse than that. 00:32:01.750 --> 00:32:05.560 If you looked at the P cubed, P to the fourth over 8M 00:32:05.560 --> 00:32:08.133 cubed term, that term would have a higher power of M, 00:32:08.133 --> 00:32:09.550 and you could imagine that there's 00:32:09.550 --> 00:32:12.840 some relativistic corrections in the four nucleons as well. 00:32:12.840 --> 00:32:14.590 Yeah. 00:32:14.590 --> 00:32:16.803 So we'll work basically here, at lowest order 00:32:16.803 --> 00:32:18.220 in the relativistic corrections so 00:32:18.220 --> 00:32:20.470 that you could put in the relativistic corrections, 00:32:20.470 --> 00:32:20.970 as well. 00:32:24.060 --> 00:32:25.800 All right. 00:32:25.800 --> 00:32:27.990 So we're basically stopping at that order, which 00:32:27.990 --> 00:32:29.448 is equivalent to quantum mechanics, 00:32:29.448 --> 00:32:31.320 but we could go further if we wanted to 00:32:31.320 --> 00:32:33.330 in this effective theory. 00:32:33.330 --> 00:32:37.980 So now, let's think about other loop diagrams in the theory, 00:32:37.980 --> 00:32:39.480 without relativistic corrections, 00:32:39.480 --> 00:32:40.800 just with these interactions. 00:32:40.800 --> 00:32:43.890 And let's insert the 2N and the 2M derivative operator 00:32:43.890 --> 00:32:45.510 on these vertices. 00:32:45.510 --> 00:32:48.720 You see all the loops look like this, they're all bubbles. 00:32:48.720 --> 00:32:50.820 And that's the same reason in HQET 00:32:50.820 --> 00:32:53.070 that you don't have any diagrams that kind of-- you 00:32:53.070 --> 00:32:55.770 don't have diagrams that look like this, because these types 00:32:55.770 --> 00:32:58.365 of diagrams involve antiparticles 00:32:58.365 --> 00:32:59.490 and we only have particles. 00:33:03.717 --> 00:33:05.800 So that's the beauty of a non-relativistic theory, 00:33:05.800 --> 00:33:07.650 you don't have any diagrams like that. 00:33:07.650 --> 00:33:09.830 You just have diagrams like this. 00:33:09.830 --> 00:33:12.610 And so basically, the whole theory is bubbles. 00:33:12.610 --> 00:33:13.590 The theory of bubbles. 00:33:16.290 --> 00:33:18.180 If you go through this loop diagram, 00:33:18.180 --> 00:33:20.190 you do the pole the same way. 00:33:20.190 --> 00:33:24.570 It's exactly the same two propagators. 00:33:24.570 --> 00:33:33.443 Then you get the same Q squared minus ME, minus I0. 00:33:33.443 --> 00:33:34.860 And the only difference is you get 00:33:34.860 --> 00:33:39.435 powers of Q in the numerator. 00:33:39.435 --> 00:33:41.310 And this, so this integral is one of the ones 00:33:41.310 --> 00:33:42.840 that shows up in here. 00:33:42.840 --> 00:33:46.230 And this integral, you can do the same kind of trick 00:33:46.230 --> 00:33:49.290 that you used when we were discussing field definitions, 00:33:49.290 --> 00:33:52.020 where you basically take the top and organize it 00:33:52.020 --> 00:33:55.150 by adding and subtracting. 00:33:55.150 --> 00:34:09.530 So add and subtract any like that. 00:34:09.530 --> 00:34:11.600 And now, you can think, N and M are integers, 00:34:11.600 --> 00:34:13.969 just expand this thing out. 00:34:13.969 --> 00:34:16.940 Some number of these factors, some number of these factors. 00:34:16.940 --> 00:34:19.310 But any time you get one or more of these factors, 00:34:19.310 --> 00:34:20.780 it cancels the denominator and then 00:34:20.780 --> 00:34:23.058 you end up with scaleless integrals. 00:34:23.058 --> 00:34:25.350 So basically, that means you can throw this piece away. 00:34:41.150 --> 00:34:43.580 So higher order derivatives actually just 00:34:43.580 --> 00:34:47.330 lead to ME's, whether they act inside or outside 00:34:47.330 --> 00:34:49.850 on the nucleon fields, they lead to P squared. 00:34:53.449 --> 00:34:54.415 Yeah, sorry. 00:34:54.415 --> 00:34:55.790 This M is a little M, and this is 00:34:55.790 --> 00:35:00.040 supposed to be a big M. Oh, N. Oh yeah, 00:35:00.040 --> 00:35:02.750 I am also using that too. 00:35:02.750 --> 00:35:03.800 Sorry. 00:35:03.800 --> 00:35:04.770 Yes. 00:35:04.770 --> 00:35:05.810 This is an integer. 00:35:05.810 --> 00:35:08.450 I should call it J or something. 00:35:08.450 --> 00:35:13.295 And the other M is 3 minus 2 epsilon. 00:35:28.100 --> 00:35:30.570 Yeah, it's dangerous. 00:35:30.570 --> 00:35:31.610 All right. 00:35:31.610 --> 00:35:34.220 But this means that basically, the graphs in this theory 00:35:34.220 --> 00:35:35.010 are very simple. 00:35:35.010 --> 00:35:38.390 So if we actually now consider the complete set of diagrams, 00:35:38.390 --> 00:35:39.365 or we impose-- 00:35:42.530 --> 00:35:46.640 we put the full amplitude in there 00:35:46.640 --> 00:35:47.990 and we consider these bubbles. 00:35:53.060 --> 00:35:54.620 Because of this fact that I just told 00:35:54.620 --> 00:35:57.770 you, which doesn't change when you insert more bubbles, 00:35:57.770 --> 00:36:01.640 the bubbles all decouple from each other. 00:36:01.640 --> 00:36:03.470 The contact interaction is decoupling them 00:36:03.470 --> 00:36:05.730 from each other. 00:36:05.730 --> 00:36:14.180 So here's a complete amplitude with K insertions 00:36:14.180 --> 00:36:16.430 of the coupling, and K minus 1 loops. 00:36:24.990 --> 00:36:28.245 OK, so we just completed all the loop diagrams the theory, 00:36:28.245 --> 00:36:30.120 and it's giving us something that we can sum, 00:36:30.120 --> 00:36:31.320 its a geometric series. 00:36:35.740 --> 00:36:40.540 So this is for K minus 1 loops, we can sum them up. 00:36:40.540 --> 00:36:46.150 So if you like I should have called this the K-th term. 00:36:46.150 --> 00:37:07.830 And if I sum up, cancel the I on each side, you get that. 00:37:16.270 --> 00:37:17.410 Let me round it again. 00:37:24.890 --> 00:37:25.610 Like this. 00:37:30.235 --> 00:37:34.390 So dividing out the numerator, rearranging 00:37:34.390 --> 00:37:38.150 the four pi over M's little bit, I can write it like that. 00:37:38.150 --> 00:37:40.030 And then we can identify this thing here 00:37:40.030 --> 00:37:41.320 as P cotangent delta. 00:37:45.950 --> 00:37:51.020 OK, so P cotangent delta, from our effective theory, 00:37:51.020 --> 00:37:55.925 we calculate it to be the sum over this, 00:37:55.925 --> 00:37:59.870 P2M, where now I've conveniently defined it, 00:37:59.870 --> 00:38:04.160 hatted coefficients, which are the following things. 00:38:07.540 --> 00:38:09.040 And that just cancels the M that's 00:38:09.040 --> 00:38:13.990 in the C2M's, so C2 hat scales like M to the 0. 00:38:16.960 --> 00:38:18.040 So that's just a-- 00:38:18.040 --> 00:38:22.060 is an obvious way of reorganizing things given 00:38:22.060 --> 00:38:24.610 the factors of 4 pi over M that we knew 00:38:24.610 --> 00:38:27.310 had to sit out front of this S matrix 00:38:27.310 --> 00:38:30.050 calculation of the amplitude. 00:38:30.050 --> 00:38:33.190 So we can identify P cotangent delta as something 00:38:33.190 --> 00:38:35.500 that doesn't involve any M's, and that actually is also 00:38:35.500 --> 00:38:39.440 what we would expect for non-relativistic scattering. 00:38:39.440 --> 00:38:41.200 So then you can look at different waves, 00:38:41.200 --> 00:38:45.725 and you can look at this formula, and you can-- 00:38:45.725 --> 00:38:47.600 doing two of them will show you how it works. 00:38:47.600 --> 00:38:50.260 So S wave is L equals 0. 00:39:14.030 --> 00:39:18.140 So that's something we can do a Taylor expansion in P in. 00:39:18.140 --> 00:39:20.460 And this is what the Taylor series looks like. 00:39:20.460 --> 00:39:22.530 And this has exactly the form that we wanted. 00:39:22.530 --> 00:39:24.800 This is 1-- some constant 1 over A, 00:39:24.800 --> 00:39:27.290 which is the scattering length. 00:39:27.290 --> 00:39:29.090 Some constant times P squared, which 00:39:29.090 --> 00:39:31.580 is called the effective range. 00:39:31.580 --> 00:39:33.535 And we see that expansion coming out, 00:39:33.535 --> 00:39:35.660 and we never had to specify what the potential was, 00:39:35.660 --> 00:39:38.270 because the effective theory was agnostic to what 00:39:38.270 --> 00:39:39.050 the potential was. 00:39:39.050 --> 00:39:40.490 That's the whole power of effective theory, 00:39:40.490 --> 00:39:42.890 that you don't need to know what the particles were that you 00:39:42.890 --> 00:39:45.057 were integrating out, they just give you some values 00:39:45.057 --> 00:39:46.490 for these coefficients. 00:39:46.490 --> 00:39:49.598 And those values exactly become the effective range 00:39:49.598 --> 00:39:51.140 and scattering length in this theory. 00:39:53.930 --> 00:39:56.620 AUDIENCE: So this seems very [INAUDIBLE] dependent. 00:39:56.620 --> 00:40:00.130 PROFESSOR: Yeah, we're going to talk about that. 00:40:00.130 --> 00:40:02.110 Yes. 00:40:02.110 --> 00:40:06.220 Yeah, that will encompass the second half of lecture. 00:40:18.230 --> 00:40:20.185 So we'll come to that momentarily. 00:40:23.510 --> 00:40:27.065 Let me just do one more way, just so you see. 00:40:27.065 --> 00:40:29.220 L equals 1. 00:40:29.220 --> 00:40:31.820 in the L equals 1 case, there's no C0 hat, 00:40:31.820 --> 00:40:35.970 you need the derivatives to correspond to the wave. 00:40:35.970 --> 00:40:43.220 And so in this case, you look at PQ cotangent delta, 00:40:43.220 --> 00:40:44.750 the denominator starts at C2. 00:40:47.390 --> 00:40:49.760 And the reason why there is this P to the 2L plus 1 00:40:49.760 --> 00:40:52.380 is just so that you get the extra P here, 00:40:52.380 --> 00:40:54.217 which compensates the P's downstairs, 00:40:54.217 --> 00:40:55.925 and again this thing has a Taylor series. 00:41:04.900 --> 00:41:07.990 Same story as over there, we can identify the scattering link 00:41:07.990 --> 00:41:10.750 for the P wave as the 1 over C2 hat. 00:41:13.977 --> 00:41:17.680 So all the higher partial ways work the same way. 00:41:17.680 --> 00:41:20.050 And you just have P to the 2 L plus 1 in front 00:41:20.050 --> 00:41:22.450 of your cotangent delta. 00:41:22.450 --> 00:41:22.950 OK. 00:41:27.080 --> 00:41:30.332 Here you can see how this generalizes [INAUDIBLE].. 00:41:35.160 --> 00:41:37.020 So that proves this non-relativistic quantum 00:41:37.020 --> 00:41:38.990 mechanics theorem. 00:41:52.160 --> 00:41:55.010 And we did it without having to specify what the potential was, 00:41:55.010 --> 00:41:58.010 because the potential is encoded in BC's and effectively 00:41:58.010 --> 00:42:00.647 that's like a basis expansion of the potential. 00:42:00.647 --> 00:42:02.730 But it's a fun one because it's in delta functions 00:42:02.730 --> 00:42:04.740 and derivatives of delta functions. 00:42:04.740 --> 00:42:07.010 So we're doing a local effective field 00:42:07.010 --> 00:42:09.200 theory, which is not something you'd think 00:42:09.200 --> 00:42:11.230 of ever using for a basis-- 00:42:11.230 --> 00:42:13.310 well, maybe you would, but most people 00:42:13.310 --> 00:42:16.190 wouldn't think of using basis of derivatives of delta functions 00:42:16.190 --> 00:42:18.120 for quantum mechanics. 00:42:18.120 --> 00:42:18.620 OK. 00:42:21.662 --> 00:42:23.870 Well, you'd hope we can do a little more than quantum 00:42:23.870 --> 00:42:24.578 mechanics, right? 00:42:29.000 --> 00:42:31.607 So you can think, if you like, in terms 00:42:31.607 --> 00:42:33.690 of determining the values of the C's, you can say, 00:42:33.690 --> 00:42:36.232 well, experiment tells me the values of the scattering length 00:42:36.232 --> 00:42:38.510 are 0, and that's indeed true. 00:42:42.300 --> 00:42:45.268 So this equation that C0 hat is equal to A, 00:42:45.268 --> 00:42:47.060 you could view this as a matching equation. 00:42:49.980 --> 00:42:54.770 Putting back the M, you have that. 00:42:54.770 --> 00:42:59.030 And then for C2 hat, we have r0 over 2, a squared. 00:43:07.980 --> 00:43:10.370 So pretend that experiment gives r0. 00:43:15.660 --> 00:43:17.913 And a, which they do measure, and then 00:43:17.913 --> 00:43:19.830 you know the value of your Wilson coefficients 00:43:19.830 --> 00:43:22.230 and you can start using this theory. 00:43:22.230 --> 00:43:27.750 Now, if you think about the power counting, if a 00:43:27.750 --> 00:43:30.570 and r0 are order 1 over lambda, that's 00:43:30.570 --> 00:43:32.408 the natural thing you'd expect. 00:43:32.408 --> 00:43:33.825 Then you reproduce what we expect. 00:43:43.230 --> 00:43:45.270 Whatever it was a minute ago. 00:43:50.340 --> 00:43:51.440 2M plus one. 00:43:56.810 --> 00:44:00.410 So if all the constants are scaling like whatever 00:44:00.410 --> 00:44:01.980 their dimensions are, in this case 00:44:01.980 --> 00:44:05.235 they're both dimension minus 1 in momentum units, 00:44:05.235 --> 00:44:07.610 then you would reproduce exactly with this power counting 00:44:07.610 --> 00:44:09.340 that we said. 00:44:09.340 --> 00:44:11.330 OK, so everything would be nice. 00:44:11.330 --> 00:44:13.050 But nature is not so nice. 00:44:13.050 --> 00:44:16.460 Or nature threw us in a different direction. 00:44:16.460 --> 00:44:19.293 When we actually look at the value of these constants, 00:44:19.293 --> 00:44:20.585 the scattering length is large. 00:44:27.970 --> 00:44:34.100 So C0 is large, from this point of view. 00:44:37.303 --> 00:44:38.720 Let me quote some numbers for you. 00:44:58.820 --> 00:45:03.320 So in the 1S0 channel, which is the larger one, 00:45:03.320 --> 00:45:04.700 this guy is 23 fermis. 00:45:11.990 --> 00:45:17.288 And in the 3S1 channel, it's 5 fermis, and both of these 00:45:17.288 --> 00:45:18.080 are actually large. 00:45:23.260 --> 00:45:25.010 We know they're large, look at the errors. 00:45:27.650 --> 00:45:29.570 If you want to think about momentum units, 00:45:29.570 --> 00:45:32.480 you could take 1 over a. 00:45:32.480 --> 00:45:38.940 And for this guy here, taking 1 over a is giving him, like, 00:45:38.940 --> 00:45:43.910 if I did the calculation right, minus 8.30 MeV. 00:45:43.910 --> 00:45:45.170 Oh, sorry, no. 00:45:45.170 --> 00:45:45.920 That's this guy. 00:45:48.750 --> 00:45:55.740 And this guy here is giving 1 over a, is 36 MeV. 00:45:55.740 --> 00:45:57.950 So if you thought the natural size was the pion, 00:45:57.950 --> 00:46:00.860 then you'd say, well, these constants should be 1 over pi-- 00:46:00.860 --> 00:46:04.130 that these numbers that are in MeV should be in pi. 00:46:04.130 --> 00:46:07.970 And 8 MeV is a much smaller number than 138 MeV. 00:46:07.970 --> 00:46:10.830 36 is also a smaller number than 130 MeV. 00:46:10.830 --> 00:46:17.600 So both of these guys are not natural. 00:46:17.600 --> 00:46:20.935 In particular, this one you see it's very not natural. 00:46:24.080 --> 00:46:26.620 So there's a fine tuning going on. 00:46:26.620 --> 00:46:29.082 Some kind of fine tuning from the perspective-- 00:46:37.460 --> 00:46:40.080 from our dimensional counting EFT point of view. 00:46:45.120 --> 00:46:49.870 There's a fine tuning that's making the a big. 00:46:49.870 --> 00:46:53.460 And if you look at the other guys, the r0 00:46:53.460 --> 00:46:55.770 and the other guys, they are exactly 00:46:55.770 --> 00:46:56.890 of the size you'd expect. 00:46:56.890 --> 00:46:58.223 So there's no fine tuning there. 00:47:01.440 --> 00:47:04.560 The only fine tuning is in a. 00:47:04.560 --> 00:47:06.660 And not the other ones. 00:47:06.660 --> 00:47:08.880 Just by comparing two data. 00:47:08.880 --> 00:47:10.300 OK, so how do we deal with that? 00:47:10.300 --> 00:47:12.092 It looks like we set up a defective theory. 00:47:12.092 --> 00:47:13.660 It has seems like a beautiful theory, 00:47:13.660 --> 00:47:15.910 it could describe some things about quantum mechanics, 00:47:15.910 --> 00:47:20.050 but then we learned that our power counting sucks. 00:47:20.050 --> 00:47:22.790 So what we want is actually a power counting 00:47:22.790 --> 00:47:24.188 it's a little different. 00:47:32.010 --> 00:47:38.190 Where AP is of order 1, or even AP much greater than 1, 00:47:38.190 --> 00:47:40.200 we'd like that to be allowed. 00:47:40.200 --> 00:47:44.520 Where our 0P is much less than 1. 00:47:44.520 --> 00:47:47.160 We'd like to be able to take a scattering length effectively 00:47:47.160 --> 00:47:50.310 into account to all orders, and that means basically 00:47:50.310 --> 00:47:55.170 that we want to treat C0 as relevant. 00:47:55.170 --> 00:47:58.537 We don't want 8MeV to be the limit of the lowered-- 00:47:58.537 --> 00:48:00.870 the thing that goes downstairs when we're making a power 00:48:00.870 --> 00:48:02.340 expansion, right? 00:48:02.340 --> 00:48:05.610 We'd like something like M pi to be downstairs. 00:48:05.610 --> 00:48:06.990 If we want M pi to be downstairs, 00:48:06.990 --> 00:48:09.180 you got to treat AP to all orders. 00:48:09.180 --> 00:48:12.570 And then you're just limited by M pi which is the r0 term. 00:48:12.570 --> 00:48:14.340 And that means you've got to promote C0 00:48:14.340 --> 00:48:18.060 from being irrelevant, scaling like 1 over M lambda, 00:48:18.060 --> 00:48:21.580 to something that's relevant. 00:48:21.580 --> 00:48:28.080 So this is actually a problem that 00:48:28.080 --> 00:48:30.630 occurred because we just proceeded and started 00:48:30.630 --> 00:48:34.820 calculating, and we used effectively the MS bar scheme 00:48:34.820 --> 00:48:36.780 in dimensional regularization. 00:48:36.780 --> 00:48:39.720 We saw powers and fractions, we did nothing. 00:48:39.720 --> 00:48:41.970 So let's try another scheme. 00:48:48.440 --> 00:48:51.260 It's a little more physical. 00:48:51.260 --> 00:48:54.800 Called offshell momentum subtraction. 00:48:54.800 --> 00:48:57.530 So what is offshell momentum subtraction? 00:48:57.530 --> 00:49:01.220 It says take the amplitude, and in this non-relativistic 00:49:01.220 --> 00:49:05.990 theory, you take P to be at some imaginary point 00:49:05.990 --> 00:49:10.302 so that you avoid any cuts. 00:49:10.302 --> 00:49:11.135 And you can define-- 00:49:15.160 --> 00:49:21.520 and whatever channel we're in, in this new r scheme, 00:49:21.520 --> 00:49:24.190 you can define the amplitude of that particular point 00:49:24.190 --> 00:49:26.730 to be the tree level result. 00:49:26.730 --> 00:49:29.540 OK, so any loops, if you take them at this point, 00:49:29.540 --> 00:49:35.830 you should get back that result. So this 00:49:35.830 --> 00:49:38.080 is the analog of what you do in a relativistic theory 00:49:38.080 --> 00:49:41.050 where you would take P squared to be minus mu squared. 00:49:41.050 --> 00:49:42.550 In the non-relativistic theory, it's 00:49:42.550 --> 00:49:46.000 always P that's showing up, which is the three vector P, 00:49:46.000 --> 00:49:49.200 magnitude of the three vector. 00:49:49.200 --> 00:49:50.910 So we should assign some rule for that. 00:49:50.910 --> 00:49:53.990 And it's just, we take it to be I times mu. 00:50:00.620 --> 00:50:03.450 So how does this change our calculation? 00:50:03.450 --> 00:50:06.750 So let's go back to this one calculation. 00:50:06.750 --> 00:50:11.210 So now there's going to be some counterterm needed. 00:50:11.210 --> 00:50:14.630 It's going to be a finite counterterm. 00:50:14.630 --> 00:50:15.620 Let's see what it does. 00:50:21.990 --> 00:50:27.830 So the loop graph gave this IP, that's what this graph gave. 00:50:27.830 --> 00:50:32.000 And this has to be just such that if I set P equal to I mu, 00:50:32.000 --> 00:50:32.960 that I get 0. 00:50:32.960 --> 00:50:35.370 So this has two plus mu. 00:50:35.370 --> 00:50:39.050 So the counterterm is giving mu r, C0 squared. 00:50:39.050 --> 00:50:42.455 And that exactly makes this amplitude vanish at that point, 00:50:42.455 --> 00:50:45.080 and that's what you want because the tree level graph of the C0 00:50:45.080 --> 00:50:47.690 is already giving the right condition. 00:50:47.690 --> 00:50:48.470 OK? 00:50:48.470 --> 00:50:50.150 Is that clear to everybody? 00:50:50.150 --> 00:50:52.880 This is the correction to that, to this. 00:50:52.880 --> 00:50:56.210 The tree level graph in the amplitude here gave minus IC0, 00:50:56.210 --> 00:50:57.710 so we already got what we want. 00:50:57.710 --> 00:50:59.210 When we go to our loop, we just want 00:50:59.210 --> 00:51:01.190 to make sure it doesn't contribute, 00:51:01.190 --> 00:51:05.190 and that forces us to put the plus mu there. 00:51:05.190 --> 00:51:10.940 And so what this mu is doing is tracking the power divergence 00:51:10.940 --> 00:51:14.630 that dimreg in MS bar did not see. 00:51:17.810 --> 00:51:20.450 There was an integral is power law divergence and our cut 00:51:20.450 --> 00:51:22.910 off, mu r, has appeared in the numerator 00:51:22.910 --> 00:51:24.981 and has tracked that divergence. 00:51:28.140 --> 00:51:29.970 So we're doing a standard thing here where 00:51:29.970 --> 00:51:36.240 we split the coefficient into the bare coefficient 00:51:36.240 --> 00:51:38.910 and to our more renormalized and countertrend piece. 00:51:38.910 --> 00:51:41.310 And unlike MS bar in this particular normalization 00:51:41.310 --> 00:51:44.085 scheme, there's a finite correction there. 00:51:44.085 --> 00:51:45.960 And we have a renormalization group equation. 00:52:00.540 --> 00:52:08.290 And if you work out what that is from the counterterm, 00:52:08.290 --> 00:52:12.480 it turns out that only this one loop, it's one loop exact. 00:52:12.480 --> 00:52:15.150 Showing that is a little more work than I've done, 00:52:15.150 --> 00:52:18.690 but it turns out that the beta function is one loop exact, 00:52:18.690 --> 00:52:20.520 higher bubbles don't give any contribution 00:52:20.520 --> 00:52:24.540 to this beta function, and this is the anomalous dimension. 00:52:24.540 --> 00:52:26.820 So we could just calculate all those bubbles, 00:52:26.820 --> 00:52:28.840 and pose the same type of thing. 00:52:28.840 --> 00:52:31.898 And I've given you the reference that does that. 00:52:31.898 --> 00:52:33.690 So there's a renormalization group equation 00:52:33.690 --> 00:52:36.060 in this scheme which we didn't have in MS bar. 00:52:36.060 --> 00:52:38.820 In MS bar it was scale independent. 00:52:38.820 --> 00:52:41.250 We also have a connection to the MS bar scheme 00:52:41.250 --> 00:52:44.970 because if mu r was 0, that corresponds to what 00:52:44.970 --> 00:52:48.090 the MS bar result, right? 00:52:48.090 --> 00:52:53.010 So C0 of 0 is where you can think of putting your boundary 00:52:53.010 --> 00:52:57.900 condition, which is matching to the experiment, which is the a. 00:52:57.900 --> 00:53:00.210 And the advantage of this offshell subtraction 00:53:00.210 --> 00:53:03.190 is that we have a mu, and we can go somewhere else. 00:53:03.190 --> 00:53:07.740 And if you look at the solution, of the RG 00:53:07.740 --> 00:53:11.557 with that boundary condition, you see something interesting. 00:53:20.330 --> 00:53:22.840 So this is the result for the coefficient. 00:53:22.840 --> 00:53:24.570 One over mu r, minus 1 over a. 00:53:28.240 --> 00:53:31.760 So if mu is of order P, which is much greater than 1 00:53:31.760 --> 00:53:36.460 over a, like we want, then the right counting for the C, 00:53:36.460 --> 00:53:40.210 which is a function of mu, is 1 over M mu. 00:53:43.700 --> 00:53:45.950 OK? 00:53:45.950 --> 00:53:48.140 And so what we've done here is we've 00:53:48.140 --> 00:53:54.770 swapped 1 over the physics scale that we're integrating out 00:53:54.770 --> 00:53:56.960 for 1 over the scale mu, which we're 00:53:56.960 --> 00:54:00.100 taking to be of order the physics we're keeping. 00:54:00.100 --> 00:54:02.120 OK? 00:54:02.120 --> 00:54:03.440 And this is relevant now. 00:54:03.440 --> 00:54:05.680 This is a relevant coupling with that change. 00:54:09.730 --> 00:54:11.620 We've made it much bigger by just 00:54:11.620 --> 00:54:13.970 switching to the physical renormalization scheme. 00:54:13.970 --> 00:54:16.210 And if we take mu of order p in that scheme, 00:54:16.210 --> 00:54:19.030 this all of a sudden becomes an order 1 effect 00:54:19.030 --> 00:54:20.740 with leading order in the Lagrangian, 00:54:20.740 --> 00:54:25.710 because we have effectively P's downstairs in the coupling. 00:54:25.710 --> 00:54:26.570 OK? 00:54:26.570 --> 00:54:30.100 So this scheme allows us a way of thinking 00:54:30.100 --> 00:54:31.890 in the effective theory way of thinking 00:54:31.890 --> 00:54:33.640 of having a power counting where we'd have 00:54:33.640 --> 00:54:35.843 to keep the C0 to all orders. 00:54:35.843 --> 00:54:37.260 AUDIENCE: I thought it was order-- 00:54:37.260 --> 00:54:39.350 AP was order 1. 00:54:39.350 --> 00:54:40.820 PROFESSOR: Yeah, AP is order 1. 00:54:44.990 --> 00:54:46.760 So if you look at the Lagrangian, 00:54:46.760 --> 00:54:53.180 then you have to go through the counting of how many powers 00:54:53.180 --> 00:54:55.100 of P the nucleon field has. 00:54:55.100 --> 00:54:59.210 Which you could do in the same way that we did with the mass. 00:54:59.210 --> 00:55:01.340 And if you do that with the four-nucleon operator 00:55:01.340 --> 00:55:03.530 with an extra power of P downstairs, 00:55:03.530 --> 00:55:08.600 you will find it has the same P scaling as the kinetic term. 00:55:08.600 --> 00:55:09.230 OK? 00:55:09.230 --> 00:55:10.933 I didn't go through that, but-- 00:55:10.933 --> 00:55:13.100 AUDIENCE: So that justifies this, even when P is not 00:55:13.100 --> 00:55:14.555 much bigger than 1 over a? 00:55:14.555 --> 00:55:15.680 Is that what you're saying? 00:55:15.680 --> 00:55:18.350 PROFESSOR: Yeah, so if P is much bigger than 1 over a, or P 00:55:18.350 --> 00:55:24.980 is even order or 1 over a, which is a sort of also-- 00:55:24.980 --> 00:55:27.560 it's also fine with this counting. 00:55:27.560 --> 00:55:29.570 If P is of order 1 over a, these two terms 00:55:29.570 --> 00:55:32.750 are sort of comparably big, but you could also 00:55:32.750 --> 00:55:34.940 use this approach for that case, as long 00:55:34.940 --> 00:55:40.700 as you're not in the case where P is much less than 1 over a. 00:55:40.700 --> 00:55:42.890 So if you're in that case, then you 00:55:42.890 --> 00:55:45.420 should really think about expanding this out some sense. 00:55:45.420 --> 00:55:47.670 And then you're getting back to the end MS bar result, 00:55:47.670 --> 00:55:50.640 the MS bar result would have been fine. 00:55:50.640 --> 00:55:52.650 But to get away from the MS bar result, 00:55:52.650 --> 00:55:56.068 and think about physics, mu of order P, 00:55:56.068 --> 00:55:57.860 we could use this scheme instead of MS bar, 00:55:57.860 --> 00:56:00.650 and then we actually see that we get a reasonable power 00:56:00.650 --> 00:56:01.450 counting. 00:56:01.450 --> 00:56:04.800 AUDIENCE: OK, Because my concern is, what about when mu is like, 00:56:04.800 --> 00:56:07.092 what about this pole that you're going to-- 00:56:07.092 --> 00:56:10.181 PROFESSOR: Yeah, we'll talk about the pole, yeah. 00:56:10.181 --> 00:56:12.580 It's coming up. 00:56:12.580 --> 00:56:14.800 All right. 00:56:14.800 --> 00:56:16.780 So it's interesting to think about this 00:56:16.780 --> 00:56:19.075 from our renormalization group point of view, which 00:56:19.075 --> 00:56:20.950 is kind of what I was doing when I wrote down 00:56:20.950 --> 00:56:22.630 a beta function that I was-- 00:56:22.630 --> 00:56:25.870 here I was just after this solution, 00:56:25.870 --> 00:56:30.740 because from that solution I got the power counting that I want. 00:56:30.740 --> 00:56:33.413 Which is that the C0 term is the same size as the kinetic term, 00:56:33.413 --> 00:56:34.330 and both are relevant. 00:56:43.980 --> 00:56:46.910 So we can do that just like we counted P's for [INAUDIBLE] 00:56:46.910 --> 00:56:48.500 theory, or just like we counted-- 00:56:51.430 --> 00:57:08.950 [? just comment. ?] These guys are all relevant 00:57:08.950 --> 00:57:13.780 as long as we count this mu as 1 over P. 1 over mu is 1 over P. 00:57:13.780 --> 00:57:15.700 So there's another way of thinking about this, 00:57:15.700 --> 00:57:17.408 and thinking about these different cases, 00:57:17.408 --> 00:57:20.690 and that's just to think about the beta function itself. 00:57:20.690 --> 00:57:24.730 So if you look at the beta function for the C0 coupling, 00:57:24.730 --> 00:57:27.430 and we just put in the solution, plug this back 00:57:27.430 --> 00:57:37.243 into that equation here, with some constant out front, 00:57:37.243 --> 00:57:38.410 and then it looks like this. 00:57:38.410 --> 00:57:43.430 8 times mu, 1 minus a mu, squared. 00:57:43.430 --> 00:57:45.880 If we want to talk about all possible values of a, 00:57:45.880 --> 00:57:49.580 well a can go from minus infinity to plus infinity. 00:57:49.580 --> 00:57:52.030 So it's useful if we want to draw this to map it 00:57:52.030 --> 00:57:53.110 to a compact interval. 00:57:55.760 --> 00:57:56.560 So let's do that. 00:58:04.680 --> 00:58:06.950 Move the tangent. 00:58:06.950 --> 00:58:09.800 And let's just plot beta as a function of x. 00:58:21.555 --> 00:58:23.805 So there's three values, actually where beta vanishes. 00:58:27.150 --> 00:58:28.650 There's one value where it blows up. 00:58:32.330 --> 00:58:38.150 So this is-- here is x equals minus 1. 00:58:38.150 --> 00:58:43.170 This is x equals 0, this is x equals plus 1. 00:58:45.840 --> 00:58:53.640 And what it looks like dips down here, goes there, blows up, 00:58:53.640 --> 00:58:57.930 then it comes back down, it goes like that. 00:58:57.930 --> 00:59:00.930 So that's what the beta function looks like. 00:59:00.930 --> 00:59:03.290 So this point here corresponds if you 00:59:03.290 --> 00:59:05.630 think of being at fixed mu. 00:59:05.630 --> 00:59:08.240 Say you're studying the physics at fixed mu. 00:59:08.240 --> 00:59:11.720 This point corresponds to a equals minus infinity, 00:59:11.720 --> 00:59:14.870 this point corresponds to a equals zero, 00:59:14.870 --> 00:59:18.002 and this point corresponds to a equals plus infinity. 00:59:24.400 --> 00:59:26.980 So there's three points where the beta function vanishes, 00:59:26.980 --> 00:59:29.168 if you think about a space for fixed mu. 00:59:41.590 --> 00:59:42.370 Use some color. 00:59:53.250 --> 00:59:55.450 So you can ask about nature. 00:59:55.450 --> 00:59:57.450 So nature told us the value of a, so what 00:59:57.450 --> 00:59:59.200 does that correspond to? 00:59:59.200 --> 01:00:01.830 So taking some value of P and mapping it 01:00:01.830 --> 01:00:04.823 to some value of x, kind of generically the kind of point 01:00:04.823 --> 01:00:06.240 that you're interested in is here. 01:00:06.240 --> 01:00:09.120 This is a1S0, sits there. 01:00:09.120 --> 01:00:16.420 And then a for the 3S1 makes it kind of here. 01:00:16.420 --> 01:00:19.350 So what's going on in this case is that you're not 01:00:19.350 --> 01:00:22.800 near this fixed point, you're actually close to this one. 01:00:22.800 --> 01:00:24.630 And you're not near this fixed point, 01:00:24.630 --> 01:00:26.505 well actually there's an infinity in between, 01:00:26.505 --> 01:00:28.510 you're closer to this one. 01:00:28.510 --> 01:00:32.940 This size you should think of as 8 MeV, generically, 01:00:32.940 --> 01:00:35.580 and then this is like the 8 but there's also a pole in between. 01:00:38.440 --> 01:00:39.670 So three fixed points. 01:00:42.778 --> 01:00:44.320 When we do perturbation theory and we 01:00:44.320 --> 01:00:46.445 expand about fixed points, and one way 01:00:46.445 --> 01:00:48.820 of saying what was wrong with dimensional analysis was it 01:00:48.820 --> 01:00:51.340 was just expanding about the wrong fixed point. 01:00:51.340 --> 01:00:51.910 The pink one. 01:00:55.300 --> 01:01:02.730 So a equals zero, was the non-interactive one 01:01:02.730 --> 01:01:04.480 where we just had the relevant interaction 01:01:04.480 --> 01:01:08.080 being the kinetic term, but none of the interaction terms 01:01:08.080 --> 01:01:09.580 are relevant. 01:01:09.580 --> 01:01:13.060 And a equals plus or minus infinity 01:01:13.060 --> 01:01:15.040 are interacting fixed points, where 01:01:15.040 --> 01:01:17.170 you have an interaction that is relevant. 01:01:22.370 --> 01:01:24.550 So you can think about that in the following sense. 01:01:24.550 --> 01:01:27.070 Classically, what a is measuring is kind of the interaction 01:01:27.070 --> 01:01:30.250 size, if you have classical scattering across sections 01:01:30.250 --> 01:01:31.700 4 pi a squared. 01:01:31.700 --> 01:01:34.858 And if a is small-- 01:01:34.858 --> 01:01:38.420 if a is either very small or very big, 01:01:38.420 --> 01:01:41.290 then basically it's the same on all scales 01:01:41.290 --> 01:01:44.290 because it's either infinity or 0 01:01:44.290 --> 01:01:46.510 and it looks the same to particles 01:01:46.510 --> 01:01:48.190 of all different momenta. 01:01:48.190 --> 01:01:49.150 OK? 01:01:49.150 --> 01:01:51.940 And that's one way of thinking about these fixed points. 01:01:51.940 --> 01:01:53.680 That the physics can't-- 01:01:53.680 --> 01:01:57.220 the physics-- yeah. 01:01:57.220 --> 01:01:58.720 Just what I said. 01:01:58.720 --> 01:02:00.880 So what about this infinity? 01:02:00.880 --> 01:02:02.518 That's also interesting. 01:02:10.486 --> 01:02:16.370 So when a is one over mu, beta goes to infinity. 01:02:16.370 --> 01:02:19.910 And this actually-- there is a reflection 01:02:19.910 --> 01:02:23.480 of this in the theory, it corresponds to a bound state. 01:02:28.580 --> 01:02:30.920 Which I think we'll talk about next time, but. 01:02:33.440 --> 01:02:35.090 So there's a bound state in the theory, 01:02:35.090 --> 01:02:37.310 and actually if you start from this side, 01:02:37.310 --> 01:02:39.110 you can never see that bound because you're 01:02:39.110 --> 01:02:39.980 doing perturbation theory. 01:02:39.980 --> 01:02:42.200 You don't see perturbation theory and bound states. 01:02:42.200 --> 01:02:44.210 If you start from this side, the bound state 01:02:44.210 --> 01:02:46.160 is actually just in the theory. 01:02:46.160 --> 01:02:48.605 We'll talk about that next time. 01:02:48.605 --> 01:02:50.480 And so it's a state in the theory, you can go 01:02:50.480 --> 01:02:52.400 and you could find the pole in your amplitude, 01:02:52.400 --> 01:02:53.900 it's just there. 01:02:53.900 --> 01:02:56.180 Corresponds to a physical state of the spectrum, 01:02:56.180 --> 01:02:57.388 and it's called the deuteron. 01:03:14.620 --> 01:03:16.120 So we have a non-interactive theory, 01:03:16.120 --> 01:03:19.990 we have some non-trivial amplitude, 01:03:19.990 --> 01:03:22.540 and this deuteron is a pole in that amplitude. 01:04:09.310 --> 01:04:13.890 And you never see a pole if you use the perturbation theory. 01:04:13.890 --> 01:04:16.350 And if you actually look at the energy, the binding energy 01:04:16.350 --> 01:04:18.773 of this state, it's also small. 01:04:18.773 --> 01:04:20.190 Characteristically small, and it's 01:04:20.190 --> 01:04:24.030 actually related to the fact that the binding energy is 01:04:24.030 --> 01:04:26.820 small just related to sort of the natural size of this a. 01:04:26.820 --> 01:04:28.260 We'll talk about that next time. 01:04:34.850 --> 01:04:38.290 So what I was saying before about the theory at all scales 01:04:38.290 --> 01:04:41.170 looking the same, when you have a equals 0 of course 01:04:41.170 --> 01:04:42.550 it's not interacting. 01:04:42.550 --> 01:04:44.770 Or when a is equal to plus minus infinity, 01:04:44.770 --> 01:04:47.770 looks the same at all scales. 01:04:47.770 --> 01:04:50.998 That means that scale invariant. 01:04:50.998 --> 01:04:52.540 So it's a scale invariant theory when 01:04:52.540 --> 01:04:55.540 a is at these fixed points. 01:04:55.540 --> 01:04:58.600 And something I worked on was the fact 01:04:58.600 --> 01:05:02.650 that these points are actually conformal fixed points. 01:05:12.570 --> 01:05:16.600 So that-- there's a conformal symmetry of the theory that 01:05:16.600 --> 01:05:20.090 exists at those points, we'll talk about that a little bit. 01:05:20.090 --> 01:05:22.750 There's another symmetry, too, which I have to mention. 01:05:25.480 --> 01:05:29.140 And not only because I also worked on this one, 01:05:29.140 --> 01:05:32.070 because it's good to enumerate all the symmetries. 01:05:35.360 --> 01:05:37.880 There's actually a combined spin, isospin symmetry, 01:05:37.880 --> 01:05:41.027 that turns into an Su4 in this limit. 01:05:41.027 --> 01:05:42.860 So much like in heavy quark effective theory 01:05:42.860 --> 01:05:45.818 where the mass was big, new symmetries popped up. 01:05:45.818 --> 01:05:48.110 Same thing happens here, when the scattering lengths go 01:05:48.110 --> 01:05:50.552 to infinity, and you go over to those fixed points, 01:05:50.552 --> 01:05:52.010 there's new symmetries that pop up. 01:05:52.010 --> 01:05:53.927 One's a conformal symmetry and the other one's 01:05:53.927 --> 01:05:55.190 a spin, isospin symmetry. 01:06:09.810 --> 01:06:12.060 All right. 01:06:12.060 --> 01:06:16.200 So this looks interesting. 01:06:16.200 --> 01:06:18.398 You could ask the question, did this pick-- 01:06:18.398 --> 01:06:20.190 did this physical picture that we developed 01:06:20.190 --> 01:06:22.050 depend on picking this renormalization 01:06:22.050 --> 01:06:24.240 scheme that I told you about? 01:06:24.240 --> 01:06:26.670 We kind of gave up on dimreg, we went over to-- well, 01:06:26.670 --> 01:06:28.110 we gave up on MS bar, we went over 01:06:28.110 --> 01:06:31.502 to this scheme which was offshell momentum subtraction. 01:06:31.502 --> 01:06:33.960 In general, people don't like offshell momentum subtraction 01:06:33.960 --> 01:06:35.793 because it makes calculations more difficult 01:06:35.793 --> 01:06:37.001 once you go to higher orders. 01:06:37.001 --> 01:06:39.210 Well here, are the calculations are not so difficult, 01:06:39.210 --> 01:06:40.320 so we could do them, but-- 01:06:40.320 --> 01:06:42.750 you might be interested in adding pions to this theory, 01:06:42.750 --> 01:06:45.420 or coupling external currents like photons, 01:06:45.420 --> 01:06:47.910 and then the calculations would get more difficult. 01:06:47.910 --> 01:06:51.180 And you'd like, for example, to have a dimreg MS bar type 01:06:51.180 --> 01:06:53.190 description of this power counting 01:06:53.190 --> 01:06:55.620 rather than a kind of minimal subtraction. 01:06:55.620 --> 01:06:57.210 Could I get, could I kind of dress up 01:06:57.210 --> 01:07:00.300 minimal subtraction to get the same physical picture? 01:07:00.300 --> 01:07:03.210 That's a reasonable question to ask. 01:07:03.210 --> 01:07:04.290 The answer is you can. 01:07:10.737 --> 01:07:13.070 So there's something called power divergence subtraction 01:07:13.070 --> 01:07:15.650 scheme, different scheme than MS bar. 01:07:22.244 --> 01:07:25.780 So the PDS scheme. 01:07:25.780 --> 01:07:31.060 And what it says is, don't just subtract poles at D equals 4, 01:07:31.060 --> 01:07:37.510 like you do in MS bar, which are corresponding to logs 01:07:37.510 --> 01:07:42.580 at the cut off, but also subtract poles and D equals 3. 01:07:42.580 --> 01:07:45.090 And dimreg knows about power law divergences 01:07:45.090 --> 01:07:46.840 and they're just poles at different places 01:07:46.840 --> 01:07:48.530 in the dimensions. 01:07:48.530 --> 01:07:50.560 And so if we subtract poles at D equals three, 01:07:50.560 --> 01:07:52.700 we can track the power law divergences in that way. 01:07:52.700 --> 01:07:54.700 And it's the power of divergence that's actually 01:07:54.700 --> 01:07:56.410 causing, if you want to think of it 01:07:56.410 --> 01:07:58.495 as a change to the anomalous dimension, where 01:07:58.495 --> 01:08:00.370 the anomalous dimension was saying this thing 01:08:00.370 --> 01:08:03.250 was irrelevant, to changing it to something relevant, 01:08:03.250 --> 01:08:05.000 you need a big change for that to happen. 01:08:05.000 --> 01:08:06.730 And the big change that's occurring 01:08:06.730 --> 01:08:09.100 is coming from a power law divergence here. 01:08:09.100 --> 01:08:11.650 That's what sort of allowed you to jump, if you like, 01:08:11.650 --> 01:08:15.730 from this fixed point to this one. 01:08:15.730 --> 01:08:17.950 The renormalization group, including the power law 01:08:17.950 --> 01:08:20.937 divergence, allows you to even flow between those points. 01:08:20.937 --> 01:08:23.020 Usually we think that power law divergences aren't 01:08:23.020 --> 01:08:25.562 doing anything, here's an example where they are. 01:08:25.562 --> 01:08:26.979 They're not doing anything as long 01:08:26.979 --> 01:08:28.812 as you know you're at the right fixed point. 01:08:28.812 --> 01:08:31.078 If you're describing the right physics around one 01:08:31.078 --> 01:08:33.370 of these fixed points, you can concentrate on the logs, 01:08:33.370 --> 01:08:35.715 but if you don't know where you are then the power 01:08:35.715 --> 01:08:37.090 law divergences could be crucial. 01:08:39.609 --> 01:08:41.490 All right, how does this scheme work? 01:08:45.250 --> 01:08:49.240 So this is a dimreg-type scheme. 01:08:49.240 --> 01:08:52.930 So we're going to get the power of mu 01:08:52.930 --> 01:08:55.510 from the mu to the two epsilon that we have out front. 01:09:07.520 --> 01:09:11.587 So if I just write this guy down in D dimensions, 01:09:11.587 --> 01:09:12.670 here's what it looks like. 01:09:31.140 --> 01:09:33.770 And I've normalized mu slightly differently than we usually 01:09:33.770 --> 01:09:37.680 do just because it's convenient for this scheme to do that. 01:09:37.680 --> 01:09:40.160 So it's not exactly the same as MS bar, it's mu over 2 01:09:40.160 --> 01:09:41.270 that I'm putting in. 01:09:41.270 --> 01:09:43.640 Other than that, it's the same kind of set up as MS bar. 01:09:51.584 --> 01:09:54.269 You'll see why I want to put that 2 there in a minute. 01:09:59.030 --> 01:10:01.608 OK, so this is just the result that we would write down 01:10:01.608 --> 01:10:02.900 for dimensional regularization. 01:10:02.900 --> 01:10:04.865 Dimensional regularization is not a scheme. 01:10:04.865 --> 01:10:08.930 A scheme has to do with what we subtract. 01:10:08.930 --> 01:10:10.613 Dimreg is just how we regulate. 01:10:21.740 --> 01:10:25.160 So now, look at D equals 4. 01:10:25.160 --> 01:10:31.370 So in D equals 4, OK. 01:10:31.370 --> 01:10:32.495 There's a bunch of factors. 01:10:38.315 --> 01:10:40.940 This is just giving, this is the answer I quoted to you before. 01:10:44.480 --> 01:10:46.700 Something finite. 01:10:46.700 --> 01:10:54.160 And if we look at D equals three, 01:10:54.160 --> 01:10:56.285 then we have a pole because of that gamma function. 01:11:02.030 --> 01:11:04.370 And I've put the 2 in here just to cancel that 2. 01:11:07.520 --> 01:11:11.990 And so, what this scheme says is to add a part subtraction 01:11:11.990 --> 01:11:14.090 for this guy. 01:11:14.090 --> 01:11:16.950 So what we do is, we add a counter term. 01:11:16.950 --> 01:11:25.640 It looks like minus IM over 4 pi, mu, got one power of mu. 01:11:25.640 --> 01:11:29.480 Over 3 minus D. C0 squared. 01:11:32.430 --> 01:11:36.710 And then if we take the graph, plus the counterterm and we 01:11:36.710 --> 01:11:38.840 set D equal four, which is where we actually want 01:11:38.840 --> 01:11:45.320 to do physics, lo and behold. 01:11:45.320 --> 01:11:51.500 In this approach, you get actually the same answer 01:11:51.500 --> 01:11:53.510 as in our offshell momentum subtraction scheme. 01:11:53.510 --> 01:11:55.610 And that's just really because this scheme tracks the power 01:11:55.610 --> 01:11:57.170 correction, the power divergence, 01:11:57.170 --> 01:12:00.250 just like the offshell momentum subtraction did. 01:12:00.250 --> 01:12:03.500 So we just invented a dimreg style 01:12:03.500 --> 01:12:05.965 of looking for poles that can track the same physics, 01:12:05.965 --> 01:12:07.340 and we just have to look at poles 01:12:07.340 --> 01:12:11.440 in D equals 3 rather than poles in D equals 4, OK? 01:12:13.887 --> 01:12:16.220 And this is easier in general than the offshell momentum 01:12:16.220 --> 01:12:19.880 subtraction scheme, although for basically everything 01:12:19.880 --> 01:12:22.650 we're talking about today you could do either one. 01:12:22.650 --> 01:12:25.848 Now, where is the predictive power of this effective theory? 01:12:25.848 --> 01:12:27.890 So far, we've just kind of cooked things together 01:12:27.890 --> 01:12:30.350 to make the C0 do what we want. 01:12:30.350 --> 01:12:32.420 Well, we didn't completely cook things together. 01:12:32.420 --> 01:12:34.370 We switched to another scheme and it kind of popped out 01:12:34.370 --> 01:12:35.745 naturally, but you can say, well, 01:12:35.745 --> 01:12:39.178 why not explore three other schemes and see if they work? 01:12:39.178 --> 01:12:41.720 But let's just imagine that we got things to work, as we just 01:12:41.720 --> 01:12:44.210 did in two different ways by tracking the power law 01:12:44.210 --> 01:12:45.320 divergence. 01:12:45.320 --> 01:12:47.750 The predictive power becomes from now the fact 01:12:47.750 --> 01:12:51.710 that if I say that's my fine tuning, that C0 got enhanced, 01:12:51.710 --> 01:12:54.230 I can now predict the size of all other operators 01:12:54.230 --> 01:12:55.310 in the theory. 01:12:55.310 --> 01:12:57.887 And other operators like C2 and C4, 01:12:57.887 --> 01:13:00.470 the power counting we assigned to them previously is not true. 01:13:00.470 --> 01:13:01.980 We have to figure it out. 01:13:01.980 --> 01:13:04.610 But we can figure that out once we know 01:13:04.610 --> 01:13:09.160 what approach we should use. 01:13:09.160 --> 01:13:15.667 OK, so this is same as above. 01:13:15.667 --> 01:13:18.250 I won't go through it, but you know, same anomalous dimension, 01:13:18.250 --> 01:13:19.420 et cetera. 01:13:19.420 --> 01:13:22.240 And it's easier in general. 01:13:22.240 --> 01:13:25.990 Let's think about C2 mu. 01:13:25.990 --> 01:13:29.170 So if you look at C2, the first kind 01:13:29.170 --> 01:13:33.160 of diagram you could think about would be a guy with one C2, 01:13:33.160 --> 01:13:36.070 and then this is the first type of loop diagram 01:13:36.070 --> 01:13:38.240 you might think about. 01:13:38.240 --> 01:13:41.960 And these guys have a P squared because C2 gave a P squared. 01:13:41.960 --> 01:13:43.840 So they have an extra P squared. 01:13:43.840 --> 01:13:45.010 And they diverge. 01:13:45.010 --> 01:13:46.690 They also have a power divergence. 01:13:46.690 --> 01:13:48.280 So if you calculate in either one 01:13:48.280 --> 01:13:50.320 of these schemes, offshell momentum 01:13:50.320 --> 01:13:56.927 subtraction, or this PDS scheme, these guys get a divergence. 01:13:56.927 --> 01:13:58.510 And again, it's a power law divergence 01:13:58.510 --> 01:14:01.190 so there's a mu here. 01:14:01.190 --> 01:14:06.190 And you get a beta function, that's that. 01:14:06.190 --> 01:14:08.800 2C0 C2. 01:14:08.800 --> 01:14:11.040 So if you take the boundary condition 01:14:11.040 --> 01:14:15.670 C2 of 0 which is our MS bar result. 4 pi over M. A 01:14:15.670 --> 01:14:17.320 squared, r0. 01:14:17.320 --> 01:14:30.010 And you solve this, you find C2 of mu is 4 pi over M. 1 01:14:30.010 --> 01:14:34.870 over mu minus 1 over a, squared. 01:14:34.870 --> 01:14:36.010 r0 over 2. 01:14:41.700 --> 01:14:44.165 So there's two-- we previously, with our counting, 01:14:44.165 --> 01:14:45.540 when we were counting dimensions, 01:14:45.540 --> 01:14:49.080 we would have said C0 goes like 1 over lambda, 01:14:49.080 --> 01:14:51.900 C2 goes like one over lambda cubed. 01:14:51.900 --> 01:14:54.480 We had 2M plus 1 lambdas. 01:14:54.480 --> 01:14:57.300 What we've just discovered is that yes, 01:14:57.300 --> 01:15:00.700 there's a lambda from this r0, that's a 1 over lambda, 01:15:00.700 --> 01:15:03.570 but the other two lambdas are really mu's. 01:15:03.570 --> 01:15:06.900 So this operator is also enhanced, 01:15:06.900 --> 01:15:09.797 and it's enhanced by two powers. 01:15:09.797 --> 01:15:11.380 So once you know leading order theory, 01:15:11.380 --> 01:15:13.770 you should be able to determine the power counting of all 01:15:13.770 --> 01:15:15.352 of the other operators, especially 01:15:15.352 --> 01:15:16.560 if they're not relevant ones. 01:15:16.560 --> 01:15:18.450 You have to get the relevant part right, 01:15:18.450 --> 01:15:20.040 and then you can use that Lagrangian 01:15:20.040 --> 01:15:22.697 to predict all the scaling for everything else. 01:15:22.697 --> 01:15:24.030 And that's what we've just done. 01:15:38.970 --> 01:15:41.850 Gone to 1 over mu squared lambda. 01:15:41.850 --> 01:15:42.360 OK? 01:15:42.360 --> 01:15:45.223 And those powers of mu we see in the scheme, 01:15:45.223 --> 01:15:46.890 and the 1 over lambda comes from the r0. 01:15:54.000 --> 01:15:54.500 OK. 01:15:54.500 --> 01:15:59.780 So what the RGE actually does is it tells us-- 01:15:59.780 --> 01:16:02.990 one way of thinking about it is that it tells us 01:16:02.990 --> 01:16:04.640 the enhancement, due to fine tuning, 01:16:04.640 --> 01:16:05.990 of all operators in the theory. 01:16:09.607 --> 01:16:11.690 And that's really because the fine tuning was just 01:16:11.690 --> 01:16:14.180 a change of our power counting, and we 01:16:14.180 --> 01:16:16.930 have to propagate that change everywhere. 01:16:22.930 --> 01:16:25.360 And we can do that, and it's the beta functions 01:16:25.360 --> 01:16:28.870 that tell us how to propagate the fine tuning. 01:16:28.870 --> 01:16:34.246 So if you keep going, you can do C2K of mu. 01:16:34.246 --> 01:16:36.070 You find an anomalous dimension. 01:16:39.680 --> 01:16:43.060 This theory is kind of nice, you can basically 01:16:43.060 --> 01:16:44.490 do all the calculations, so. 01:16:48.076 --> 01:16:50.500 When you go to CK, you get contributions 01:16:50.500 --> 01:16:54.250 from various lower order coefficients, 01:16:54.250 --> 01:16:56.365 and it's one loop exact so you only have pairs. 01:16:59.365 --> 01:17:00.740 And then you can kind of contrast 01:17:00.740 --> 01:17:04.880 what's going on in a naive power counting where 01:17:04.880 --> 01:17:06.230 P is much less than 1 over a. 01:17:14.720 --> 01:17:17.750 Let's just go up to C4. 01:17:17.750 --> 01:17:21.200 Versus this kind of improved power 01:17:21.200 --> 01:17:26.540 counting, which is valid when PA is greater than our order 1. 01:17:30.995 --> 01:17:34.820 So C0 hat it went like 1 over mu, no suppression there. 01:17:34.820 --> 01:17:37.370 C2 hat goes like one over mu squared lambda, 01:17:37.370 --> 01:17:39.840 and that actually is irrelevant. 01:17:39.840 --> 01:17:41.480 But it's just irrelevant by one power. 01:17:46.300 --> 01:17:49.950 So relative to this guy, it's down by a P over lambda. 01:17:49.950 --> 01:17:52.020 And then interesting things start 01:17:52.020 --> 01:17:54.660 to happen with the higher ones, at least 01:17:54.660 --> 01:17:56.386 from an RGE perspective. 01:17:59.493 --> 01:18:01.410 So these guys start to get more than one term. 01:18:05.358 --> 01:18:07.650 But this guy actually doesn't introduce a new constant. 01:18:13.458 --> 01:18:16.000 There's a piece of the anomalous dimension of this guy that's 01:18:16.000 --> 01:18:17.458 actually just fixed by the constant 01:18:17.458 --> 01:18:20.170 that you already had here, and then there's a new piece. 01:18:20.170 --> 01:18:22.350 So the new piece is down by two powers of lambda. 01:18:26.545 --> 01:18:29.170 And that's encoding things about the amplitude, actually, but-- 01:18:32.240 --> 01:18:33.800 OK, so that's just a little table 01:18:33.800 --> 01:18:35.342 to kind of convince you that once you 01:18:35.342 --> 01:18:37.370 have a beta function that you can compute 01:18:37.370 --> 01:18:39.260 for the coefficients, you can quickly 01:18:39.260 --> 01:18:41.570 propagate this enhancement from the fine tuning 01:18:41.570 --> 01:18:42.860 to the rest of your theory. 01:18:42.860 --> 01:18:45.320 I.e., figure out what the power counting 01:18:45.320 --> 01:18:47.020 is for all the operators. 01:18:47.020 --> 01:18:49.640 AUDIENCE: So every time there is a power law divergence, 01:18:49.640 --> 01:18:51.482 should I be worried if I'm using MS bar, 01:18:51.482 --> 01:18:53.940 should I be worried that the power counting could be wrong? 01:18:53.940 --> 01:18:56.160 PROFESSOR: Every time-- 01:18:56.160 --> 01:18:58.160 Yeah, every time there's a power law divergence, 01:18:58.160 --> 01:18:59.660 it's worth thinking about whether it 01:18:59.660 --> 01:19:06.220 had some physical impact on what you're doing, I think. 01:19:11.042 --> 01:19:13.500 If you know you're expanding-- if you can convince yourself 01:19:13.500 --> 01:19:15.930 that you're expanding around the right fixed point then 01:19:15.930 --> 01:19:16.530 you're OK. 01:19:16.530 --> 01:19:20.520 That's my equivalence claim. 01:19:20.520 --> 01:19:22.020 But you don't necessarily know that. 01:19:22.020 --> 01:19:24.110 So let's go back to our amplitude 01:19:24.110 --> 01:19:26.930 and see what's going on here, and see 01:19:26.930 --> 01:19:29.890 what it looks like with this power counting. 01:19:29.890 --> 01:19:33.290 And so it's really just a different expansion 01:19:33.290 --> 01:19:34.760 of that amplitude that we had. 01:19:41.420 --> 01:19:48.380 And in either the PDS scheme or the power diversion subtraction 01:19:48.380 --> 01:19:54.620 scheme, we end up with this amplitude in the case where 01:19:54.620 --> 01:19:56.670 we would use that scheme. 01:19:56.670 --> 01:19:59.948 So you can see in PDS that if I set this coefficient to 0, 01:19:59.948 --> 01:20:01.490 the denominator becomes 1, and then I 01:20:01.490 --> 01:20:04.580 get with the offshell momentum subtraction scheme. 01:20:04.580 --> 01:20:07.640 In PDS is a little harder see that it gives just that, 01:20:07.640 --> 01:20:10.080 but it actually gives the same thing in either scheme. 01:20:10.080 --> 01:20:14.270 And if you think about what type of expansion you're doing here, 01:20:14.270 --> 01:20:15.930 you're keeping C0 to all orders. 01:20:15.930 --> 01:20:23.720 So your amplitude at lowest order is just this, 01:20:23.720 --> 01:20:38.510 and then the C2 term looks like that. 01:20:38.510 --> 01:20:40.040 And then there's some higher terms 01:20:40.040 --> 01:20:43.770 which I wrote in my notes, but I want right here. 01:20:43.770 --> 01:20:47.270 And what this is, this here is some kind of interaction. 01:20:47.270 --> 01:20:50.030 I'll make it a bigger circle, which 01:20:50.030 --> 01:20:53.900 sums up all the bubbles with C0's in them. 01:20:53.900 --> 01:20:56.780 That's what's happened here. 01:20:56.780 --> 01:21:01.610 And this here, if you like, is like taking C2 and then 01:21:01.610 --> 01:21:04.535 dressing it with bubbles on either side. 01:21:04.535 --> 01:21:07.700 So there's two, there's bubbles. 01:21:07.700 --> 01:21:11.523 The bubbles on the other side. 01:21:11.523 --> 01:21:12.815 And then bubbles on both sides. 01:21:16.500 --> 01:21:18.480 So we calculate these loop graphs 01:21:18.480 --> 01:21:19.980 and these are the amplitudes we get, 01:21:19.980 --> 01:21:21.355 and that's because we're treating 01:21:21.355 --> 01:21:23.645 the C0 coupling to all orders, we're summing it up. 01:21:23.645 --> 01:21:25.770 And actually, each of these amplitudes, if you look 01:21:25.770 --> 01:21:28.590 at the RGE, is mu independent. 01:21:28.590 --> 01:21:31.115 Explicitly mu independent. 01:21:31.115 --> 01:21:32.490 So it's like perturbation theory, 01:21:32.490 --> 01:21:33.960 where we're doing a momentum expansion, 01:21:33.960 --> 01:21:35.877 and order by order in that momentum expansion, 01:21:35.877 --> 01:21:39.300 the amplitude is independent of the scale mu. 01:21:39.300 --> 01:21:41.160 The only purpose of the scale mu is 01:21:41.160 --> 01:21:45.155 to help us think about power counting of these operators. 01:21:45.155 --> 01:21:47.530 In the end of the day, when we make physical predictions, 01:21:47.530 --> 01:21:50.790 then getting mu independent answers. 01:21:50.790 --> 01:21:51.840 OK. 01:21:51.840 --> 01:21:55.813 And this is like organizing, if you like. 01:21:55.813 --> 01:21:56.980 And you put it in terms of-- 01:22:01.380 --> 01:22:07.255 put it back in terms of a, this is 01:22:07.255 --> 01:22:08.880 like organizing the theory in this way, 01:22:08.880 --> 01:22:12.900 where you keep all powers of AP, and that makes it very clear 01:22:12.900 --> 01:22:14.910 that it's mu independent. 01:22:14.910 --> 01:22:17.100 Now, this part of the theory is so 01:22:17.100 --> 01:22:19.380 simple you could have figured that out just 01:22:19.380 --> 01:22:22.380 by writing the top line down in terms of a's and P's and just 01:22:22.380 --> 01:22:24.180 writing this line down. 01:22:24.180 --> 01:22:26.730 But you could also use what I've been talking about 01:22:26.730 --> 01:22:31.260 to figure out, for example, say I coupled an external photon 01:22:31.260 --> 01:22:33.150 to my four nucleon operators. 01:22:33.150 --> 01:22:34.830 How big is this? 01:22:34.830 --> 01:22:35.640 OK. 01:22:35.640 --> 01:22:38.370 Well, it actually gets enhanced from the fact the scattering 01:22:38.370 --> 01:22:39.995 length is large, and you can figure out 01:22:39.995 --> 01:22:43.650 how important this effect is, and when people do things 01:22:43.650 --> 01:22:48.030 like deuteron formation in the sun and stuff like that, 01:22:48.030 --> 01:22:51.330 they use this effective theory to do higher order calculations 01:22:51.330 --> 01:22:55.040 and make precision predictions for deuteron physics. 01:22:55.040 --> 01:22:56.910 So it's not just a toy model. 01:22:56.910 --> 01:23:00.750 It's actually something that has a real impact on some physics. 01:23:00.750 --> 01:23:02.813 We'll talk a little bit more about it next time. 01:23:02.813 --> 01:23:04.980 We'll talk a little bit about the conformal symmetry 01:23:04.980 --> 01:23:07.105 and I'll talk a little bit more about the deuterons 01:23:07.105 --> 01:23:09.210 since that's something interesting in this theory, 01:23:09.210 --> 01:23:13.330 and then we'll go on from there. 01:23:13.330 --> 01:23:14.760 So, any questions? 01:23:17.388 --> 01:23:18.880 It's cool stuff. 01:23:18.880 --> 01:23:21.310 Simple to do calculations. 01:23:21.310 --> 01:23:23.850 It's kind of interesting to think about.