1 00:00:00,000 --> 00:00:02,520 The following content is provided under a Creative 2 00:00:02,520 --> 00:00:03,970 Commons license. 3 00:00:03,970 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,660 continue to offer high quality educational resources for free. 5 00:00:10,660 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,190 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,190 --> 00:00:18,370 at ocw.mit.edu. 8 00:00:23,370 --> 00:00:24,403 PROFESSOR: Apparently. 9 00:00:28,190 --> 00:00:29,210 We'll start slow. 10 00:00:29,210 --> 00:00:31,088 So last time we were talking, we just 11 00:00:31,088 --> 00:00:32,630 started talking about effective field 12 00:00:32,630 --> 00:00:34,310 theory with a fine tuning. 13 00:00:34,310 --> 00:00:36,620 And what actually that means takes 14 00:00:36,620 --> 00:00:38,060 a little bit of discussion. 15 00:00:41,840 --> 00:00:44,380 So what you could mean by a fine tuning 16 00:00:44,380 --> 00:00:47,350 is that you have something that's irrelevant. 17 00:00:47,350 --> 00:00:49,610 You look at the operator, you think it's irrelevant, 18 00:00:49,610 --> 00:00:50,530 but it's not. 19 00:00:50,530 --> 00:00:51,340 It's relevant. 20 00:00:51,340 --> 00:00:53,800 Something that you should include even at lowest order 21 00:00:53,800 --> 00:00:55,300 in your power accounting. 22 00:00:55,300 --> 00:00:57,815 But saying something is irrelevant 23 00:00:57,815 --> 00:01:00,190 means that you have a power counting, that you understand 24 00:01:00,190 --> 00:01:02,650 the power counting for the theory, what the correct power 25 00:01:02,650 --> 00:01:04,010 counting is. 26 00:01:04,010 --> 00:01:05,710 So in this example that I'll give, 27 00:01:05,710 --> 00:01:07,960 what "irrelevant" will mean is that you basically 28 00:01:07,960 --> 00:01:10,480 do a dimensional power counting, which 29 00:01:10,480 --> 00:01:11,980 is how we usually think of defining 30 00:01:11,980 --> 00:01:13,450 irrelevant and relevant. 31 00:01:13,450 --> 00:01:15,520 Do a dimensional power counting and you end up 32 00:01:15,520 --> 00:01:17,047 finding an operator that-- 33 00:01:17,047 --> 00:01:18,880 you find that the operator looks irrelevant, 34 00:01:18,880 --> 00:01:20,920 but when you do calculations you can 35 00:01:20,920 --> 00:01:22,618 see that should be relevant. 36 00:01:22,618 --> 00:01:24,160 And that means, really, what it means 37 00:01:24,160 --> 00:01:26,433 is that this natural power counting of dimensions 38 00:01:26,433 --> 00:01:27,850 is not the right one, and you have 39 00:01:27,850 --> 00:01:29,692 to do something more complicated. 40 00:01:29,692 --> 00:01:31,150 But it is still is a sense in which 41 00:01:31,150 --> 00:01:37,110 it can be thought of as a fine tuning, because as you'll see, 42 00:01:37,110 --> 00:01:39,330 the changing of the power counting from the naive one 43 00:01:39,330 --> 00:01:41,700 to the more complicated power counting 44 00:01:41,700 --> 00:01:46,330 involves some kind of tuning, if you like. 45 00:01:46,330 --> 00:01:47,850 And it really, in this case, we'll 46 00:01:47,850 --> 00:01:49,730 see actually that it corresponds not 47 00:01:49,730 --> 00:01:52,230 to expanding around the trivial fixed point, where you would 48 00:01:52,230 --> 00:01:54,060 have a free theory, but expanding 49 00:01:54,060 --> 00:01:55,560 around an interactive fixed point. 50 00:01:58,455 --> 00:01:59,830 So it'll be a little non-trivial, 51 00:01:59,830 --> 00:02:02,080 but we'll be doing this in the context of two 52 00:02:02,080 --> 00:02:03,490 nucleon effective field theory. 53 00:02:03,490 --> 00:02:05,710 And the advantage of this is that the nucleons are 54 00:02:05,710 --> 00:02:07,390 going to be non-relativistic. 55 00:02:07,390 --> 00:02:09,857 So P is going to be much less than M pi. 56 00:02:09,857 --> 00:02:11,440 It's going to be a very simple theory. 57 00:02:11,440 --> 00:02:14,620 Everything that's an exchange particle gets integrated out. 58 00:02:14,620 --> 00:02:16,690 It's just a theory of contact interactions, 59 00:02:16,690 --> 00:02:19,210 and derivatives of contact interactions. 60 00:02:19,210 --> 00:02:20,800 And because it's non-relativistic, 61 00:02:20,800 --> 00:02:22,510 we can actually calculate all the loops 62 00:02:22,510 --> 00:02:24,093 to all orders and perturbation theory, 63 00:02:24,093 --> 00:02:25,490 and we'll do that in a minute. 64 00:02:25,490 --> 00:02:28,820 So this theory, we can calculate a lot of stuff. 65 00:02:28,820 --> 00:02:32,980 And so we'll actually be able to see how this non-trivial fine 66 00:02:32,980 --> 00:02:35,713 tuning works, and explore it from multiple directions, 67 00:02:35,713 --> 00:02:37,380 and we'll be sure that what we're saying 68 00:02:37,380 --> 00:02:38,500 is actually correct. 69 00:02:41,340 --> 00:02:43,720 So it can be a lesson for understanding 70 00:02:43,720 --> 00:02:46,330 some of the concepts and other effective field theories 71 00:02:46,330 --> 00:02:49,360 with the fine tuning, which you might want to design, 72 00:02:49,360 --> 00:02:53,407 where you don't have as much ability to calculate. 73 00:02:53,407 --> 00:02:55,240 All right, so let's start off with something 74 00:02:55,240 --> 00:02:59,650 simple, which is elastic scattering. 75 00:02:59,650 --> 00:03:01,930 And that's mostly what we're going to talk about. 76 00:03:07,350 --> 00:03:12,300 Two particles in, two particles out. 77 00:03:12,300 --> 00:03:15,430 Center of mass frame. 78 00:03:15,430 --> 00:03:21,183 They scatter to some P's coming in, P primes going out. 79 00:03:21,183 --> 00:03:23,100 And this is basically a problem that you could 80 00:03:23,100 --> 00:03:24,390 treat with quantum mechanics. 81 00:03:27,342 --> 00:03:31,810 It's like non-relativistic scattering. 82 00:03:31,810 --> 00:03:41,650 So if you have a single partial wave, 83 00:03:41,650 --> 00:03:43,290 then this scattering is described 84 00:03:43,290 --> 00:03:45,930 by a phase shift, delta. 85 00:03:45,930 --> 00:03:51,870 And the relation of the phase shift to the amplitude, 86 00:03:51,870 --> 00:03:57,600 with our normalization for the amplitude, is this. 87 00:03:57,600 --> 00:03:59,700 So this is the S-matrix, it's just a phase, 88 00:03:59,700 --> 00:04:02,670 and that's the relation of the S-matrix to the amplitude. 89 00:04:02,670 --> 00:04:05,922 And this thing is the amplitude. 90 00:04:05,922 --> 00:04:07,380 And I guess the other thing we know 91 00:04:07,380 --> 00:04:12,960 is that, by energy conservation, the magnitude of P 92 00:04:12,960 --> 00:04:16,529 is equal to the magnitude of P prime. 93 00:04:16,529 --> 00:04:19,860 All right, so if we rearrange this equation, 94 00:04:19,860 --> 00:04:23,760 and we write it as A and put the phase, solve for a. 95 00:04:29,736 --> 00:04:30,810 Could do that. 96 00:04:34,550 --> 00:04:37,363 So that gives that equation, which we 97 00:04:37,363 --> 00:04:38,655 can rearrange a little further. 98 00:04:47,954 --> 00:05:00,690 Now, there's kind of a conventional way of writing it, 99 00:05:00,690 --> 00:05:02,420 which is that the amplitude should 100 00:05:02,420 --> 00:05:06,440 be given by 1 over something that's P cotangent delta. 101 00:05:06,440 --> 00:05:10,370 Scattering angle, or that S-matrix angle, 102 00:05:10,370 --> 00:05:13,440 and then minus this IP. 103 00:05:13,440 --> 00:05:14,000 OK? 104 00:05:14,000 --> 00:05:15,860 So the I of-- 105 00:05:15,860 --> 00:05:19,250 the I's just-- shows up here in this part, 106 00:05:19,250 --> 00:05:21,660 and that's the complex part of the amplitude. 107 00:05:21,660 --> 00:05:23,810 That's basically going to be related to unitarity, 108 00:05:23,810 --> 00:05:26,960 that that IP is there. 109 00:05:26,960 --> 00:05:30,326 This S-matrix is obviously unitary. 110 00:05:30,326 --> 00:05:33,830 This digress is 1. 111 00:05:33,830 --> 00:05:37,760 All right, so let me tell you something 112 00:05:37,760 --> 00:05:44,000 about non-relativistic scattering, 113 00:05:44,000 --> 00:05:46,700 which on the face of it, looks kind of non-trivial. 114 00:06:10,990 --> 00:06:13,690 So this thing, P cotangent delta. 115 00:06:13,690 --> 00:06:16,450 So here, I was doing a single partial wave. 116 00:06:16,450 --> 00:06:18,460 Yeah, OK, no, it's fine. 117 00:06:18,460 --> 00:06:20,590 So what is this L? 118 00:06:20,590 --> 00:06:22,360 This L is the partial I am considering. 119 00:06:22,360 --> 00:06:23,690 S wave, P wave. 120 00:06:23,690 --> 00:06:27,310 So L is 0 for the S wave, L is 1 for the P wave. 121 00:06:27,310 --> 00:06:30,100 And the statement is that if you have a short range potential, 122 00:06:30,100 --> 00:06:35,860 and you pick a wave, then this P cotangent delta 123 00:06:35,860 --> 00:06:38,590 with the appropriate power of P can be a Taylor series 124 00:06:38,590 --> 00:06:41,398 expansion of P. OK? 125 00:06:41,398 --> 00:06:42,940 And this is actually something that's 126 00:06:42,940 --> 00:06:46,000 quite difficult to prove in quantum mechanics, 127 00:06:46,000 --> 00:06:49,043 this particular fact. 128 00:06:49,043 --> 00:06:50,960 And it's called the effective range expansion. 129 00:07:00,807 --> 00:07:03,390 It's difficult to prove because when you do quantum mechanics, 130 00:07:03,390 --> 00:07:05,250 you pick a potential. 131 00:07:05,250 --> 00:07:08,140 And I'm saying that this is true for any potential. 132 00:07:08,140 --> 00:07:11,140 So if you start doing quantum mechanics with some potential, 133 00:07:11,140 --> 00:07:13,380 you've got to prove that you can put it in this form 134 00:07:13,380 --> 00:07:16,510 irrespective of what the choice of that potential would be. 135 00:07:16,510 --> 00:07:18,360 And that makes it a little bit tricky 136 00:07:18,360 --> 00:07:22,002 to do in a quantum mechanical setup. 137 00:07:22,002 --> 00:07:23,460 But we'll see actually that this is 138 00:07:23,460 --> 00:07:25,950 very easy to prove from an effective field theory setup. 139 00:07:34,104 --> 00:07:36,280 So, as a way of getting into this effective field 140 00:07:36,280 --> 00:07:40,450 theory of two nucleons, let's prove this fact. 141 00:07:40,450 --> 00:07:42,320 So what is the Lagrangian for this theory? 142 00:07:46,930 --> 00:07:49,980 There's no-- it's not a gauge theory, 143 00:07:49,980 --> 00:07:53,920 so we just have ordinary derivatives. 144 00:07:53,920 --> 00:07:56,170 So if you like, you can think of what I'm writing here 145 00:07:56,170 --> 00:07:58,780 as kind of like our IV.D term, except I think 146 00:07:58,780 --> 00:08:00,790 the center of mass frame, and this 147 00:08:00,790 --> 00:08:02,560 would be like the kinetic energy term, 148 00:08:02,560 --> 00:08:06,040 but now it's just partial squared with no D, et cetera. 149 00:08:20,370 --> 00:08:23,028 And then there's a bunch of contact interactions. 150 00:08:23,028 --> 00:08:24,570 So there's a whole bunch of operators 151 00:08:24,570 --> 00:08:27,540 that are involved in nucleon and fields 152 00:08:27,540 --> 00:08:30,450 with some Wilson coefficients. 153 00:08:30,450 --> 00:08:34,350 The notation here is this S is kind 154 00:08:34,350 --> 00:08:36,850 of a pseudonym for the channel. 155 00:08:36,850 --> 00:08:38,909 So this S here-- 156 00:08:38,909 --> 00:08:41,970 maybe it should be a script S or something, 157 00:08:41,970 --> 00:08:43,679 is telling me what channel I'm in, 158 00:08:43,679 --> 00:08:46,980 and if in a spectroscopic notation, 159 00:08:46,980 --> 00:08:52,717 you'd say you're in the 2S plus 1 LJ channel. 160 00:08:52,717 --> 00:08:54,300 So this would be the angular momentum, 161 00:08:54,300 --> 00:08:55,470 total momentum in the spin. 162 00:08:58,970 --> 00:09:04,010 And these operators here, for our purposes, 163 00:09:04,010 --> 00:09:07,892 are four nuclear fields with 2M derivatives. 164 00:09:14,300 --> 00:09:17,650 Now, this is not really the complete theory 165 00:09:17,650 --> 00:09:18,650 for a couple of reasons. 166 00:09:18,650 --> 00:09:20,210 Well, there's higher order relativistic corrections 167 00:09:20,210 --> 00:09:21,560 indicated by the dots. 168 00:09:21,560 --> 00:09:24,230 There would also be dots over here 169 00:09:24,230 --> 00:09:26,660 That could have to do with having more nucleon fields. 170 00:09:26,660 --> 00:09:28,340 For example, I could have-- 171 00:09:28,340 --> 00:09:30,612 instead of just four, I could have six. 172 00:09:30,612 --> 00:09:32,570 But I don't need to worry about those operators 173 00:09:32,570 --> 00:09:37,113 for two-to-two scattering, so I'm leaving them out. 174 00:09:37,113 --> 00:09:38,780 So this is actually the complete theory, 175 00:09:38,780 --> 00:09:45,060 if we include these dots here, for two-to-two scattering. 176 00:09:45,060 --> 00:09:55,100 And this nucleon field, [? its ?] spin [? at ?] half, 177 00:09:55,100 --> 00:09:56,630 and it's isospin been at half too, 178 00:09:56,630 --> 00:09:58,610 so it includes both the proton and the neutron. 179 00:10:04,680 --> 00:10:13,260 Nucleons are fermions, and that implies, actually, a relation, 180 00:10:13,260 --> 00:10:15,850 because the wave function has to be anti-symmetric. 181 00:10:15,850 --> 00:10:23,040 And so actually, you know that you can associate isospin 182 00:10:23,040 --> 00:10:25,410 and the angular momentum in the following 183 00:10:25,410 --> 00:10:27,370 way because of this fact. 184 00:10:27,370 --> 00:10:31,770 So all the isotriplets have -1 to the S plus L even, 185 00:10:31,770 --> 00:10:37,770 and the isosinglets have -1 to the S plus L odd. 186 00:10:37,770 --> 00:10:40,230 So that cuts down by a factor of two 187 00:10:40,230 --> 00:10:42,990 the number of combinations you have to consider. 188 00:10:42,990 --> 00:10:45,000 And basically, what this theory has 189 00:10:45,000 --> 00:10:49,320 is for some given channel in and some given channel 190 00:10:49,320 --> 00:11:01,700 out, which I could denote in general different, 191 00:11:01,700 --> 00:11:04,220 we get operators that just will have some power 192 00:11:04,220 --> 00:11:05,810 of the center of mass momentum. 193 00:11:05,810 --> 00:11:06,750 P to the 2M. 194 00:11:09,500 --> 00:11:13,670 And actually, just by angular momentum conservation, 195 00:11:13,670 --> 00:11:16,460 that has to be the same as that. 196 00:11:16,460 --> 00:11:19,010 And so all that can change is the L's. 197 00:11:19,010 --> 00:11:24,680 And so if S is 0, S is either 0 or 1, 198 00:11:24,680 --> 00:11:27,500 because we have 2 spin half particles. 199 00:11:27,500 --> 00:11:31,010 If S is zero, L will be equal to L prime 200 00:11:31,010 --> 00:11:34,130 because J is equal to J prime, and there's 201 00:11:34,130 --> 00:11:36,110 no shift of the spin. 202 00:11:36,110 --> 00:11:39,920 So that's one possibility, and if S is 1, then-- 203 00:11:42,630 --> 00:11:45,450 so S here is the same, if S is 1, 204 00:11:45,450 --> 00:11:55,480 then you can have L minus L prime which is 2, so, or 0. 205 00:11:55,480 --> 00:11:56,220 OK? 206 00:11:56,220 --> 00:11:57,720 You're shifting by one unit, and you 207 00:11:57,720 --> 00:12:03,390 can compensate either by having minus L prime be 0 or 2. 208 00:12:08,880 --> 00:12:14,930 OK, so we conserve J. So we can enumerate 209 00:12:14,930 --> 00:12:16,340 all the possible partial waves. 210 00:12:16,340 --> 00:12:19,530 We'll mostly focus on the S wave. 211 00:12:19,530 --> 00:12:26,360 So, let me write out some of these operators for you. 212 00:12:33,540 --> 00:12:36,290 So the first operator has no derivatives, 213 00:12:36,290 --> 00:12:42,220 and I can write it in a way that makes the partial wave explicit 214 00:12:42,220 --> 00:12:43,786 if I do the following. 215 00:12:50,940 --> 00:12:54,135 And then there would be some derivative operator. 216 00:12:54,135 --> 00:12:55,760 And I'm going to pick the normalization 217 00:12:55,760 --> 00:12:59,930 to make our lives as easy as possible, 218 00:12:59,930 --> 00:13:02,440 as we usually do when we're setting up the operator basis. 219 00:13:22,660 --> 00:13:29,242 There's the first two guys where this derivative operator 220 00:13:29,242 --> 00:13:33,736 is like, [? grad squared ?] to the left. 221 00:13:33,736 --> 00:13:36,857 [? Grad ?] left dot [? grad ?] right. 222 00:13:36,857 --> 00:13:38,190 That [? grad ?] go to the right. 223 00:13:41,980 --> 00:13:44,720 And these P's, if you look at them, 224 00:13:44,720 --> 00:13:48,080 they're just matrices in the spin and isospin space. 225 00:13:48,080 --> 00:13:53,260 So the two we're focusing on are the S waves. 226 00:13:53,260 --> 00:13:55,570 And in the S wave you either have 1S0 or 3S1. 227 00:14:16,310 --> 00:14:18,350 And so we've encoded all the, sort of, 228 00:14:18,350 --> 00:14:21,230 complexity in just these matrices, which kind of just 229 00:14:21,230 --> 00:14:26,460 go along for the ride, and I'll tell you what they are. 230 00:14:26,460 --> 00:14:33,980 So I sigma 2 projects you onto a spin singlet, and I tau 2 tau 231 00:14:33,980 --> 00:14:37,290 I projects you onto an isotriplet. 232 00:14:37,290 --> 00:14:43,700 And then likewise, 3S1, which is a spin triplet, 233 00:14:43,700 --> 00:14:50,540 you put I sigma 2 sigma I. I tau 2. 234 00:14:50,540 --> 00:14:52,815 I tau 2 and I sigma 2 are just because of the way 235 00:14:52,815 --> 00:14:53,690 I wrote the operator. 236 00:14:53,690 --> 00:14:56,060 I wrote it, instead of writing N dagger N, 237 00:14:56,060 --> 00:14:58,958 I wrote N transpose N, all dagger. 238 00:14:58,958 --> 00:15:00,500 And that means, basically, you should 239 00:15:00,500 --> 00:15:02,167 think about the way this operator works, 240 00:15:02,167 --> 00:15:05,960 is it annihilates two nucleons in a particular spin wave, 241 00:15:05,960 --> 00:15:09,680 or a particular spin, isospin channel, 242 00:15:09,680 --> 00:15:13,180 and then creates them again. 243 00:15:13,180 --> 00:15:18,340 So annihilate, create. 244 00:15:18,340 --> 00:15:20,080 So I just put the two fields that 245 00:15:20,080 --> 00:15:21,587 are doing the annihilating together, 246 00:15:21,587 --> 00:15:23,920 and the two fields that are doing the creating together. 247 00:15:23,920 --> 00:15:26,560 And that's nice because you're annihilating them 248 00:15:26,560 --> 00:15:27,570 in a particular channel. 249 00:15:32,110 --> 00:15:36,190 So with those conventions, our Feynman Rules 250 00:15:36,190 --> 00:15:39,500 are particularly simple. 251 00:15:39,500 --> 00:15:43,600 If we just have a C0 in some channel, 252 00:15:43,600 --> 00:15:47,020 then the Feynman Rule is just minus IC0, 253 00:15:47,020 --> 00:15:52,870 and if we have one of these higher C2 operators 254 00:15:52,870 --> 00:15:56,320 in the center of mass frame, it's just minus IC2P squared. 255 00:16:02,172 --> 00:16:03,630 In the center of mass frame, that's 256 00:16:03,630 --> 00:16:04,620 the center of mass momentum. 257 00:16:04,620 --> 00:16:06,370 And remember, in the center of mass frame, 258 00:16:06,370 --> 00:16:08,670 P squared was equal to P prime squared. 259 00:16:08,670 --> 00:16:12,120 And so we can actually just write down the Feynman Rule 260 00:16:12,120 --> 00:16:13,890 for the complete set of operators 261 00:16:13,890 --> 00:16:16,129 there if we adopt this convention. 262 00:16:20,350 --> 00:16:22,693 So if you insert a guy with 2M derivatives-- 263 00:16:22,693 --> 00:16:24,360 derivatives always have to come in pairs 264 00:16:24,360 --> 00:16:27,140 because of angular momentum. 265 00:16:35,077 --> 00:16:36,660 You just have that Feynman Rule, sum's 266 00:16:36,660 --> 00:16:43,300 over the number of derivatives. 267 00:16:43,300 --> 00:16:45,080 So this is the complete, in this theory, 268 00:16:45,080 --> 00:16:53,682 this is the complete tree-level amplitude 269 00:16:53,682 --> 00:16:55,140 from those interactions over there. 270 00:16:57,970 --> 00:17:01,285 It's very nice theory. 271 00:17:01,285 --> 00:17:01,785 Simple. 272 00:17:04,660 --> 00:17:05,847 What about loops? 273 00:17:05,847 --> 00:17:07,180 We are going to need some loops. 274 00:17:10,950 --> 00:17:14,609 So let's look at the following loop, 275 00:17:14,609 --> 00:17:17,251 and I'll start by looking at just E equals 0. 276 00:17:22,069 --> 00:17:24,050 Let's just take it, take the nuclear arms 277 00:17:24,050 --> 00:17:26,599 and then scatter them again. 278 00:17:26,599 --> 00:17:28,460 So, in terms of the momentum flow 279 00:17:28,460 --> 00:17:31,070 I have some Q going this way, and then I 280 00:17:31,070 --> 00:17:36,800 have minus Q going that way, that's my loop momenta. 281 00:17:36,800 --> 00:17:39,453 So I get 2 coupling C0. 282 00:17:39,453 --> 00:17:40,800 Want these to be 0's. 283 00:17:43,680 --> 00:17:44,180 Ergo, DDQ. 284 00:17:48,061 --> 00:17:56,580 And if I just kept the HQET type term in my kinetic term, 285 00:17:56,580 --> 00:17:59,220 then it would look like that, OK? 286 00:17:59,220 --> 00:18:00,060 So this is just-- 287 00:18:03,090 --> 00:18:06,840 for the moment, if we just keep partial DT 288 00:18:06,840 --> 00:18:11,970 in the kinetic term, which is what we were doing in HQET, 289 00:18:11,970 --> 00:18:14,200 then we would get that. 290 00:18:14,200 --> 00:18:15,750 And that's a bad integral. 291 00:18:15,750 --> 00:18:17,912 It's got a pinch singularity. 292 00:18:17,912 --> 00:18:19,120 It's an ill-defined interval. 293 00:18:26,400 --> 00:18:28,620 So just from that little algebra, 294 00:18:28,620 --> 00:18:30,855 we see that actually keeping just the partial D 295 00:18:30,855 --> 00:18:32,940 by DT in the kinetic term is not the right thing 296 00:18:32,940 --> 00:18:34,920 to do for this theory. 297 00:18:34,920 --> 00:18:39,330 And that's because the kinetic energy is a relevant operator 298 00:18:39,330 --> 00:18:40,743 in quantum mechanics. 299 00:18:45,090 --> 00:18:47,160 Whenever you write down the Schrodinger equation, 300 00:18:47,160 --> 00:18:47,700 you kept it. 301 00:18:54,417 --> 00:18:56,000 So the right power counting and should 302 00:18:56,000 --> 00:19:00,100 have E, which is of order P squared over 2M. 303 00:19:00,100 --> 00:19:02,675 So the partial T term and the [? grad ?] scored over M term 304 00:19:02,675 --> 00:19:03,675 should be the same size. 305 00:19:13,580 --> 00:19:16,720 So this is generically true of two heavy particles. 306 00:19:25,030 --> 00:19:32,350 They have a different power counting for the kinetic term, 307 00:19:32,350 --> 00:19:33,120 than HQET. 308 00:19:39,452 --> 00:19:41,410 Any two heavy particles, whether it's too heavy 309 00:19:41,410 --> 00:19:44,350 quarks, two heavy nucleons, two heavy anything. 310 00:19:44,350 --> 00:19:46,580 All right, P should be [? order ?] P squared over 2M. 311 00:19:46,580 --> 00:19:49,157 AUDIENCE: E being the kinetic energy? 312 00:19:49,157 --> 00:19:50,740 PROFESSOR: So really, what I mean here 313 00:19:50,740 --> 00:19:54,400 is just the partial T term in the action 314 00:19:54,400 --> 00:19:56,990 over there should be the same size as the [? grad ?] squared 315 00:19:56,990 --> 00:19:57,490 over 2M. 316 00:20:04,140 --> 00:20:05,620 Yes, I have pulled out the mass. 317 00:20:05,620 --> 00:20:06,870 Yeah. 318 00:20:06,870 --> 00:20:10,260 So just like in HQET, to get this partial T 319 00:20:10,260 --> 00:20:12,750 we pulled out the mass, and we have 320 00:20:12,750 --> 00:20:15,192 a kind of non-relativistic type expansion. 321 00:20:15,192 --> 00:20:17,400 And the difference here is that we need the partial T 322 00:20:17,400 --> 00:20:18,775 to be of [? order grad squared ?] 323 00:20:18,775 --> 00:20:22,410 over 2M, and that leads to effectively counting velocities 324 00:20:22,410 --> 00:20:25,260 rather than-- 325 00:20:25,260 --> 00:20:29,190 because you have to count energies different than P's. 326 00:20:29,190 --> 00:20:31,410 We won't spend too long talking about 327 00:20:31,410 --> 00:20:34,080 that because we have other things to discuss here, 328 00:20:34,080 --> 00:20:36,870 but this is a whole interesting subject in itself. 329 00:20:36,870 --> 00:20:39,172 The power counting and what it means. 330 00:20:39,172 --> 00:20:41,130 One thing that's kind of interesting here which 331 00:20:41,130 --> 00:20:44,370 we won't cover, which I can't help but mention, 332 00:20:44,370 --> 00:20:47,050 is that say you did quarks, which was a gauge theory. 333 00:20:47,050 --> 00:20:48,720 This is not a gauge theory that we're talking about, 334 00:20:48,720 --> 00:20:51,303 but let's-- but you could do a gauge theory that has this type 335 00:20:51,303 --> 00:20:54,090 of power counting and it has exactly the same problem. 336 00:20:54,090 --> 00:20:58,140 Just replace these heavy nucleons by quarks, 337 00:20:58,140 --> 00:21:01,410 and replace this dot here by cooling potential. 338 00:21:01,410 --> 00:21:05,130 Exactly the same problem if you try to use HQET. 339 00:21:05,130 --> 00:21:07,170 And in that theory, too, you need 340 00:21:07,170 --> 00:21:08,670 E to be of order P squared over 2, 341 00:21:08,670 --> 00:21:11,120 and that's called non-relativistic QCD. 342 00:21:11,120 --> 00:21:12,870 Or you could do heavy electrons, where you 343 00:21:12,870 --> 00:21:15,990 have QED as the gauge theory. 344 00:21:15,990 --> 00:21:19,200 Non-relativistic QED, same issue. 345 00:21:19,200 --> 00:21:22,650 E has to be of order P squared over M. And in gauge theory, 346 00:21:22,650 --> 00:21:24,780 there's even a further complication, which 347 00:21:24,780 --> 00:21:27,330 is basically that there's gauge particles that 348 00:21:27,330 --> 00:21:29,940 want to talk to E, and there's gauge particles that 349 00:21:29,940 --> 00:21:33,010 want to talk to P, and those are different sizes. 350 00:21:33,010 --> 00:21:35,820 So you have something called ultra-soft photons 351 00:21:35,820 --> 00:21:38,700 and soft photons that are the gauge particles for E, 352 00:21:38,700 --> 00:21:42,660 and the gauge particles for P. Kind of an interesting theory. 353 00:21:42,660 --> 00:21:44,320 We want to have time to talk about it. 354 00:21:44,320 --> 00:21:47,240 Somebody wants to talk about it for their project, 355 00:21:47,240 --> 00:21:48,090 it's kind of fun. 356 00:21:52,980 --> 00:21:56,790 OK, so we have to keep P squared over 2M. 357 00:21:56,790 --> 00:21:59,610 At least knowing a little quantum mechanics, 358 00:21:59,610 --> 00:22:01,650 we know that that's true. 359 00:22:01,650 --> 00:22:03,450 Or running into this pinch singularity, 360 00:22:03,450 --> 00:22:08,810 we see that trying to do something different than that 361 00:22:08,810 --> 00:22:09,560 leads to problems. 362 00:22:21,080 --> 00:22:23,380 So let's redo our calculation here, but now keeping 363 00:22:23,380 --> 00:22:27,380 that term, and see what we get. 364 00:22:27,380 --> 00:22:29,740 Same bubble diagram. 365 00:22:29,740 --> 00:22:34,060 Let me send in on each of these legs E over 2, 366 00:22:34,060 --> 00:22:36,070 so E is the total energy that I'm sending in. 367 00:22:40,528 --> 00:22:42,228 Convenient normalization. 368 00:22:47,710 --> 00:22:51,400 Let's see how having the kinetic energy fixes this pinch pole. 369 00:23:16,390 --> 00:23:18,610 OK, so if you look at the poles in the complex plane 370 00:23:18,610 --> 00:23:22,010 here, what's happened is you've split them like this. 371 00:23:22,010 --> 00:23:25,160 So that you've moved them along the real axis. 372 00:23:25,160 --> 00:23:27,280 And so now you can just think of a contour, 373 00:23:27,280 --> 00:23:32,860 for example, if you want to think in the complex Q0 plane, 374 00:23:32,860 --> 00:23:37,200 you can think of doing a contour integral like that. 375 00:23:37,200 --> 00:23:40,640 Everything is well defined, convergence at infinity, 376 00:23:40,640 --> 00:23:42,370 everybody's happy. 377 00:23:42,370 --> 00:23:46,200 So we can close above, pick the polar. 378 00:23:46,200 --> 00:23:55,227 Q0 is E over 2, minus Q squared over 2M, plus I0. 379 00:23:55,227 --> 00:23:56,560 Plug it back into the other one. 380 00:24:03,760 --> 00:24:08,110 My notation is that N is going to be D minus 1. 381 00:24:08,110 --> 00:24:10,150 So when I do one of the integrals by contour, 382 00:24:10,150 --> 00:24:11,140 I go down a dimension. 383 00:24:11,140 --> 00:24:13,480 I'll call that N. So it's N here. 384 00:24:24,630 --> 00:24:25,740 This integral we can do. 385 00:24:33,905 --> 00:24:35,655 You have to be careful about the epsilons. 386 00:24:45,930 --> 00:24:48,240 Because they tell us tell us whether it's minus IP 387 00:24:48,240 --> 00:24:50,805 or plus IP. 388 00:24:50,805 --> 00:24:53,870 ME is set up in my convention. 389 00:24:53,870 --> 00:24:56,450 ME is P squared. 390 00:24:56,450 --> 00:24:59,948 So this is giving a P, but it's giving either a minus 391 00:24:59,948 --> 00:25:01,990 IP or a plus IP, depending on the sign of the I0, 392 00:25:01,990 --> 00:25:05,800 but I know it's a minus IP. 393 00:25:05,800 --> 00:25:07,240 So I used dimreg here. 394 00:25:11,878 --> 00:25:13,420 Because if you look at this integral, 395 00:25:13,420 --> 00:25:16,240 there's three pairs of Q upstairs, two downstairs. 396 00:25:16,240 --> 00:25:18,458 So it's power law divergent. 397 00:25:18,458 --> 00:25:20,500 But we don't see power law divergences in dimreg, 398 00:25:20,500 --> 00:25:23,155 we just get this finite answer. 399 00:25:29,120 --> 00:25:32,030 And actually, that finite answer is exactly the imaginary part 400 00:25:32,030 --> 00:25:34,670 that you need if you want to cut to graph, 401 00:25:34,670 --> 00:25:37,970 and say that the cut of the forward scattering 402 00:25:37,970 --> 00:25:42,360 is the same as this amplitude squared. 403 00:25:42,360 --> 00:25:44,390 So it's exactly, in essence, this 404 00:25:44,390 --> 00:25:47,090 is the piece that you need to be there by unitary. 405 00:25:50,210 --> 00:25:51,867 If you were trying to keep the E, 406 00:25:51,867 --> 00:25:53,450 you could think, well, maybe if I just 407 00:25:53,450 --> 00:25:54,950 kept the E in this calculation, it 408 00:25:54,950 --> 00:25:56,610 would solve this pinch problem. 409 00:25:56,610 --> 00:26:00,050 But it doesn't really do it because if you really 410 00:26:00,050 --> 00:26:03,290 stick with the partial D by DT as your leading order action, 411 00:26:03,290 --> 00:26:06,830 then the equations of motion are equal 0, so. 412 00:26:06,830 --> 00:26:10,145 You have to take equal 0 to go on [INAUDIBLE].. 413 00:26:10,145 --> 00:26:12,020 So you can't really avoid the pinch that way, 414 00:26:12,020 --> 00:26:15,980 you really have to take this kinetic term. 415 00:26:15,980 --> 00:26:18,470 you have to take the kinetic term to have both the partial 416 00:26:18,470 --> 00:26:21,440 DT and the [? grad ?] [? squared ?] over 2M. 417 00:26:21,440 --> 00:26:23,510 OK, any questions so far? 418 00:26:28,417 --> 00:26:30,000 All right, well there's something here 419 00:26:30,000 --> 00:26:31,680 that might bother you. 420 00:26:31,680 --> 00:26:34,350 We've got an M upstairs. 421 00:26:34,350 --> 00:26:35,730 M is big. 422 00:26:35,730 --> 00:26:39,970 M is the mass the nucleon, and it's appearing upstairs. 423 00:26:39,970 --> 00:26:41,250 That's always a bad sign. 424 00:26:43,920 --> 00:26:45,450 Usually a bad sign. 425 00:26:45,450 --> 00:26:48,720 Well, at least that's something we should worry about. 426 00:26:48,720 --> 00:26:50,490 So let's figure out where all the M's are. 427 00:26:53,910 --> 00:26:57,540 Let's count all the M's in the theory, 428 00:26:57,540 --> 00:26:59,025 holding the momentum fixed. 429 00:27:04,960 --> 00:27:07,410 So if we hold the momentum fixed, 430 00:27:07,410 --> 00:27:11,520 then [? grad ?] has no M's. 431 00:27:11,520 --> 00:27:17,340 But partial T scales like 1 over M. 432 00:27:17,340 --> 00:27:25,360 And correspondingly, T scales like M. OK. 433 00:27:25,360 --> 00:27:30,050 X and [? grad ?] don't scale but partial T and T do. 434 00:27:30,050 --> 00:27:32,950 And that's exactly what makes these two terms the same size. 435 00:27:38,300 --> 00:27:42,640 So we can ask, what is the scaling of our nucleon field? 436 00:27:42,640 --> 00:27:47,860 And if we want to do that, we go back to our action 437 00:27:47,860 --> 00:27:52,410 and we just say the action shouldn't have any scaling. 438 00:27:52,410 --> 00:27:55,900 T has a scaling, so there's an M in here. 439 00:27:55,900 --> 00:27:58,690 This guy here, the whole thing scales nicely, 1 over M. 440 00:27:58,690 --> 00:28:00,770 From what we just said over there. 441 00:28:00,770 --> 00:28:04,630 This M cancels that 1 over M, so this guy 442 00:28:04,630 --> 00:28:08,390 is therefore M to the 0. 443 00:28:08,390 --> 00:28:13,000 The nucleon field has no scaling with them. 444 00:28:13,000 --> 00:28:15,370 So then, once you've done the kinetic term to figure out 445 00:28:15,370 --> 00:28:17,260 the scaling of the nucleon field, you can do, 446 00:28:17,260 --> 00:28:20,875 then, any other interaction. 447 00:28:20,875 --> 00:28:22,750 So that the power counting, the kinetic term, 448 00:28:22,750 --> 00:28:26,080 is always which determines the power counting of the field. 449 00:28:26,080 --> 00:28:28,443 That's true in any theory, any effective theory, 450 00:28:28,443 --> 00:28:29,110 because you're-- 451 00:28:32,080 --> 00:28:34,180 that's the basis of the fluctuations you're 452 00:28:34,180 --> 00:28:37,578 describing by that field. 453 00:28:37,578 --> 00:28:39,370 And once you've fixed that counting, you've 454 00:28:39,370 --> 00:28:41,120 got all the counting you need, and now you 455 00:28:41,120 --> 00:28:44,590 can go count other operators like the one with 2M 456 00:28:44,590 --> 00:28:48,100 derivatives. 457 00:28:48,100 --> 00:28:50,290 And this guy here just has-- 458 00:28:50,290 --> 00:28:53,410 you can have just vector derivatives, no time 459 00:28:53,410 --> 00:28:55,840 derivatives, just vector derivatives, 2M of them, 460 00:28:55,840 --> 00:28:56,830 and nucleon fields. 461 00:28:56,830 --> 00:29:00,640 So there's no M's in there, that's M to the 0. 462 00:29:00,640 --> 00:29:05,260 And this guy here is an M. So this guy here 463 00:29:05,260 --> 00:29:17,380 is C2M, must be a 1 over M. So therefore, any C2M scales like 464 00:29:17,380 --> 00:29:24,490 1 over M. And now you see it's not 465 00:29:24,490 --> 00:29:26,680 such a problem, because here I have 2 C0's, 466 00:29:26,680 --> 00:29:28,270 each one of them scales like 1 over M, 467 00:29:28,270 --> 00:29:30,730 I pick up one more M in the numerator, 468 00:29:30,730 --> 00:29:33,790 and that just makes the whole thing feel like one over M. 469 00:29:33,790 --> 00:29:36,220 Which is the same order that the tree level was scaling, 470 00:29:36,220 --> 00:29:39,550 the tree level with scaling like 1 over M. 471 00:29:39,550 --> 00:29:44,230 So loop diagrams and the tree level both scale like 1 over M, 472 00:29:44,230 --> 00:29:46,030 and that's actually generically true 473 00:29:46,030 --> 00:29:48,560 because every time I add a loop I add a coupling, 474 00:29:48,560 --> 00:29:51,350 and so those two M's can compensate each other. 475 00:29:51,350 --> 00:29:52,690 So there's no issue with M's. 476 00:30:05,090 --> 00:30:10,700 So the one that graph is the same size as tree level, 477 00:30:10,700 --> 00:30:16,190 and they both go like 1 over M. If you now 478 00:30:16,190 --> 00:30:22,730 do dimension counting, you can say given that we've identified 479 00:30:22,730 --> 00:30:26,990 that the C has an M in it, if you 480 00:30:26,990 --> 00:30:31,880 ask about the dimensions of C, -2 minus 2M. 481 00:30:31,880 --> 00:30:36,390 Because the dimensions of the nucleon field are 3 halves. 482 00:30:36,390 --> 00:30:39,080 And we're doing some expansion in P much less than lambda, 483 00:30:39,080 --> 00:30:40,850 so just by dimensional power counting 484 00:30:40,850 --> 00:30:47,710 we expect that the coefficients would be of the following size. 485 00:30:47,710 --> 00:30:51,200 We know that there's an M and that it's the only M, 486 00:30:51,200 --> 00:30:52,910 and all the rest of the dimensions 487 00:30:52,910 --> 00:30:54,980 you should think of as being made up 488 00:30:54,980 --> 00:30:56,910 by the stuff you integrated it out. 489 00:30:56,910 --> 00:30:58,700 So it could be the pion, for example, 490 00:30:58,700 --> 00:31:00,540 setting the scale lambda. 491 00:31:00,540 --> 00:31:02,300 So what you'd expect for the C2M's is 492 00:31:02,300 --> 00:31:05,060 that this is how big they are, and that the derivatives are 493 00:31:05,060 --> 00:31:07,998 then being suppressed because of these lambdas 494 00:31:07,998 --> 00:31:10,040 and you're expanding for P much less than lambda. 495 00:31:13,540 --> 00:31:14,590 OK. 496 00:31:14,590 --> 00:31:15,690 Are we happy so far? 497 00:31:20,470 --> 00:31:22,120 All right, and we'll see when you 498 00:31:22,120 --> 00:31:24,740 do matching calculations, this M [? law's ?] always there. 499 00:31:24,740 --> 00:31:29,770 And this is just a nice, elegant way of figuring that out. 500 00:31:29,770 --> 00:31:34,445 Because this is a very simple argument. 501 00:31:34,445 --> 00:31:34,945 All right. 502 00:31:34,945 --> 00:31:36,903 AUDIENCE: I don't know if I totally understand. 503 00:31:36,903 --> 00:31:39,820 How do you know that C2M-- 504 00:31:39,820 --> 00:31:41,380 PROFESSOR: Goes like 1 over M? 505 00:31:41,380 --> 00:31:42,160 AUDIENCE: Yeah-- 506 00:31:42,160 --> 00:31:43,860 PROFESSOR: So the kinetic term-- 507 00:31:43,860 --> 00:31:47,350 so at first you know this, right? 508 00:31:47,350 --> 00:31:48,550 AUDIENCE: Yeah. 509 00:31:48,550 --> 00:31:49,690 PROFESSOR: Yeah, so I see. 510 00:31:49,690 --> 00:31:50,860 You're worried about whether there could be 511 00:31:50,860 --> 00:31:52,080 1 over M squared corrections? 512 00:31:52,080 --> 00:31:52,705 AUDIENCE: Yeah. 513 00:31:52,705 --> 00:31:53,020 PROFESSOR: Yeah. 514 00:31:53,020 --> 00:31:55,240 In principle, there could be 1 over M squared corrections 515 00:31:55,240 --> 00:31:57,282 from relativistic corrections, so that this would 516 00:31:57,282 --> 00:31:58,810 be the leading order term. 517 00:31:58,810 --> 00:31:59,500 Yeah. 518 00:31:59,500 --> 00:32:01,750 Not worse than that. 519 00:32:01,750 --> 00:32:05,560 If you looked at the P cubed, P to the fourth over 8M 520 00:32:05,560 --> 00:32:08,133 cubed term, that term would have a higher power of M, 521 00:32:08,133 --> 00:32:09,550 and you could imagine that there's 522 00:32:09,550 --> 00:32:12,840 some relativistic corrections in the four nucleons as well. 523 00:32:12,840 --> 00:32:14,590 Yeah. 524 00:32:14,590 --> 00:32:16,803 So we'll work basically here, at lowest order 525 00:32:16,803 --> 00:32:18,220 in the relativistic corrections so 526 00:32:18,220 --> 00:32:20,470 that you could put in the relativistic corrections, 527 00:32:20,470 --> 00:32:20,970 as well. 528 00:32:24,060 --> 00:32:25,800 All right. 529 00:32:25,800 --> 00:32:27,990 So we're basically stopping at that order, which 530 00:32:27,990 --> 00:32:29,448 is equivalent to quantum mechanics, 531 00:32:29,448 --> 00:32:31,320 but we could go further if we wanted to 532 00:32:31,320 --> 00:32:33,330 in this effective theory. 533 00:32:33,330 --> 00:32:37,980 So now, let's think about other loop diagrams in the theory, 534 00:32:37,980 --> 00:32:39,480 without relativistic corrections, 535 00:32:39,480 --> 00:32:40,800 just with these interactions. 536 00:32:40,800 --> 00:32:43,890 And let's insert the 2N and the 2M derivative operator 537 00:32:43,890 --> 00:32:45,510 on these vertices. 538 00:32:45,510 --> 00:32:48,720 You see all the loops look like this, they're all bubbles. 539 00:32:48,720 --> 00:32:50,820 And that's the same reason in HQET 540 00:32:50,820 --> 00:32:53,070 that you don't have any diagrams that kind of-- you 541 00:32:53,070 --> 00:32:55,770 don't have diagrams that look like this, because these types 542 00:32:55,770 --> 00:32:58,365 of diagrams involve antiparticles 543 00:32:58,365 --> 00:32:59,490 and we only have particles. 544 00:33:03,717 --> 00:33:05,800 So that's the beauty of a non-relativistic theory, 545 00:33:05,800 --> 00:33:07,650 you don't have any diagrams like that. 546 00:33:07,650 --> 00:33:09,830 You just have diagrams like this. 547 00:33:09,830 --> 00:33:12,610 And so basically, the whole theory is bubbles. 548 00:33:12,610 --> 00:33:13,590 The theory of bubbles. 549 00:33:16,290 --> 00:33:18,180 If you go through this loop diagram, 550 00:33:18,180 --> 00:33:20,190 you do the pole the same way. 551 00:33:20,190 --> 00:33:24,570 It's exactly the same two propagators. 552 00:33:24,570 --> 00:33:33,443 Then you get the same Q squared minus ME, minus I0. 553 00:33:33,443 --> 00:33:34,860 And the only difference is you get 554 00:33:34,860 --> 00:33:39,435 powers of Q in the numerator. 555 00:33:39,435 --> 00:33:41,310 And this, so this integral is one of the ones 556 00:33:41,310 --> 00:33:42,840 that shows up in here. 557 00:33:42,840 --> 00:33:46,230 And this integral, you can do the same kind of trick 558 00:33:46,230 --> 00:33:49,290 that you used when we were discussing field definitions, 559 00:33:49,290 --> 00:33:52,020 where you basically take the top and organize it 560 00:33:52,020 --> 00:33:55,150 by adding and subtracting. 561 00:33:55,150 --> 00:34:09,530 So add and subtract any like that. 562 00:34:09,530 --> 00:34:11,600 And now, you can think, N and M are integers, 563 00:34:11,600 --> 00:34:13,969 just expand this thing out. 564 00:34:13,969 --> 00:34:16,940 Some number of these factors, some number of these factors. 565 00:34:16,940 --> 00:34:19,310 But any time you get one or more of these factors, 566 00:34:19,310 --> 00:34:20,780 it cancels the denominator and then 567 00:34:20,780 --> 00:34:23,058 you end up with scaleless integrals. 568 00:34:23,058 --> 00:34:25,350 So basically, that means you can throw this piece away. 569 00:34:41,150 --> 00:34:43,580 So higher order derivatives actually just 570 00:34:43,580 --> 00:34:47,330 lead to ME's, whether they act inside or outside 571 00:34:47,330 --> 00:34:49,850 on the nucleon fields, they lead to P squared. 572 00:34:53,449 --> 00:34:54,415 Yeah, sorry. 573 00:34:54,415 --> 00:34:55,790 This M is a little M, and this is 574 00:34:55,790 --> 00:35:00,040 supposed to be a big M. Oh, N. Oh yeah, 575 00:35:00,040 --> 00:35:02,750 I am also using that too. 576 00:35:02,750 --> 00:35:03,800 Sorry. 577 00:35:03,800 --> 00:35:04,770 Yes. 578 00:35:04,770 --> 00:35:05,810 This is an integer. 579 00:35:05,810 --> 00:35:08,450 I should call it J or something. 580 00:35:08,450 --> 00:35:13,295 And the other M is 3 minus 2 epsilon. 581 00:35:28,100 --> 00:35:30,570 Yeah, it's dangerous. 582 00:35:30,570 --> 00:35:31,610 All right. 583 00:35:31,610 --> 00:35:34,220 But this means that basically, the graphs in this theory 584 00:35:34,220 --> 00:35:35,010 are very simple. 585 00:35:35,010 --> 00:35:38,390 So if we actually now consider the complete set of diagrams, 586 00:35:38,390 --> 00:35:39,365 or we impose-- 587 00:35:42,530 --> 00:35:46,640 we put the full amplitude in there 588 00:35:46,640 --> 00:35:47,990 and we consider these bubbles. 589 00:35:53,060 --> 00:35:54,620 Because of this fact that I just told 590 00:35:54,620 --> 00:35:57,770 you, which doesn't change when you insert more bubbles, 591 00:35:57,770 --> 00:36:01,640 the bubbles all decouple from each other. 592 00:36:01,640 --> 00:36:03,470 The contact interaction is decoupling them 593 00:36:03,470 --> 00:36:05,730 from each other. 594 00:36:05,730 --> 00:36:14,180 So here's a complete amplitude with K insertions 595 00:36:14,180 --> 00:36:16,430 of the coupling, and K minus 1 loops. 596 00:36:24,990 --> 00:36:28,245 OK, so we just completed all the loop diagrams the theory, 597 00:36:28,245 --> 00:36:30,120 and it's giving us something that we can sum, 598 00:36:30,120 --> 00:36:31,320 its a geometric series. 599 00:36:35,740 --> 00:36:40,540 So this is for K minus 1 loops, we can sum them up. 600 00:36:40,540 --> 00:36:46,150 So if you like I should have called this the K-th term. 601 00:36:46,150 --> 00:37:07,830 And if I sum up, cancel the I on each side, you get that. 602 00:37:16,270 --> 00:37:17,410 Let me round it again. 603 00:37:24,890 --> 00:37:25,610 Like this. 604 00:37:30,235 --> 00:37:34,390 So dividing out the numerator, rearranging 605 00:37:34,390 --> 00:37:38,150 the four pi over M's little bit, I can write it like that. 606 00:37:38,150 --> 00:37:40,030 And then we can identify this thing here 607 00:37:40,030 --> 00:37:41,320 as P cotangent delta. 608 00:37:45,950 --> 00:37:51,020 OK, so P cotangent delta, from our effective theory, 609 00:37:51,020 --> 00:37:55,925 we calculate it to be the sum over this, 610 00:37:55,925 --> 00:37:59,870 P2M, where now I've conveniently defined it, 611 00:37:59,870 --> 00:38:04,160 hatted coefficients, which are the following things. 612 00:38:07,540 --> 00:38:09,040 And that just cancels the M that's 613 00:38:09,040 --> 00:38:13,990 in the C2M's, so C2 hat scales like M to the 0. 614 00:38:16,960 --> 00:38:18,040 So that's just a-- 615 00:38:18,040 --> 00:38:22,060 is an obvious way of reorganizing things given 616 00:38:22,060 --> 00:38:24,610 the factors of 4 pi over M that we knew 617 00:38:24,610 --> 00:38:27,310 had to sit out front of this S matrix 618 00:38:27,310 --> 00:38:30,050 calculation of the amplitude. 619 00:38:30,050 --> 00:38:33,190 So we can identify P cotangent delta as something 620 00:38:33,190 --> 00:38:35,500 that doesn't involve any M's, and that actually is also 621 00:38:35,500 --> 00:38:39,440 what we would expect for non-relativistic scattering. 622 00:38:39,440 --> 00:38:41,200 So then you can look at different waves, 623 00:38:41,200 --> 00:38:45,725 and you can look at this formula, and you can-- 624 00:38:45,725 --> 00:38:47,600 doing two of them will show you how it works. 625 00:38:47,600 --> 00:38:50,260 So S wave is L equals 0. 626 00:39:14,030 --> 00:39:18,140 So that's something we can do a Taylor expansion in P in. 627 00:39:18,140 --> 00:39:20,460 And this is what the Taylor series looks like. 628 00:39:20,460 --> 00:39:22,530 And this has exactly the form that we wanted. 629 00:39:22,530 --> 00:39:24,800 This is 1-- some constant 1 over A, 630 00:39:24,800 --> 00:39:27,290 which is the scattering length. 631 00:39:27,290 --> 00:39:29,090 Some constant times P squared, which 632 00:39:29,090 --> 00:39:31,580 is called the effective range. 633 00:39:31,580 --> 00:39:33,535 And we see that expansion coming out, 634 00:39:33,535 --> 00:39:35,660 and we never had to specify what the potential was, 635 00:39:35,660 --> 00:39:38,270 because the effective theory was agnostic to what 636 00:39:38,270 --> 00:39:39,050 the potential was. 637 00:39:39,050 --> 00:39:40,490 That's the whole power of effective theory, 638 00:39:40,490 --> 00:39:42,890 that you don't need to know what the particles were that you 639 00:39:42,890 --> 00:39:45,057 were integrating out, they just give you some values 640 00:39:45,057 --> 00:39:46,490 for these coefficients. 641 00:39:46,490 --> 00:39:49,598 And those values exactly become the effective range 642 00:39:49,598 --> 00:39:51,140 and scattering length in this theory. 643 00:39:53,930 --> 00:39:56,620 AUDIENCE: So this seems very [INAUDIBLE] dependent. 644 00:39:56,620 --> 00:40:00,130 PROFESSOR: Yeah, we're going to talk about that. 645 00:40:00,130 --> 00:40:02,110 Yes. 646 00:40:02,110 --> 00:40:06,220 Yeah, that will encompass the second half of lecture. 647 00:40:18,230 --> 00:40:20,185 So we'll come to that momentarily. 648 00:40:23,510 --> 00:40:27,065 Let me just do one more way, just so you see. 649 00:40:27,065 --> 00:40:29,220 L equals 1. 650 00:40:29,220 --> 00:40:31,820 in the L equals 1 case, there's no C0 hat, 651 00:40:31,820 --> 00:40:35,970 you need the derivatives to correspond to the wave. 652 00:40:35,970 --> 00:40:43,220 And so in this case, you look at PQ cotangent delta, 653 00:40:43,220 --> 00:40:44,750 the denominator starts at C2. 654 00:40:47,390 --> 00:40:49,760 And the reason why there is this P to the 2L plus 1 655 00:40:49,760 --> 00:40:52,380 is just so that you get the extra P here, 656 00:40:52,380 --> 00:40:54,217 which compensates the P's downstairs, 657 00:40:54,217 --> 00:40:55,925 and again this thing has a Taylor series. 658 00:41:04,900 --> 00:41:07,990 Same story as over there, we can identify the scattering link 659 00:41:07,990 --> 00:41:10,750 for the P wave as the 1 over C2 hat. 660 00:41:13,977 --> 00:41:17,680 So all the higher partial ways work the same way. 661 00:41:17,680 --> 00:41:20,050 And you just have P to the 2 L plus 1 in front 662 00:41:20,050 --> 00:41:22,450 of your cotangent delta. 663 00:41:22,450 --> 00:41:22,950 OK. 664 00:41:27,080 --> 00:41:30,332 Here you can see how this generalizes [INAUDIBLE].. 665 00:41:35,160 --> 00:41:37,020 So that proves this non-relativistic quantum 666 00:41:37,020 --> 00:41:38,990 mechanics theorem. 667 00:41:52,160 --> 00:41:55,010 And we did it without having to specify what the potential was, 668 00:41:55,010 --> 00:41:58,010 because the potential is encoded in BC's and effectively 669 00:41:58,010 --> 00:42:00,647 that's like a basis expansion of the potential. 670 00:42:00,647 --> 00:42:02,730 But it's a fun one because it's in delta functions 671 00:42:02,730 --> 00:42:04,740 and derivatives of delta functions. 672 00:42:04,740 --> 00:42:07,010 So we're doing a local effective field 673 00:42:07,010 --> 00:42:09,200 theory, which is not something you'd think 674 00:42:09,200 --> 00:42:11,230 of ever using for a basis-- 675 00:42:11,230 --> 00:42:13,310 well, maybe you would, but most people 676 00:42:13,310 --> 00:42:16,190 wouldn't think of using basis of derivatives of delta functions 677 00:42:16,190 --> 00:42:18,120 for quantum mechanics. 678 00:42:18,120 --> 00:42:18,620 OK. 679 00:42:21,662 --> 00:42:23,870 Well, you'd hope we can do a little more than quantum 680 00:42:23,870 --> 00:42:24,578 mechanics, right? 681 00:42:29,000 --> 00:42:31,607 So you can think, if you like, in terms 682 00:42:31,607 --> 00:42:33,690 of determining the values of the C's, you can say, 683 00:42:33,690 --> 00:42:36,232 well, experiment tells me the values of the scattering length 684 00:42:36,232 --> 00:42:38,510 are 0, and that's indeed true. 685 00:42:42,300 --> 00:42:45,268 So this equation that C0 hat is equal to A, 686 00:42:45,268 --> 00:42:47,060 you could view this as a matching equation. 687 00:42:49,980 --> 00:42:54,770 Putting back the M, you have that. 688 00:42:54,770 --> 00:42:59,030 And then for C2 hat, we have r0 over 2, a squared. 689 00:43:07,980 --> 00:43:10,370 So pretend that experiment gives r0. 690 00:43:15,660 --> 00:43:17,913 And a, which they do measure, and then 691 00:43:17,913 --> 00:43:19,830 you know the value of your Wilson coefficients 692 00:43:19,830 --> 00:43:22,230 and you can start using this theory. 693 00:43:22,230 --> 00:43:27,750 Now, if you think about the power counting, if a 694 00:43:27,750 --> 00:43:30,570 and r0 are order 1 over lambda, that's 695 00:43:30,570 --> 00:43:32,408 the natural thing you'd expect. 696 00:43:32,408 --> 00:43:33,825 Then you reproduce what we expect. 697 00:43:43,230 --> 00:43:45,270 Whatever it was a minute ago. 698 00:43:50,340 --> 00:43:51,440 2M plus one. 699 00:43:56,810 --> 00:44:00,410 So if all the constants are scaling like whatever 700 00:44:00,410 --> 00:44:01,980 their dimensions are, in this case 701 00:44:01,980 --> 00:44:05,235 they're both dimension minus 1 in momentum units, 702 00:44:05,235 --> 00:44:07,610 then you would reproduce exactly with this power counting 703 00:44:07,610 --> 00:44:09,340 that we said. 704 00:44:09,340 --> 00:44:11,330 OK, so everything would be nice. 705 00:44:11,330 --> 00:44:13,050 But nature is not so nice. 706 00:44:13,050 --> 00:44:16,460 Or nature threw us in a different direction. 707 00:44:16,460 --> 00:44:19,293 When we actually look at the value of these constants, 708 00:44:19,293 --> 00:44:20,585 the scattering length is large. 709 00:44:27,970 --> 00:44:34,100 So C0 is large, from this point of view. 710 00:44:37,303 --> 00:44:38,720 Let me quote some numbers for you. 711 00:44:58,820 --> 00:45:03,320 So in the 1S0 channel, which is the larger one, 712 00:45:03,320 --> 00:45:04,700 this guy is 23 fermis. 713 00:45:11,990 --> 00:45:17,288 And in the 3S1 channel, it's 5 fermis, and both of these 714 00:45:17,288 --> 00:45:18,080 are actually large. 715 00:45:23,260 --> 00:45:25,010 We know they're large, look at the errors. 716 00:45:27,650 --> 00:45:29,570 If you want to think about momentum units, 717 00:45:29,570 --> 00:45:32,480 you could take 1 over a. 718 00:45:32,480 --> 00:45:38,940 And for this guy here, taking 1 over a is giving him, like, 719 00:45:38,940 --> 00:45:43,910 if I did the calculation right, minus 8.30 MeV. 720 00:45:43,910 --> 00:45:45,170 Oh, sorry, no. 721 00:45:45,170 --> 00:45:45,920 That's this guy. 722 00:45:48,750 --> 00:45:55,740 And this guy here is giving 1 over a, is 36 MeV. 723 00:45:55,740 --> 00:45:57,950 So if you thought the natural size was the pion, 724 00:45:57,950 --> 00:46:00,860 then you'd say, well, these constants should be 1 over pi-- 725 00:46:00,860 --> 00:46:04,130 that these numbers that are in MeV should be in pi. 726 00:46:04,130 --> 00:46:07,970 And 8 MeV is a much smaller number than 138 MeV. 727 00:46:07,970 --> 00:46:10,830 36 is also a smaller number than 130 MeV. 728 00:46:10,830 --> 00:46:17,600 So both of these guys are not natural. 729 00:46:17,600 --> 00:46:20,935 In particular, this one you see it's very not natural. 730 00:46:24,080 --> 00:46:26,620 So there's a fine tuning going on. 731 00:46:26,620 --> 00:46:29,082 Some kind of fine tuning from the perspective-- 732 00:46:37,460 --> 00:46:40,080 from our dimensional counting EFT point of view. 733 00:46:45,120 --> 00:46:49,870 There's a fine tuning that's making the a big. 734 00:46:49,870 --> 00:46:53,460 And if you look at the other guys, the r0 735 00:46:53,460 --> 00:46:55,770 and the other guys, they are exactly 736 00:46:55,770 --> 00:46:56,890 of the size you'd expect. 737 00:46:56,890 --> 00:46:58,223 So there's no fine tuning there. 738 00:47:01,440 --> 00:47:04,560 The only fine tuning is in a. 739 00:47:04,560 --> 00:47:06,660 And not the other ones. 740 00:47:06,660 --> 00:47:08,880 Just by comparing two data. 741 00:47:08,880 --> 00:47:10,300 OK, so how do we deal with that? 742 00:47:10,300 --> 00:47:12,092 It looks like we set up a defective theory. 743 00:47:12,092 --> 00:47:13,660 It has seems like a beautiful theory, 744 00:47:13,660 --> 00:47:15,910 it could describe some things about quantum mechanics, 745 00:47:15,910 --> 00:47:20,050 but then we learned that our power counting sucks. 746 00:47:20,050 --> 00:47:22,790 So what we want is actually a power counting 747 00:47:22,790 --> 00:47:24,188 it's a little different. 748 00:47:32,010 --> 00:47:38,190 Where AP is of order 1, or even AP much greater than 1, 749 00:47:38,190 --> 00:47:40,200 we'd like that to be allowed. 750 00:47:40,200 --> 00:47:44,520 Where our 0P is much less than 1. 751 00:47:44,520 --> 00:47:47,160 We'd like to be able to take a scattering length effectively 752 00:47:47,160 --> 00:47:50,310 into account to all orders, and that means basically 753 00:47:50,310 --> 00:47:55,170 that we want to treat C0 as relevant. 754 00:47:55,170 --> 00:47:58,537 We don't want 8MeV to be the limit of the lowered-- 755 00:47:58,537 --> 00:48:00,870 the thing that goes downstairs when we're making a power 756 00:48:00,870 --> 00:48:02,340 expansion, right? 757 00:48:02,340 --> 00:48:05,610 We'd like something like M pi to be downstairs. 758 00:48:05,610 --> 00:48:06,990 If we want M pi to be downstairs, 759 00:48:06,990 --> 00:48:09,180 you got to treat AP to all orders. 760 00:48:09,180 --> 00:48:12,570 And then you're just limited by M pi which is the r0 term. 761 00:48:12,570 --> 00:48:14,340 And that means you've got to promote C0 762 00:48:14,340 --> 00:48:18,060 from being irrelevant, scaling like 1 over M lambda, 763 00:48:18,060 --> 00:48:21,580 to something that's relevant. 764 00:48:21,580 --> 00:48:28,080 So this is actually a problem that 765 00:48:28,080 --> 00:48:30,630 occurred because we just proceeded and started 766 00:48:30,630 --> 00:48:34,820 calculating, and we used effectively the MS bar scheme 767 00:48:34,820 --> 00:48:36,780 in dimensional regularization. 768 00:48:36,780 --> 00:48:39,720 We saw powers and fractions, we did nothing. 769 00:48:39,720 --> 00:48:41,970 So let's try another scheme. 770 00:48:48,440 --> 00:48:51,260 It's a little more physical. 771 00:48:51,260 --> 00:48:54,800 Called offshell momentum subtraction. 772 00:48:54,800 --> 00:48:57,530 So what is offshell momentum subtraction? 773 00:48:57,530 --> 00:49:01,220 It says take the amplitude, and in this non-relativistic 774 00:49:01,220 --> 00:49:05,990 theory, you take P to be at some imaginary point 775 00:49:05,990 --> 00:49:10,302 so that you avoid any cuts. 776 00:49:10,302 --> 00:49:11,135 And you can define-- 777 00:49:15,160 --> 00:49:21,520 and whatever channel we're in, in this new r scheme, 778 00:49:21,520 --> 00:49:24,190 you can define the amplitude of that particular point 779 00:49:24,190 --> 00:49:26,730 to be the tree level result. 780 00:49:26,730 --> 00:49:29,540 OK, so any loops, if you take them at this point, 781 00:49:29,540 --> 00:49:35,830 you should get back that result. So this 782 00:49:35,830 --> 00:49:38,080 is the analog of what you do in a relativistic theory 783 00:49:38,080 --> 00:49:41,050 where you would take P squared to be minus mu squared. 784 00:49:41,050 --> 00:49:42,550 In the non-relativistic theory, it's 785 00:49:42,550 --> 00:49:46,000 always P that's showing up, which is the three vector P, 786 00:49:46,000 --> 00:49:49,200 magnitude of the three vector. 787 00:49:49,200 --> 00:49:50,910 So we should assign some rule for that. 788 00:49:50,910 --> 00:49:53,990 And it's just, we take it to be I times mu. 789 00:50:00,620 --> 00:50:03,450 So how does this change our calculation? 790 00:50:03,450 --> 00:50:06,750 So let's go back to this one calculation. 791 00:50:06,750 --> 00:50:11,210 So now there's going to be some counterterm needed. 792 00:50:11,210 --> 00:50:14,630 It's going to be a finite counterterm. 793 00:50:14,630 --> 00:50:15,620 Let's see what it does. 794 00:50:21,990 --> 00:50:27,830 So the loop graph gave this IP, that's what this graph gave. 795 00:50:27,830 --> 00:50:32,000 And this has to be just such that if I set P equal to I mu, 796 00:50:32,000 --> 00:50:32,960 that I get 0. 797 00:50:32,960 --> 00:50:35,370 So this has two plus mu. 798 00:50:35,370 --> 00:50:39,050 So the counterterm is giving mu r, C0 squared. 799 00:50:39,050 --> 00:50:42,455 And that exactly makes this amplitude vanish at that point, 800 00:50:42,455 --> 00:50:45,080 and that's what you want because the tree level graph of the C0 801 00:50:45,080 --> 00:50:47,690 is already giving the right condition. 802 00:50:47,690 --> 00:50:48,470 OK? 803 00:50:48,470 --> 00:50:50,150 Is that clear to everybody? 804 00:50:50,150 --> 00:50:52,880 This is the correction to that, to this. 805 00:50:52,880 --> 00:50:56,210 The tree level graph in the amplitude here gave minus IC0, 806 00:50:56,210 --> 00:50:57,710 so we already got what we want. 807 00:50:57,710 --> 00:50:59,210 When we go to our loop, we just want 808 00:50:59,210 --> 00:51:01,190 to make sure it doesn't contribute, 809 00:51:01,190 --> 00:51:05,190 and that forces us to put the plus mu there. 810 00:51:05,190 --> 00:51:10,940 And so what this mu is doing is tracking the power divergence 811 00:51:10,940 --> 00:51:14,630 that dimreg in MS bar did not see. 812 00:51:17,810 --> 00:51:20,450 There was an integral is power law divergence and our cut 813 00:51:20,450 --> 00:51:22,910 off, mu r, has appeared in the numerator 814 00:51:22,910 --> 00:51:24,981 and has tracked that divergence. 815 00:51:28,140 --> 00:51:29,970 So we're doing a standard thing here where 816 00:51:29,970 --> 00:51:36,240 we split the coefficient into the bare coefficient 817 00:51:36,240 --> 00:51:38,910 and to our more renormalized and countertrend piece. 818 00:51:38,910 --> 00:51:41,310 And unlike MS bar in this particular normalization 819 00:51:41,310 --> 00:51:44,085 scheme, there's a finite correction there. 820 00:51:44,085 --> 00:51:45,960 And we have a renormalization group equation. 821 00:52:00,540 --> 00:52:08,290 And if you work out what that is from the counterterm, 822 00:52:08,290 --> 00:52:12,480 it turns out that only this one loop, it's one loop exact. 823 00:52:12,480 --> 00:52:15,150 Showing that is a little more work than I've done, 824 00:52:15,150 --> 00:52:18,690 but it turns out that the beta function is one loop exact, 825 00:52:18,690 --> 00:52:20,520 higher bubbles don't give any contribution 826 00:52:20,520 --> 00:52:24,540 to this beta function, and this is the anomalous dimension. 827 00:52:24,540 --> 00:52:26,820 So we could just calculate all those bubbles, 828 00:52:26,820 --> 00:52:28,840 and pose the same type of thing. 829 00:52:28,840 --> 00:52:31,898 And I've given you the reference that does that. 830 00:52:31,898 --> 00:52:33,690 So there's a renormalization group equation 831 00:52:33,690 --> 00:52:36,060 in this scheme which we didn't have in MS bar. 832 00:52:36,060 --> 00:52:38,820 In MS bar it was scale independent. 833 00:52:38,820 --> 00:52:41,250 We also have a connection to the MS bar scheme 834 00:52:41,250 --> 00:52:44,970 because if mu r was 0, that corresponds to what 835 00:52:44,970 --> 00:52:48,090 the MS bar result, right? 836 00:52:48,090 --> 00:52:53,010 So C0 of 0 is where you can think of putting your boundary 837 00:52:53,010 --> 00:52:57,900 condition, which is matching to the experiment, which is the a. 838 00:52:57,900 --> 00:53:00,210 And the advantage of this offshell subtraction 839 00:53:00,210 --> 00:53:03,190 is that we have a mu, and we can go somewhere else. 840 00:53:03,190 --> 00:53:07,740 And if you look at the solution, of the RG 841 00:53:07,740 --> 00:53:11,557 with that boundary condition, you see something interesting. 842 00:53:20,330 --> 00:53:22,840 So this is the result for the coefficient. 843 00:53:22,840 --> 00:53:24,570 One over mu r, minus 1 over a. 844 00:53:28,240 --> 00:53:31,760 So if mu is of order P, which is much greater than 1 845 00:53:31,760 --> 00:53:36,460 over a, like we want, then the right counting for the C, 846 00:53:36,460 --> 00:53:40,210 which is a function of mu, is 1 over M mu. 847 00:53:43,700 --> 00:53:45,950 OK? 848 00:53:45,950 --> 00:53:48,140 And so what we've done here is we've 849 00:53:48,140 --> 00:53:54,770 swapped 1 over the physics scale that we're integrating out 850 00:53:54,770 --> 00:53:56,960 for 1 over the scale mu, which we're 851 00:53:56,960 --> 00:54:00,100 taking to be of order the physics we're keeping. 852 00:54:00,100 --> 00:54:02,120 OK? 853 00:54:02,120 --> 00:54:03,440 And this is relevant now. 854 00:54:03,440 --> 00:54:05,680 This is a relevant coupling with that change. 855 00:54:09,730 --> 00:54:11,620 We've made it much bigger by just 856 00:54:11,620 --> 00:54:13,970 switching to the physical renormalization scheme. 857 00:54:13,970 --> 00:54:16,210 And if we take mu of order p in that scheme, 858 00:54:16,210 --> 00:54:19,030 this all of a sudden becomes an order 1 effect 859 00:54:19,030 --> 00:54:20,740 with leading order in the Lagrangian, 860 00:54:20,740 --> 00:54:25,710 because we have effectively P's downstairs in the coupling. 861 00:54:25,710 --> 00:54:26,570 OK? 862 00:54:26,570 --> 00:54:30,100 So this scheme allows us a way of thinking 863 00:54:30,100 --> 00:54:31,890 in the effective theory way of thinking 864 00:54:31,890 --> 00:54:33,640 of having a power counting where we'd have 865 00:54:33,640 --> 00:54:35,843 to keep the C0 to all orders. 866 00:54:35,843 --> 00:54:37,260 AUDIENCE: I thought it was order-- 867 00:54:37,260 --> 00:54:39,350 AP was order 1. 868 00:54:39,350 --> 00:54:40,820 PROFESSOR: Yeah, AP is order 1. 869 00:54:44,990 --> 00:54:46,760 So if you look at the Lagrangian, 870 00:54:46,760 --> 00:54:53,180 then you have to go through the counting of how many powers 871 00:54:53,180 --> 00:54:55,100 of P the nucleon field has. 872 00:54:55,100 --> 00:54:59,210 Which you could do in the same way that we did with the mass. 873 00:54:59,210 --> 00:55:01,340 And if you do that with the four-nucleon operator 874 00:55:01,340 --> 00:55:03,530 with an extra power of P downstairs, 875 00:55:03,530 --> 00:55:08,600 you will find it has the same P scaling as the kinetic term. 876 00:55:08,600 --> 00:55:09,230 OK? 877 00:55:09,230 --> 00:55:10,933 I didn't go through that, but-- 878 00:55:10,933 --> 00:55:13,100 AUDIENCE: So that justifies this, even when P is not 879 00:55:13,100 --> 00:55:14,555 much bigger than 1 over a? 880 00:55:14,555 --> 00:55:15,680 Is that what you're saying? 881 00:55:15,680 --> 00:55:18,350 PROFESSOR: Yeah, so if P is much bigger than 1 over a, or P 882 00:55:18,350 --> 00:55:24,980 is even order or 1 over a, which is a sort of also-- 883 00:55:24,980 --> 00:55:27,560 it's also fine with this counting. 884 00:55:27,560 --> 00:55:29,570 If P is of order 1 over a, these two terms 885 00:55:29,570 --> 00:55:32,750 are sort of comparably big, but you could also 886 00:55:32,750 --> 00:55:34,940 use this approach for that case, as long 887 00:55:34,940 --> 00:55:40,700 as you're not in the case where P is much less than 1 over a. 888 00:55:40,700 --> 00:55:42,890 So if you're in that case, then you 889 00:55:42,890 --> 00:55:45,420 should really think about expanding this out some sense. 890 00:55:45,420 --> 00:55:47,670 And then you're getting back to the end MS bar result, 891 00:55:47,670 --> 00:55:50,640 the MS bar result would have been fine. 892 00:55:50,640 --> 00:55:52,650 But to get away from the MS bar result, 893 00:55:52,650 --> 00:55:56,068 and think about physics, mu of order P, 894 00:55:56,068 --> 00:55:57,860 we could use this scheme instead of MS bar, 895 00:55:57,860 --> 00:56:00,650 and then we actually see that we get a reasonable power 896 00:56:00,650 --> 00:56:01,450 counting. 897 00:56:01,450 --> 00:56:04,800 AUDIENCE: OK, Because my concern is, what about when mu is like, 898 00:56:04,800 --> 00:56:07,092 what about this pole that you're going to-- 899 00:56:07,092 --> 00:56:10,181 PROFESSOR: Yeah, we'll talk about the pole, yeah. 900 00:56:10,181 --> 00:56:12,580 It's coming up. 901 00:56:12,580 --> 00:56:14,800 All right. 902 00:56:14,800 --> 00:56:16,780 So it's interesting to think about this 903 00:56:16,780 --> 00:56:19,075 from our renormalization group point of view, which 904 00:56:19,075 --> 00:56:20,950 is kind of what I was doing when I wrote down 905 00:56:20,950 --> 00:56:22,630 a beta function that I was-- 906 00:56:22,630 --> 00:56:25,870 here I was just after this solution, 907 00:56:25,870 --> 00:56:30,740 because from that solution I got the power counting that I want. 908 00:56:30,740 --> 00:56:33,413 Which is that the C0 term is the same size as the kinetic term, 909 00:56:33,413 --> 00:56:34,330 and both are relevant. 910 00:56:43,980 --> 00:56:46,910 So we can do that just like we counted P's for [INAUDIBLE] 911 00:56:46,910 --> 00:56:48,500 theory, or just like we counted-- 912 00:56:51,430 --> 00:57:08,950 [? just comment. ?] These guys are all relevant 913 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. 914 00:57:13,780 --> 00:57:15,700 So there's another way of thinking about this, 915 00:57:15,700 --> 00:57:17,408 and thinking about these different cases, 916 00:57:17,408 --> 00:57:20,690 and that's just to think about the beta function itself. 917 00:57:20,690 --> 00:57:24,730 So if you look at the beta function for the C0 coupling, 918 00:57:24,730 --> 00:57:27,430 and we just put in the solution, plug this back 919 00:57:27,430 --> 00:57:37,243 into that equation here, with some constant out front, 920 00:57:37,243 --> 00:57:38,410 and then it looks like this. 921 00:57:38,410 --> 00:57:43,430 8 times mu, 1 minus a mu, squared. 922 00:57:43,430 --> 00:57:45,880 If we want to talk about all possible values of a, 923 00:57:45,880 --> 00:57:49,580 well a can go from minus infinity to plus infinity. 924 00:57:49,580 --> 00:57:52,030 So it's useful if we want to draw this to map it 925 00:57:52,030 --> 00:57:53,110 to a compact interval. 926 00:57:55,760 --> 00:57:56,560 So let's do that. 927 00:58:04,680 --> 00:58:06,950 Move the tangent. 928 00:58:06,950 --> 00:58:09,800 And let's just plot beta as a function of x. 929 00:58:21,555 --> 00:58:23,805 So there's three values, actually where beta vanishes. 930 00:58:27,150 --> 00:58:28,650 There's one value where it blows up. 931 00:58:32,330 --> 00:58:38,150 So this is-- here is x equals minus 1. 932 00:58:38,150 --> 00:58:43,170 This is x equals 0, this is x equals plus 1. 933 00:58:45,840 --> 00:58:53,640 And what it looks like dips down here, goes there, blows up, 934 00:58:53,640 --> 00:58:57,930 then it comes back down, it goes like that. 935 00:58:57,930 --> 00:59:00,930 So that's what the beta function looks like. 936 00:59:00,930 --> 00:59:03,290 So this point here corresponds if you 937 00:59:03,290 --> 00:59:05,630 think of being at fixed mu. 938 00:59:05,630 --> 00:59:08,240 Say you're studying the physics at fixed mu. 939 00:59:08,240 --> 00:59:11,720 This point corresponds to a equals minus infinity, 940 00:59:11,720 --> 00:59:14,870 this point corresponds to a equals zero, 941 00:59:14,870 --> 00:59:18,002 and this point corresponds to a equals plus infinity. 942 00:59:24,400 --> 00:59:26,980 So there's three points where the beta function vanishes, 943 00:59:26,980 --> 00:59:29,168 if you think about a space for fixed mu. 944 00:59:41,590 --> 00:59:42,370 Use some color. 945 00:59:53,250 --> 00:59:55,450 So you can ask about nature. 946 00:59:55,450 --> 00:59:57,450 So nature told us the value of a, so what 947 00:59:57,450 --> 00:59:59,200 does that correspond to? 948 00:59:59,200 --> 01:00:01,830 So taking some value of P and mapping it 949 01:00:01,830 --> 01:00:04,823 to some value of x, kind of generically the kind of point 950 01:00:04,823 --> 01:00:06,240 that you're interested in is here. 951 01:00:06,240 --> 01:00:09,120 This is a1S0, sits there. 952 01:00:09,120 --> 01:00:16,420 And then a for the 3S1 makes it kind of here. 953 01:00:16,420 --> 01:00:19,350 So what's going on in this case is that you're not 954 01:00:19,350 --> 01:00:22,800 near this fixed point, you're actually close to this one. 955 01:00:22,800 --> 01:00:24,630 And you're not near this fixed point, 956 01:00:24,630 --> 01:00:26,505 well actually there's an infinity in between, 957 01:00:26,505 --> 01:00:28,510 you're closer to this one. 958 01:00:28,510 --> 01:00:32,940 This size you should think of as 8 MeV, generically, 959 01:00:32,940 --> 01:00:35,580 and then this is like the 8 but there's also a pole in between. 960 01:00:38,440 --> 01:00:39,670 So three fixed points. 961 01:00:42,778 --> 01:00:44,320 When we do perturbation theory and we 962 01:00:44,320 --> 01:00:46,445 expand about fixed points, and one way 963 01:00:46,445 --> 01:00:48,820 of saying what was wrong with dimensional analysis was it 964 01:00:48,820 --> 01:00:51,340 was just expanding about the wrong fixed point. 965 01:00:51,340 --> 01:00:51,910 The pink one. 966 01:00:55,300 --> 01:01:02,730 So a equals zero, was the non-interactive one 967 01:01:02,730 --> 01:01:04,480 where we just had the relevant interaction 968 01:01:04,480 --> 01:01:08,080 being the kinetic term, but none of the interaction terms 969 01:01:08,080 --> 01:01:09,580 are relevant. 970 01:01:09,580 --> 01:01:13,060 And a equals plus or minus infinity 971 01:01:13,060 --> 01:01:15,040 are interacting fixed points, where 972 01:01:15,040 --> 01:01:17,170 you have an interaction that is relevant. 973 01:01:22,370 --> 01:01:24,550 So you can think about that in the following sense. 974 01:01:24,550 --> 01:01:27,070 Classically, what a is measuring is kind of the interaction 975 01:01:27,070 --> 01:01:30,250 size, if you have classical scattering across sections 976 01:01:30,250 --> 01:01:31,700 4 pi a squared. 977 01:01:31,700 --> 01:01:34,858 And if a is small-- 978 01:01:34,858 --> 01:01:38,420 if a is either very small or very big, 979 01:01:38,420 --> 01:01:41,290 then basically it's the same on all scales 980 01:01:41,290 --> 01:01:44,290 because it's either infinity or 0 981 01:01:44,290 --> 01:01:46,510 and it looks the same to particles 982 01:01:46,510 --> 01:01:48,190 of all different momenta. 983 01:01:48,190 --> 01:01:49,150 OK? 984 01:01:49,150 --> 01:01:51,940 And that's one way of thinking about these fixed points. 985 01:01:51,940 --> 01:01:53,680 That the physics can't-- 986 01:01:53,680 --> 01:01:57,220 the physics-- yeah. 987 01:01:57,220 --> 01:01:58,720 Just what I said. 988 01:01:58,720 --> 01:02:00,880 So what about this infinity? 989 01:02:00,880 --> 01:02:02,518 That's also interesting. 990 01:02:10,486 --> 01:02:16,370 So when a is one over mu, beta goes to infinity. 991 01:02:16,370 --> 01:02:19,910 And this actually-- there is a reflection 992 01:02:19,910 --> 01:02:23,480 of this in the theory, it corresponds to a bound state. 993 01:02:28,580 --> 01:02:30,920 Which I think we'll talk about next time, but. 994 01:02:33,440 --> 01:02:35,090 So there's a bound state in the theory, 995 01:02:35,090 --> 01:02:37,310 and actually if you start from this side, 996 01:02:37,310 --> 01:02:39,110 you can never see that bound because you're 997 01:02:39,110 --> 01:02:39,980 doing perturbation theory. 998 01:02:39,980 --> 01:02:42,200 You don't see perturbation theory and bound states. 999 01:02:42,200 --> 01:02:44,210 If you start from this side, the bound state 1000 01:02:44,210 --> 01:02:46,160 is actually just in the theory. 1001 01:02:46,160 --> 01:02:48,605 We'll talk about that next time. 1002 01:02:48,605 --> 01:02:50,480 And so it's a state in the theory, you can go 1003 01:02:50,480 --> 01:02:52,400 and you could find the pole in your amplitude, 1004 01:02:52,400 --> 01:02:53,900 it's just there. 1005 01:02:53,900 --> 01:02:56,180 Corresponds to a physical state of the spectrum, 1006 01:02:56,180 --> 01:02:57,388 and it's called the deuteron. 1007 01:03:14,620 --> 01:03:16,120 So we have a non-interactive theory, 1008 01:03:16,120 --> 01:03:19,990 we have some non-trivial amplitude, 1009 01:03:19,990 --> 01:03:22,540 and this deuteron is a pole in that amplitude. 1010 01:04:09,310 --> 01:04:13,890 And you never see a pole if you use the perturbation theory. 1011 01:04:13,890 --> 01:04:16,350 And if you actually look at the energy, the binding energy 1012 01:04:16,350 --> 01:04:18,773 of this state, it's also small. 1013 01:04:18,773 --> 01:04:20,190 Characteristically small, and it's 1014 01:04:20,190 --> 01:04:24,030 actually related to the fact that the binding energy is 1015 01:04:24,030 --> 01:04:26,820 small just related to sort of the natural size of this a. 1016 01:04:26,820 --> 01:04:28,260 We'll talk about that next time. 1017 01:04:34,850 --> 01:04:38,290 So what I was saying before about the theory at all scales 1018 01:04:38,290 --> 01:04:41,170 looking the same, when you have a equals 0 of course 1019 01:04:41,170 --> 01:04:42,550 it's not interacting. 1020 01:04:42,550 --> 01:04:44,770 Or when a is equal to plus minus infinity, 1021 01:04:44,770 --> 01:04:47,770 looks the same at all scales. 1022 01:04:47,770 --> 01:04:50,998 That means that scale invariant. 1023 01:04:50,998 --> 01:04:52,540 So it's a scale invariant theory when 1024 01:04:52,540 --> 01:04:55,540 a is at these fixed points. 1025 01:04:55,540 --> 01:04:58,600 And something I worked on was the fact 1026 01:04:58,600 --> 01:05:02,650 that these points are actually conformal fixed points. 1027 01:05:12,570 --> 01:05:16,600 So that-- there's a conformal symmetry of the theory that 1028 01:05:16,600 --> 01:05:20,090 exists at those points, we'll talk about that a little bit. 1029 01:05:20,090 --> 01:05:22,750 There's another symmetry, too, which I have to mention. 1030 01:05:25,480 --> 01:05:29,140 And not only because I also worked on this one, 1031 01:05:29,140 --> 01:05:32,070 because it's good to enumerate all the symmetries. 1032 01:05:35,360 --> 01:05:37,880 There's actually a combined spin, isospin symmetry, 1033 01:05:37,880 --> 01:05:41,027 that turns into an Su4 in this limit. 1034 01:05:41,027 --> 01:05:42,860 So much like in heavy quark effective theory 1035 01:05:42,860 --> 01:05:45,818 where the mass was big, new symmetries popped up. 1036 01:05:45,818 --> 01:05:48,110 Same thing happens here, when the scattering lengths go 1037 01:05:48,110 --> 01:05:50,552 to infinity, and you go over to those fixed points, 1038 01:05:50,552 --> 01:05:52,010 there's new symmetries that pop up. 1039 01:05:52,010 --> 01:05:53,927 One's a conformal symmetry and the other one's 1040 01:05:53,927 --> 01:05:55,190 a spin, isospin symmetry. 1041 01:06:09,810 --> 01:06:12,060 All right. 1042 01:06:12,060 --> 01:06:16,200 So this looks interesting. 1043 01:06:16,200 --> 01:06:18,398 You could ask the question, did this pick-- 1044 01:06:18,398 --> 01:06:20,190 did this physical picture that we developed 1045 01:06:20,190 --> 01:06:22,050 depend on picking this renormalization 1046 01:06:22,050 --> 01:06:24,240 scheme that I told you about? 1047 01:06:24,240 --> 01:06:26,670 We kind of gave up on dimreg, we went over to-- well, 1048 01:06:26,670 --> 01:06:28,110 we gave up on MS bar, we went over 1049 01:06:28,110 --> 01:06:31,502 to this scheme which was offshell momentum subtraction. 1050 01:06:31,502 --> 01:06:33,960 In general, people don't like offshell momentum subtraction 1051 01:06:33,960 --> 01:06:35,793 because it makes calculations more difficult 1052 01:06:35,793 --> 01:06:37,001 once you go to higher orders. 1053 01:06:37,001 --> 01:06:39,210 Well here, are the calculations are not so difficult, 1054 01:06:39,210 --> 01:06:40,320 so we could do them, but-- 1055 01:06:40,320 --> 01:06:42,750 you might be interested in adding pions to this theory, 1056 01:06:42,750 --> 01:06:45,420 or coupling external currents like photons, 1057 01:06:45,420 --> 01:06:47,910 and then the calculations would get more difficult. 1058 01:06:47,910 --> 01:06:51,180 And you'd like, for example, to have a dimreg MS bar type 1059 01:06:51,180 --> 01:06:53,190 description of this power counting 1060 01:06:53,190 --> 01:06:55,620 rather than a kind of minimal subtraction. 1061 01:06:55,620 --> 01:06:57,210 Could I get, could I kind of dress up 1062 01:06:57,210 --> 01:07:00,300 minimal subtraction to get the same physical picture? 1063 01:07:00,300 --> 01:07:03,210 That's a reasonable question to ask. 1064 01:07:03,210 --> 01:07:04,290 The answer is you can. 1065 01:07:10,737 --> 01:07:13,070 So there's something called power divergence subtraction 1066 01:07:13,070 --> 01:07:15,650 scheme, different scheme than MS bar. 1067 01:07:22,244 --> 01:07:25,780 So the PDS scheme. 1068 01:07:25,780 --> 01:07:31,060 And what it says is, don't just subtract poles at D equals 4, 1069 01:07:31,060 --> 01:07:37,510 like you do in MS bar, which are corresponding to logs 1070 01:07:37,510 --> 01:07:42,580 at the cut off, but also subtract poles and D equals 3. 1071 01:07:42,580 --> 01:07:45,090 And dimreg knows about power law divergences 1072 01:07:45,090 --> 01:07:46,840 and they're just poles at different places 1073 01:07:46,840 --> 01:07:48,530 in the dimensions. 1074 01:07:48,530 --> 01:07:50,560 And so if we subtract poles at D equals three, 1075 01:07:50,560 --> 01:07:52,700 we can track the power law divergences in that way. 1076 01:07:52,700 --> 01:07:54,700 And it's the power of divergence that's actually 1077 01:07:54,700 --> 01:07:56,410 causing, if you want to think of it 1078 01:07:56,410 --> 01:07:58,495 as a change to the anomalous dimension, where 1079 01:07:58,495 --> 01:08:00,370 the anomalous dimension was saying this thing 1080 01:08:00,370 --> 01:08:03,250 was irrelevant, to changing it to something relevant, 1081 01:08:03,250 --> 01:08:05,000 you need a big change for that to happen. 1082 01:08:05,000 --> 01:08:06,730 And the big change that's occurring 1083 01:08:06,730 --> 01:08:09,100 is coming from a power law divergence here. 1084 01:08:09,100 --> 01:08:11,650 That's what sort of allowed you to jump, if you like, 1085 01:08:11,650 --> 01:08:15,730 from this fixed point to this one. 1086 01:08:15,730 --> 01:08:17,950 The renormalization group, including the power law 1087 01:08:17,950 --> 01:08:20,937 divergence, allows you to even flow between those points. 1088 01:08:20,937 --> 01:08:23,020 Usually we think that power law divergences aren't 1089 01:08:23,020 --> 01:08:25,562 doing anything, here's an example where they are. 1090 01:08:25,562 --> 01:08:26,979 They're not doing anything as long 1091 01:08:26,979 --> 01:08:28,812 as you know you're at the right fixed point. 1092 01:08:28,812 --> 01:08:31,078 If you're describing the right physics around one 1093 01:08:31,078 --> 01:08:33,370 of these fixed points, you can concentrate on the logs, 1094 01:08:33,370 --> 01:08:35,715 but if you don't know where you are then the power 1095 01:08:35,715 --> 01:08:37,090 law divergences could be crucial. 1096 01:08:39,609 --> 01:08:41,490 All right, how does this scheme work? 1097 01:08:45,250 --> 01:08:49,240 So this is a dimreg-type scheme. 1098 01:08:49,240 --> 01:08:52,930 So we're going to get the power of mu 1099 01:08:52,930 --> 01:08:55,510 from the mu to the two epsilon that we have out front. 1100 01:09:07,520 --> 01:09:11,587 So if I just write this guy down in D dimensions, 1101 01:09:11,587 --> 01:09:12,670 here's what it looks like. 1102 01:09:31,140 --> 01:09:33,770 And I've normalized mu slightly differently than we usually 1103 01:09:33,770 --> 01:09:37,680 do just because it's convenient for this scheme to do that. 1104 01:09:37,680 --> 01:09:40,160 So it's not exactly the same as MS bar, it's mu over 2 1105 01:09:40,160 --> 01:09:41,270 that I'm putting in. 1106 01:09:41,270 --> 01:09:43,640 Other than that, it's the same kind of set up as MS bar. 1107 01:09:51,584 --> 01:09:54,269 You'll see why I want to put that 2 there in a minute. 1108 01:09:59,030 --> 01:10:01,608 OK, so this is just the result that we would write down 1109 01:10:01,608 --> 01:10:02,900 for dimensional regularization. 1110 01:10:02,900 --> 01:10:04,865 Dimensional regularization is not a scheme. 1111 01:10:04,865 --> 01:10:08,930 A scheme has to do with what we subtract. 1112 01:10:08,930 --> 01:10:10,613 Dimreg is just how we regulate. 1113 01:10:21,740 --> 01:10:25,160 So now, look at D equals 4. 1114 01:10:25,160 --> 01:10:31,370 So in D equals 4, OK. 1115 01:10:31,370 --> 01:10:32,495 There's a bunch of factors. 1116 01:10:38,315 --> 01:10:40,940 This is just giving, this is the answer I quoted to you before. 1117 01:10:44,480 --> 01:10:46,700 Something finite. 1118 01:10:46,700 --> 01:10:54,160 And if we look at D equals three, 1119 01:10:54,160 --> 01:10:56,285 then we have a pole because of that gamma function. 1120 01:11:02,030 --> 01:11:04,370 And I've put the 2 in here just to cancel that 2. 1121 01:11:07,520 --> 01:11:11,990 And so, what this scheme says is to add a part subtraction 1122 01:11:11,990 --> 01:11:14,090 for this guy. 1123 01:11:14,090 --> 01:11:16,950 So what we do is, we add a counter term. 1124 01:11:16,950 --> 01:11:25,640 It looks like minus IM over 4 pi, mu, got one power of mu. 1125 01:11:25,640 --> 01:11:29,480 Over 3 minus D. C0 squared. 1126 01:11:32,430 --> 01:11:36,710 And then if we take the graph, plus the counterterm and we 1127 01:11:36,710 --> 01:11:38,840 set D equal four, which is where we actually want 1128 01:11:38,840 --> 01:11:45,320 to do physics, lo and behold. 1129 01:11:45,320 --> 01:11:51,500 In this approach, you get actually the same answer 1130 01:11:51,500 --> 01:11:53,510 as in our offshell momentum subtraction scheme. 1131 01:11:53,510 --> 01:11:55,610 And that's just really because this scheme tracks the power 1132 01:11:55,610 --> 01:11:57,170 correction, the power divergence, 1133 01:11:57,170 --> 01:12:00,250 just like the offshell momentum subtraction did. 1134 01:12:00,250 --> 01:12:03,500 So we just invented a dimreg style 1135 01:12:03,500 --> 01:12:05,965 of looking for poles that can track the same physics, 1136 01:12:05,965 --> 01:12:07,340 and we just have to look at poles 1137 01:12:07,340 --> 01:12:11,440 in D equals 3 rather than poles in D equals 4, OK? 1138 01:12:13,887 --> 01:12:16,220 And this is easier in general than the offshell momentum 1139 01:12:16,220 --> 01:12:19,880 subtraction scheme, although for basically everything 1140 01:12:19,880 --> 01:12:22,650 we're talking about today you could do either one. 1141 01:12:22,650 --> 01:12:25,848 Now, where is the predictive power of this effective theory? 1142 01:12:25,848 --> 01:12:27,890 So far, we've just kind of cooked things together 1143 01:12:27,890 --> 01:12:30,350 to make the C0 do what we want. 1144 01:12:30,350 --> 01:12:32,420 Well, we didn't completely cook things together. 1145 01:12:32,420 --> 01:12:34,370 We switched to another scheme and it kind of popped out 1146 01:12:34,370 --> 01:12:35,745 naturally, but you can say, well, 1147 01:12:35,745 --> 01:12:39,178 why not explore three other schemes and see if they work? 1148 01:12:39,178 --> 01:12:41,720 But let's just imagine that we got things to work, as we just 1149 01:12:41,720 --> 01:12:44,210 did in two different ways by tracking the power law 1150 01:12:44,210 --> 01:12:45,320 divergence. 1151 01:12:45,320 --> 01:12:47,750 The predictive power becomes from now the fact 1152 01:12:47,750 --> 01:12:51,710 that if I say that's my fine tuning, that C0 got enhanced, 1153 01:12:51,710 --> 01:12:54,230 I can now predict the size of all other operators 1154 01:12:54,230 --> 01:12:55,310 in the theory. 1155 01:12:55,310 --> 01:12:57,887 And other operators like C2 and C4, 1156 01:12:57,887 --> 01:13:00,470 the power counting we assigned to them previously is not true. 1157 01:13:00,470 --> 01:13:01,980 We have to figure it out. 1158 01:13:01,980 --> 01:13:04,610 But we can figure that out once we know 1159 01:13:04,610 --> 01:13:09,160 what approach we should use. 1160 01:13:09,160 --> 01:13:15,667 OK, so this is same as above. 1161 01:13:15,667 --> 01:13:18,250 I won't go through it, but you know, same anomalous dimension, 1162 01:13:18,250 --> 01:13:19,420 et cetera. 1163 01:13:19,420 --> 01:13:22,240 And it's easier in general. 1164 01:13:22,240 --> 01:13:25,990 Let's think about C2 mu. 1165 01:13:25,990 --> 01:13:29,170 So if you look at C2, the first kind 1166 01:13:29,170 --> 01:13:33,160 of diagram you could think about would be a guy with one C2, 1167 01:13:33,160 --> 01:13:36,070 and then this is the first type of loop diagram 1168 01:13:36,070 --> 01:13:38,240 you might think about. 1169 01:13:38,240 --> 01:13:41,960 And these guys have a P squared because C2 gave a P squared. 1170 01:13:41,960 --> 01:13:43,840 So they have an extra P squared. 1171 01:13:43,840 --> 01:13:45,010 And they diverge. 1172 01:13:45,010 --> 01:13:46,690 They also have a power divergence. 1173 01:13:46,690 --> 01:13:48,280 So if you calculate in either one 1174 01:13:48,280 --> 01:13:50,320 of these schemes, offshell momentum 1175 01:13:50,320 --> 01:13:56,927 subtraction, or this PDS scheme, these guys get a divergence. 1176 01:13:56,927 --> 01:13:58,510 And again, it's a power law divergence 1177 01:13:58,510 --> 01:14:01,190 so there's a mu here. 1178 01:14:01,190 --> 01:14:06,190 And you get a beta function, that's that. 1179 01:14:06,190 --> 01:14:08,800 2C0 C2. 1180 01:14:08,800 --> 01:14:11,040 So if you take the boundary condition 1181 01:14:11,040 --> 01:14:15,670 C2 of 0 which is our MS bar result. 4 pi over M. A 1182 01:14:15,670 --> 01:14:17,320 squared, r0. 1183 01:14:17,320 --> 01:14:30,010 And you solve this, you find C2 of mu is 4 pi over M. 1 1184 01:14:30,010 --> 01:14:34,870 over mu minus 1 over a, squared. 1185 01:14:34,870 --> 01:14:36,010 r0 over 2. 1186 01:14:41,700 --> 01:14:44,165 So there's two-- we previously, with our counting, 1187 01:14:44,165 --> 01:14:45,540 when we were counting dimensions, 1188 01:14:45,540 --> 01:14:49,080 we would have said C0 goes like 1 over lambda, 1189 01:14:49,080 --> 01:14:51,900 C2 goes like one over lambda cubed. 1190 01:14:51,900 --> 01:14:54,480 We had 2M plus 1 lambdas. 1191 01:14:54,480 --> 01:14:57,300 What we've just discovered is that yes, 1192 01:14:57,300 --> 01:15:00,700 there's a lambda from this r0, that's a 1 over lambda, 1193 01:15:00,700 --> 01:15:03,570 but the other two lambdas are really mu's. 1194 01:15:03,570 --> 01:15:06,900 So this operator is also enhanced, 1195 01:15:06,900 --> 01:15:09,797 and it's enhanced by two powers. 1196 01:15:09,797 --> 01:15:11,380 So once you know leading order theory, 1197 01:15:11,380 --> 01:15:13,770 you should be able to determine the power counting of all 1198 01:15:13,770 --> 01:15:15,352 of the other operators, especially 1199 01:15:15,352 --> 01:15:16,560 if they're not relevant ones. 1200 01:15:16,560 --> 01:15:18,450 You have to get the relevant part right, 1201 01:15:18,450 --> 01:15:20,040 and then you can use that Lagrangian 1202 01:15:20,040 --> 01:15:22,697 to predict all the scaling for everything else. 1203 01:15:22,697 --> 01:15:24,030 And that's what we've just done. 1204 01:15:38,970 --> 01:15:41,850 Gone to 1 over mu squared lambda. 1205 01:15:41,850 --> 01:15:42,360 OK? 1206 01:15:42,360 --> 01:15:45,223 And those powers of mu we see in the scheme, 1207 01:15:45,223 --> 01:15:46,890 and the 1 over lambda comes from the r0. 1208 01:15:54,000 --> 01:15:54,500 OK. 1209 01:15:54,500 --> 01:15:59,780 So what the RGE actually does is it tells us-- 1210 01:15:59,780 --> 01:16:02,990 one way of thinking about it is that it tells us 1211 01:16:02,990 --> 01:16:04,640 the enhancement, due to fine tuning, 1212 01:16:04,640 --> 01:16:05,990 of all operators in the theory. 1213 01:16:09,607 --> 01:16:11,690 And that's really because the fine tuning was just 1214 01:16:11,690 --> 01:16:14,180 a change of our power counting, and we 1215 01:16:14,180 --> 01:16:16,930 have to propagate that change everywhere. 1216 01:16:22,930 --> 01:16:25,360 And we can do that, and it's the beta functions 1217 01:16:25,360 --> 01:16:28,870 that tell us how to propagate the fine tuning. 1218 01:16:28,870 --> 01:16:34,246 So if you keep going, you can do C2K of mu. 1219 01:16:34,246 --> 01:16:36,070 You find an anomalous dimension. 1220 01:16:39,680 --> 01:16:43,060 This theory is kind of nice, you can basically 1221 01:16:43,060 --> 01:16:44,490 do all the calculations, so. 1222 01:16:48,076 --> 01:16:50,500 When you go to CK, you get contributions 1223 01:16:50,500 --> 01:16:54,250 from various lower order coefficients, 1224 01:16:54,250 --> 01:16:56,365 and it's one loop exact so you only have pairs. 1225 01:16:59,365 --> 01:17:00,740 And then you can kind of contrast 1226 01:17:00,740 --> 01:17:04,880 what's going on in a naive power counting where 1227 01:17:04,880 --> 01:17:06,230 P is much less than 1 over a. 1228 01:17:14,720 --> 01:17:17,750 Let's just go up to C4. 1229 01:17:17,750 --> 01:17:21,200 Versus this kind of improved power 1230 01:17:21,200 --> 01:17:26,540 counting, which is valid when PA is greater than our order 1. 1231 01:17:30,995 --> 01:17:34,820 So C0 hat it went like 1 over mu, no suppression there. 1232 01:17:34,820 --> 01:17:37,370 C2 hat goes like one over mu squared lambda, 1233 01:17:37,370 --> 01:17:39,840 and that actually is irrelevant. 1234 01:17:39,840 --> 01:17:41,480 But it's just irrelevant by one power. 1235 01:17:46,300 --> 01:17:49,950 So relative to this guy, it's down by a P over lambda. 1236 01:17:49,950 --> 01:17:52,020 And then interesting things start 1237 01:17:52,020 --> 01:17:54,660 to happen with the higher ones, at least 1238 01:17:54,660 --> 01:17:56,386 from an RGE perspective. 1239 01:17:59,493 --> 01:18:01,410 So these guys start to get more than one term. 1240 01:18:05,358 --> 01:18:07,650 But this guy actually doesn't introduce a new constant. 1241 01:18:13,458 --> 01:18:16,000 There's a piece of the anomalous dimension of this guy that's 1242 01:18:16,000 --> 01:18:17,458 actually just fixed by the constant 1243 01:18:17,458 --> 01:18:20,170 that you already had here, and then there's a new piece. 1244 01:18:20,170 --> 01:18:22,350 So the new piece is down by two powers of lambda. 1245 01:18:26,545 --> 01:18:29,170 And that's encoding things about the amplitude, actually, but-- 1246 01:18:32,240 --> 01:18:33,800 OK, so that's just a little table 1247 01:18:33,800 --> 01:18:35,342 to kind of convince you that once you 1248 01:18:35,342 --> 01:18:37,370 have a beta function that you can compute 1249 01:18:37,370 --> 01:18:39,260 for the coefficients, you can quickly 1250 01:18:39,260 --> 01:18:41,570 propagate this enhancement from the fine tuning 1251 01:18:41,570 --> 01:18:42,860 to the rest of your theory. 1252 01:18:42,860 --> 01:18:45,320 I.e., figure out what the power counting 1253 01:18:45,320 --> 01:18:47,020 is for all the operators. 1254 01:18:47,020 --> 01:18:49,640 AUDIENCE: So every time there is a power law divergence, 1255 01:18:49,640 --> 01:18:51,482 should I be worried if I'm using MS bar, 1256 01:18:51,482 --> 01:18:53,940 should I be worried that the power counting could be wrong? 1257 01:18:53,940 --> 01:18:56,160 PROFESSOR: Every time-- 1258 01:18:56,160 --> 01:18:58,160 Yeah, every time there's a power law divergence, 1259 01:18:58,160 --> 01:18:59,660 it's worth thinking about whether it 1260 01:18:59,660 --> 01:19:06,220 had some physical impact on what you're doing, I think. 1261 01:19:11,042 --> 01:19:13,500 If you know you're expanding-- if you can convince yourself 1262 01:19:13,500 --> 01:19:15,930 that you're expanding around the right fixed point then 1263 01:19:15,930 --> 01:19:16,530 you're OK. 1264 01:19:16,530 --> 01:19:20,520 That's my equivalence claim. 1265 01:19:20,520 --> 01:19:22,020 But you don't necessarily know that. 1266 01:19:22,020 --> 01:19:24,110 So let's go back to our amplitude 1267 01:19:24,110 --> 01:19:26,930 and see what's going on here, and see 1268 01:19:26,930 --> 01:19:29,890 what it looks like with this power counting. 1269 01:19:29,890 --> 01:19:33,290 And so it's really just a different expansion 1270 01:19:33,290 --> 01:19:34,760 of that amplitude that we had. 1271 01:19:41,420 --> 01:19:48,380 And in either the PDS scheme or the power diversion subtraction 1272 01:19:48,380 --> 01:19:54,620 scheme, we end up with this amplitude in the case where 1273 01:19:54,620 --> 01:19:56,670 we would use that scheme. 1274 01:19:56,670 --> 01:19:59,948 So you can see in PDS that if I set this coefficient to 0, 1275 01:19:59,948 --> 01:20:01,490 the denominator becomes 1, and then I 1276 01:20:01,490 --> 01:20:04,580 get with the offshell momentum subtraction scheme. 1277 01:20:04,580 --> 01:20:07,640 In PDS is a little harder see that it gives just that, 1278 01:20:07,640 --> 01:20:10,080 but it actually gives the same thing in either scheme. 1279 01:20:10,080 --> 01:20:14,270 And if you think about what type of expansion you're doing here, 1280 01:20:14,270 --> 01:20:15,930 you're keeping C0 to all orders. 1281 01:20:15,930 --> 01:20:23,720 So your amplitude at lowest order is just this, 1282 01:20:23,720 --> 01:20:38,510 and then the C2 term looks like that. 1283 01:20:38,510 --> 01:20:40,040 And then there's some higher terms 1284 01:20:40,040 --> 01:20:43,770 which I wrote in my notes, but I want right here. 1285 01:20:43,770 --> 01:20:47,270 And what this is, this here is some kind of interaction. 1286 01:20:47,270 --> 01:20:50,030 I'll make it a bigger circle, which 1287 01:20:50,030 --> 01:20:53,900 sums up all the bubbles with C0's in them. 1288 01:20:53,900 --> 01:20:56,780 That's what's happened here. 1289 01:20:56,780 --> 01:21:01,610 And this here, if you like, is like taking C2 and then 1290 01:21:01,610 --> 01:21:04,535 dressing it with bubbles on either side. 1291 01:21:04,535 --> 01:21:07,700 So there's two, there's bubbles. 1292 01:21:07,700 --> 01:21:11,523 The bubbles on the other side. 1293 01:21:11,523 --> 01:21:12,815 And then bubbles on both sides. 1294 01:21:16,500 --> 01:21:18,480 So we calculate these loop graphs 1295 01:21:18,480 --> 01:21:19,980 and these are the amplitudes we get, 1296 01:21:19,980 --> 01:21:21,355 and that's because we're treating 1297 01:21:21,355 --> 01:21:23,645 the C0 coupling to all orders, we're summing it up. 1298 01:21:23,645 --> 01:21:25,770 And actually, each of these amplitudes, if you look 1299 01:21:25,770 --> 01:21:28,590 at the RGE, is mu independent. 1300 01:21:28,590 --> 01:21:31,115 Explicitly mu independent. 1301 01:21:31,115 --> 01:21:32,490 So it's like perturbation theory, 1302 01:21:32,490 --> 01:21:33,960 where we're doing a momentum expansion, 1303 01:21:33,960 --> 01:21:35,877 and order by order in that momentum expansion, 1304 01:21:35,877 --> 01:21:39,300 the amplitude is independent of the scale mu. 1305 01:21:39,300 --> 01:21:41,160 The only purpose of the scale mu is 1306 01:21:41,160 --> 01:21:45,155 to help us think about power counting of these operators. 1307 01:21:45,155 --> 01:21:47,530 In the end of the day, when we make physical predictions, 1308 01:21:47,530 --> 01:21:50,790 then getting mu independent answers. 1309 01:21:50,790 --> 01:21:51,840 OK. 1310 01:21:51,840 --> 01:21:55,813 And this is like organizing, if you like. 1311 01:21:55,813 --> 01:21:56,980 And you put it in terms of-- 1312 01:22:01,380 --> 01:22:07,255 put it back in terms of a, this is 1313 01:22:07,255 --> 01:22:08,880 like organizing the theory in this way, 1314 01:22:08,880 --> 01:22:12,900 where you keep all powers of AP, and that makes it very clear 1315 01:22:12,900 --> 01:22:14,910 that it's mu independent. 1316 01:22:14,910 --> 01:22:17,100 Now, this part of the theory is so 1317 01:22:17,100 --> 01:22:19,380 simple you could have figured that out just 1318 01:22:19,380 --> 01:22:22,380 by writing the top line down in terms of a's and P's and just 1319 01:22:22,380 --> 01:22:24,180 writing this line down. 1320 01:22:24,180 --> 01:22:26,730 But you could also use what I've been talking about 1321 01:22:26,730 --> 01:22:31,260 to figure out, for example, say I coupled an external photon 1322 01:22:31,260 --> 01:22:33,150 to my four nucleon operators. 1323 01:22:33,150 --> 01:22:34,830 How big is this? 1324 01:22:34,830 --> 01:22:35,640 OK. 1325 01:22:35,640 --> 01:22:38,370 Well, it actually gets enhanced from the fact the scattering 1326 01:22:38,370 --> 01:22:39,995 length is large, and you can figure out 1327 01:22:39,995 --> 01:22:43,650 how important this effect is, and when people do things 1328 01:22:43,650 --> 01:22:48,030 like deuteron formation in the sun and stuff like that, 1329 01:22:48,030 --> 01:22:51,330 they use this effective theory to do higher order calculations 1330 01:22:51,330 --> 01:22:55,040 and make precision predictions for deuteron physics. 1331 01:22:55,040 --> 01:22:56,910 So it's not just a toy model. 1332 01:22:56,910 --> 01:23:00,750 It's actually something that has a real impact on some physics. 1333 01:23:00,750 --> 01:23:02,813 We'll talk a little bit more about it next time. 1334 01:23:02,813 --> 01:23:04,980 We'll talk a little bit about the conformal symmetry 1335 01:23:04,980 --> 01:23:07,105 and I'll talk a little bit more about the deuterons 1336 01:23:07,105 --> 01:23:09,210 since that's something interesting in this theory, 1337 01:23:09,210 --> 01:23:13,330 and then we'll go on from there. 1338 01:23:13,330 --> 01:23:14,760 So, any questions? 1339 01:23:17,388 --> 01:23:18,880 It's cool stuff. 1340 01:23:18,880 --> 01:23:21,310 Simple to do calculations. 1341 01:23:21,310 --> 01:23:23,850 It's kind of interesting to think about.