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 00:00:22,380 --> 00:00:24,880 IAN STEWART: So last time, we were midway through this proof 9 00:00:24,880 --> 00:00:28,930 that the equations of motion, i.e., field redefinitions, 10 00:00:28,930 --> 00:00:32,470 can be used to simplify the theory. 11 00:00:32,470 --> 00:00:35,320 And I stated last time that really, 12 00:00:35,320 --> 00:00:38,530 all we have to worry about is the change to the Lagrangian. 13 00:00:38,530 --> 00:00:40,270 But when we looked at the path integral 14 00:00:40,270 --> 00:00:42,640 to do things properly and make a change variable in the path 15 00:00:42,640 --> 00:00:44,550 integral, it wasn't just the Lagrangian that changed. 16 00:00:44,550 --> 00:00:47,050 The Lagrangian changed, and we could set up our field 17 00:00:47,050 --> 00:00:50,200 redefinition to do what we want, but there were also changes 18 00:00:50,200 --> 00:00:51,970 to the Jacobian and the source, and that's 19 00:00:51,970 --> 00:00:56,120 what we had started to talk about last time. 20 00:00:56,120 --> 00:01:00,460 So when I look at the change the Jacobian, which is this thing, 21 00:01:00,460 --> 00:01:03,750 we could write that as a ghost Lagrangian. 22 00:01:03,750 --> 00:01:07,030 And the argument for why we don't need to worry about this 23 00:01:07,030 --> 00:01:07,795 is as follows. 24 00:01:13,880 --> 00:01:15,940 So the effective field theory is going 25 00:01:15,940 --> 00:01:19,180 to be valid for small momentum. 26 00:01:19,180 --> 00:01:24,280 There's an expansion and there's some scale introduced 27 00:01:24,280 --> 00:01:27,310 by the higher dimensional operators, lamda new, 28 00:01:27,310 --> 00:01:31,120 and we're looking at low momentum relative to that. 29 00:01:31,120 --> 00:01:34,150 In the particular case that we're dealing with, 30 00:01:34,150 --> 00:01:37,100 this parameter eta had dimensions, 31 00:01:37,100 --> 00:01:40,570 and so lambda nu was 1 over root eta. 32 00:01:40,570 --> 00:01:43,312 And we put factors of eta in front of our higher dimension 33 00:01:43,312 --> 00:01:44,770 operators, so that's like putting 1 34 00:01:44,770 --> 00:01:47,230 over lambda nus in front of those operators. 35 00:01:49,695 --> 00:01:51,070 And the reason that we don't have 36 00:01:51,070 --> 00:01:52,960 to worry so much about this ghost Lagrangian 37 00:01:52,960 --> 00:01:55,240 is that these ghosts are going to get mass 38 00:01:55,240 --> 00:01:57,220 that's of the size lambda nu. 39 00:02:20,200 --> 00:02:23,577 So what that means is that the ghost, along with perhaps 40 00:02:23,577 --> 00:02:25,660 other particles that are up at the scale lamda new 41 00:02:25,660 --> 00:02:27,860 are things that we don't have to worry about, 42 00:02:27,860 --> 00:02:30,010 and we can effectively integrate out 43 00:02:30,010 --> 00:02:32,440 the ghost from the theory in the same way 44 00:02:32,440 --> 00:02:34,720 that we would think about removing some particles that 45 00:02:34,720 --> 00:02:37,120 had masses of order lambda nu. 46 00:02:37,120 --> 00:02:39,100 So I'll show you why they get masses 47 00:02:39,100 --> 00:02:40,510 of order lambda nu in a second. 48 00:03:09,340 --> 00:03:12,310 So let's do that by way of picking a particular example. 49 00:03:12,310 --> 00:03:14,880 So far, we've kept things fairly general without specifying 50 00:03:14,880 --> 00:03:18,060 what this set of fields it's in this thing 51 00:03:18,060 --> 00:03:19,980 that we called T was. 52 00:03:19,980 --> 00:03:22,740 We just left it as an arbitrary thing. 53 00:03:22,740 --> 00:03:36,640 Let's pick a particular T. The argument here is actually 54 00:03:36,640 --> 00:03:39,340 fairly general, but I think it'd be easier to see it 55 00:03:39,340 --> 00:03:41,390 for particular example. 56 00:03:41,390 --> 00:03:47,240 So if we pick this to be T, we have 57 00:03:47,240 --> 00:03:51,146 this term which has no relation to T. That term is important. 58 00:03:54,428 --> 00:03:55,970 And then we have the terms from this. 59 00:04:04,512 --> 00:04:05,970 And so we'd have a ghost Lagrangian 60 00:04:05,970 --> 00:04:08,830 that would look like that. 61 00:04:08,830 --> 00:04:11,220 So this here is going to be the mass term, 62 00:04:11,220 --> 00:04:14,520 and so far, it doesn't look like it has the right dimensions. 63 00:04:14,520 --> 00:04:18,930 And that's because we've got our kinetic term, if you like, 64 00:04:18,930 --> 00:04:21,310 with the wrong dimensions. 65 00:04:21,310 --> 00:04:26,550 So if we want ghost fields of standard dimensions, 66 00:04:26,550 --> 00:04:29,950 or canonically normalized kinetic term in this case, 67 00:04:29,950 --> 00:04:34,230 we would take c and rescale it to c over root eta. 68 00:04:34,230 --> 00:04:38,850 And if we do that, then this does become a lambda nu. 69 00:04:49,080 --> 00:04:51,480 So that removes the eta in the interaction term here. 70 00:04:51,480 --> 00:04:54,340 It removes it from this kinetic term that happened to be there, 71 00:04:54,340 --> 00:04:56,970 and then the only place that the eta's showing up now 72 00:04:56,970 --> 00:04:59,520 in this case is in this term. 73 00:04:59,520 --> 00:05:02,550 But now you see explicitly in terms of these ghost fields 74 00:05:02,550 --> 00:05:04,688 that they have mass lambda nu. 75 00:05:14,936 --> 00:05:17,340 OK, and that's just what I was claiming. 76 00:05:20,360 --> 00:05:23,313 So the important point here was that you did need this 1. 77 00:05:23,313 --> 00:05:24,980 This 1 was playing a very important role 78 00:05:24,980 --> 00:05:25,688 in this argument. 79 00:05:25,688 --> 00:05:27,890 If the 1 wasn't there, this wouldn't work. 80 00:05:33,140 --> 00:05:34,850 And that 1 was related to something 81 00:05:34,850 --> 00:05:38,850 that was part of one of our starting assumptions, 82 00:05:38,850 --> 00:05:40,680 that when we made the field redefinition, 83 00:05:40,680 --> 00:05:44,230 that the one-particle states would stay the same. 84 00:05:44,230 --> 00:05:52,310 So if you look back at what our field redefinition was and you 85 00:05:52,310 --> 00:06:00,517 trace back where that 1 came from, 86 00:06:00,517 --> 00:06:02,600 we had a field redefinition that was of this form, 87 00:06:02,600 --> 00:06:04,910 and this guy was a function of other fields. 88 00:06:04,910 --> 00:06:07,400 But we always had this term, and that term 89 00:06:07,400 --> 00:06:09,290 was related to the presence of this 1, 90 00:06:09,290 --> 00:06:12,650 and that was important for this argument, OK. 91 00:06:34,350 --> 00:06:39,080 So there has to be a term that's linear in the new field. 92 00:06:39,080 --> 00:06:43,730 You want the same one-particle states. 93 00:06:43,730 --> 00:06:46,580 So these ghosts, as you know about ghosts, 94 00:06:46,580 --> 00:06:47,780 always appear in loops. 95 00:06:51,490 --> 00:06:53,240 And so if you want to think about removing 96 00:06:53,240 --> 00:06:55,460 this massive particle, it's like a massive particle 97 00:06:55,460 --> 00:06:56,600 that occurs in some loops. 98 00:07:00,200 --> 00:07:03,620 It's really just like that. 99 00:07:03,620 --> 00:07:05,440 So it's just like a heavy particle 100 00:07:05,440 --> 00:07:07,190 that you would remove from the theory that 101 00:07:07,190 --> 00:07:08,315 would only appear in loops. 102 00:07:12,110 --> 00:07:14,900 And we're going to discuss exactly how that works 103 00:07:14,900 --> 00:07:18,320 and how to remove heavy particles that appear in loops 104 00:07:18,320 --> 00:07:22,280 in detail a little bit later, so for now, 105 00:07:22,280 --> 00:07:26,007 that's enough detail for number two. 106 00:07:26,007 --> 00:07:27,590 So are there any questions about that? 107 00:07:32,570 --> 00:07:35,740 OK, so the change to the Jacobian 108 00:07:35,740 --> 00:07:38,170 is effectively a massive ghost that we could integrate out 109 00:07:38,170 --> 00:07:38,753 of the theory. 110 00:07:38,753 --> 00:07:40,750 When we integrate it out of the theory, 111 00:07:40,750 --> 00:07:45,010 it would shift the values of the couplings in the Lagrangian, 112 00:07:45,010 --> 00:07:48,190 but it doesn't have any impact that 113 00:07:48,190 --> 00:07:49,920 can't be absorbed in other operators 114 00:07:49,920 --> 00:07:50,920 in the effective theory. 115 00:07:57,633 --> 00:08:00,175 So then the final thing we have to worry about is the source. 116 00:08:05,900 --> 00:08:10,510 So there is this term with j phi dagger, and when we take-- 117 00:08:10,510 --> 00:08:12,280 there was an extra term with j phi dagger 118 00:08:12,280 --> 00:08:13,687 that we had last time. 119 00:08:13,687 --> 00:08:15,520 I'm going to remind you what it looked like. 120 00:08:18,790 --> 00:08:28,520 So in our path integral, we had an exponential 121 00:08:28,520 --> 00:08:30,610 of a bunch of stuff, but then there 122 00:08:30,610 --> 00:08:37,820 was one final term that was induced that involved j phi 123 00:08:37,820 --> 00:08:41,200 dagger from the field redefinition. 124 00:08:41,200 --> 00:08:43,840 So when you take derivatives with respect to j phi dagger, 125 00:08:43,840 --> 00:08:46,270 there's sort of the usual term that we want to source, 126 00:08:46,270 --> 00:08:47,833 and then there's this new term. 127 00:08:47,833 --> 00:08:49,000 So what about that new term? 128 00:08:52,000 --> 00:08:54,950 That's the final thing we have to worry about. 129 00:08:54,950 --> 00:08:57,220 And this is where it's important that we're actually 130 00:08:57,220 --> 00:09:00,070 considering observables. 131 00:09:00,070 --> 00:09:03,850 So we need to consider observables. 132 00:09:03,850 --> 00:09:08,328 So let's start by considering some Green's functions 133 00:09:08,328 --> 00:09:09,370 or time-ordered products. 134 00:09:23,210 --> 00:09:25,150 So I'm just taking a string of fields. 135 00:09:25,150 --> 00:09:28,120 I could have some other fields here as well. 136 00:09:28,120 --> 00:09:32,560 I'm going to make my life a little easier by taking the phi 137 00:09:32,560 --> 00:09:37,980 to be real, and that's just for notational simplicity. 138 00:09:41,490 --> 00:09:43,920 I could write phi dagger and phis, but. 139 00:09:56,010 --> 00:09:59,767 So when we make the field redefinition, what happens 140 00:09:59,767 --> 00:10:00,975 to that time-ordered product? 141 00:10:08,020 --> 00:10:10,697 So this is one way of thinking about these extra terms 142 00:10:10,697 --> 00:10:11,280 in the source. 143 00:10:16,672 --> 00:10:18,630 So you take functional derivatives with respect 144 00:10:18,630 --> 00:10:21,457 to the other piece that involves j phi, you get phi. 145 00:10:21,457 --> 00:10:23,040 So if you take [INAUDIBLE] that piece, 146 00:10:23,040 --> 00:10:24,082 you get the extra eta Ts. 147 00:10:26,760 --> 00:10:29,880 Or you can just think about making the field redefinition 148 00:10:29,880 --> 00:10:34,350 directly in this matrix element, and you'd 149 00:10:34,350 --> 00:10:37,488 have a matrix element that looks like that. 150 00:10:37,488 --> 00:10:39,780 So it's not at all obvious that these extra pieces that 151 00:10:39,780 --> 00:10:42,900 involve fields can't change the value of this Green's function, 152 00:10:42,900 --> 00:10:45,880 and in fact, they will change the value of that Green's 153 00:10:45,880 --> 00:10:47,753 function. 154 00:10:47,753 --> 00:10:49,170 But the important thing is that we 155 00:10:49,170 --> 00:10:54,790 have to consider observables and not just Green's functions. 156 00:10:54,790 --> 00:10:56,790 So when we're going to think about observables, 157 00:10:56,790 --> 00:11:02,940 we should think about the LSZ formula, which connects Green's 158 00:11:02,940 --> 00:11:05,010 functions to S-matrix elements. 159 00:11:07,750 --> 00:11:09,180 So let me remind you about that. 160 00:11:18,300 --> 00:11:20,850 And I'll remind you what it looks like. 161 00:11:20,850 --> 00:11:22,560 First one is fixed set of fields, 162 00:11:22,560 --> 00:11:25,870 which are scalar fields, but we could put fermions in it 163 00:11:25,870 --> 00:11:28,800 as well. 164 00:11:28,800 --> 00:11:32,180 So what this says is, if I look at this-- 165 00:11:32,180 --> 00:11:34,160 so it's an integral over the spacetime. 166 00:11:34,160 --> 00:11:35,930 I'm sticking in particular momentum 167 00:11:35,930 --> 00:11:37,870 for the particular fields. 168 00:11:37,870 --> 00:11:41,570 And if I look at it, if I look at the leading term, 169 00:11:41,570 --> 00:11:51,620 the leading pole term as the particles are taken on shell, 170 00:11:51,620 --> 00:11:54,680 then that's an observable known as the S-matrix. 171 00:12:15,040 --> 00:12:17,200 So some number of these particles are incoming. 172 00:12:17,200 --> 00:12:19,150 Some number of them are outgoing. 173 00:12:19,150 --> 00:12:22,360 That affects the sign that I put in this plus minus. 174 00:12:22,360 --> 00:12:24,955 I'm not trying to be too detailed about which ones are 175 00:12:24,955 --> 00:12:27,080 outgoing, which ones are incoming, but some of them 176 00:12:27,080 --> 00:12:30,280 are incoming, some of them are outgoing. 177 00:12:30,280 --> 00:12:37,143 And for each one, I want to strip off the leading pole. 178 00:12:37,143 --> 00:12:38,560 I want to look at the coefficient, 179 00:12:38,560 --> 00:12:40,390 and it has to have a pole, so this is 180 00:12:40,390 --> 00:12:42,280 a product over all particles. 181 00:12:42,280 --> 00:12:46,390 We really need to pull in all the different cases and all 182 00:12:46,390 --> 00:12:51,460 for all the external particles, OK. 183 00:12:51,460 --> 00:12:54,100 So if I wanted to make this an equality, 184 00:12:54,100 --> 00:12:56,380 I'd say plus dot dot dot. 185 00:12:56,380 --> 00:13:00,310 There's other terms, and the thing that's observable 186 00:13:00,310 --> 00:13:03,130 is the coefficient of these poles, not 187 00:13:03,130 --> 00:13:05,115 the Z but this thing. 188 00:13:09,480 --> 00:13:12,870 So if I make changes to my theory and they effect this 189 00:13:12,870 --> 00:13:15,607 thing but get cancelled by this thing, 190 00:13:15,607 --> 00:13:17,690 as long as they don't affect this thing, we're OK. 191 00:13:21,610 --> 00:13:24,850 And the claim is that we make this change to the source 192 00:13:24,850 --> 00:13:28,467 and it won't affect the matrix on the S, the S-matrix. 193 00:13:58,810 --> 00:14:01,440 So again, we can do some examples, 194 00:14:01,440 --> 00:14:05,520 and I think the examples are fairly quickly convincing 195 00:14:05,520 --> 00:14:07,950 that this is the case. 196 00:14:07,950 --> 00:14:10,980 So we'll do three different examples, 197 00:14:10,980 --> 00:14:14,160 first a rather trivial one. 198 00:14:14,160 --> 00:14:15,630 What if T was just by itself? 199 00:14:18,250 --> 00:14:23,910 So T is just phi, so this is just 1 plus eta phi. 200 00:14:34,760 --> 00:14:37,535 So if you think back, when we have T, 201 00:14:37,535 --> 00:14:39,410 this was the form of the terminal Lagrangian, 202 00:14:39,410 --> 00:14:43,640 so this is just eta phi del squared phi, 203 00:14:43,640 --> 00:14:47,900 so it's just changing the kinetic term for phi. 204 00:14:47,900 --> 00:14:54,590 If we look at our matrix on it, we're 205 00:14:54,590 --> 00:15:01,050 just getting a factor of 1 plus eta for each of the fields. 206 00:15:01,050 --> 00:15:04,160 So let's say we had 4 of them just so we don't have to-- 207 00:15:07,760 --> 00:15:11,380 so say I had 4 of them. 208 00:15:11,380 --> 00:15:13,642 If I had n of them, it would just be to the nth power, 209 00:15:13,642 --> 00:15:16,100 but if I have power of them, they'd be to the fourth power, 210 00:15:16,100 --> 00:15:18,170 and I'd just get this extra prefactor. 211 00:15:18,170 --> 00:15:20,870 And it looks like it's changed G. 212 00:15:20,870 --> 00:15:23,390 And indeed, it has changed the left-hand side of this 213 00:15:23,390 --> 00:15:27,920 equation, but when we calculate the z factor and we canonically 214 00:15:27,920 --> 00:15:29,700 normalize-- 215 00:15:29,700 --> 00:15:31,650 if you want to think about it that way-- 216 00:15:31,650 --> 00:15:35,190 then we would exactly cancel off these 1 plus etas. 217 00:15:35,190 --> 00:15:38,420 So the root Z here is going to be 1 plus eta as well. 218 00:15:48,210 --> 00:15:49,700 So when we look at the-- 219 00:15:49,700 --> 00:15:52,010 this is just the residue of the free propagator. 220 00:15:52,010 --> 00:15:57,255 The residue gets changed when you change the kinetic term, 221 00:15:57,255 --> 00:16:00,597 so that doesn't do anything to the S-matrix. 222 00:16:04,107 --> 00:16:05,690 So that's one way we can be protected. 223 00:16:05,690 --> 00:16:06,898 We change the left-hand side. 224 00:16:06,898 --> 00:16:10,730 It's compensated by a change to the Z and leaves S invariant. 225 00:16:10,730 --> 00:16:14,390 Let's do another example, a little more non-trivial. 226 00:16:21,080 --> 00:16:29,890 So let's just take some cubic term in our field redefinition. 227 00:16:36,950 --> 00:16:39,500 So this is like having a 5 cubed del squared phi term. 228 00:16:46,400 --> 00:16:49,960 So this extra term will give rise 229 00:16:49,960 --> 00:16:52,540 to extra terms in the primary product. 230 00:16:52,540 --> 00:16:54,700 And let me just write one of them, 231 00:16:54,700 --> 00:16:59,390 again thinking of it as a 4-point function. 232 00:16:59,390 --> 00:17:01,750 So let's just imagine I look at the change that comes 233 00:17:01,750 --> 00:17:04,510 from changing the 4th guy. 234 00:17:07,839 --> 00:17:12,530 So there's other terms from changing 5x1, 5x2 and 5x3, 235 00:17:12,530 --> 00:17:16,119 but if I work to order eta, then I'd change one of them 236 00:17:16,119 --> 00:17:21,339 at a time, and we're only working to order eta here. 237 00:17:21,339 --> 00:17:25,180 So the claim is actually that this matrix element has 238 00:17:25,180 --> 00:17:28,150 no effect on this structure. 239 00:17:28,150 --> 00:17:31,335 It can affect the dots, but it's not 240 00:17:31,335 --> 00:17:32,710 going to affect the leading term, 241 00:17:32,710 --> 00:17:35,590 and that's because having a 5 cubed 242 00:17:35,590 --> 00:17:38,120 means that you're not having a one-particle state. 243 00:17:38,120 --> 00:17:43,000 So if you try to draw it as a [INAUDIBLE] diagram, 244 00:17:43,000 --> 00:17:46,810 in the position space, you'd label these external points 245 00:17:46,810 --> 00:17:49,300 and you inject momenta there. 246 00:17:49,300 --> 00:17:51,920 And then we label point 4, but there's three fields there, 247 00:17:51,920 --> 00:17:55,830 so maybe we have to tie it up like this or something. 248 00:17:55,830 --> 00:17:58,810 And when you look at asymptotically what's 249 00:17:58,810 --> 00:18:01,000 going to happen from [INAUDIBLE] a 5 cubed, 250 00:18:01,000 --> 00:18:03,640 you do not get a single-particle pole from a 5 cubed. 251 00:18:07,800 --> 00:18:11,415 So this guy is less singular. 252 00:18:19,750 --> 00:18:27,420 There's no single-particle pole, and hence, 253 00:18:27,420 --> 00:18:28,980 gives no contribution to scatter. 254 00:18:37,720 --> 00:18:38,563 Yeah. 255 00:18:38,563 --> 00:18:40,105 AUDIENCE: As far as the theorem goes, 256 00:18:40,105 --> 00:18:45,010 this step that you're showing, it proves my hypothesis, right? 257 00:18:45,010 --> 00:18:47,315 IAN STEWART: In what sense? 258 00:18:47,315 --> 00:18:48,940 AUDIENCE: The hypothesis of the theorem 259 00:18:48,940 --> 00:18:51,315 is that [INAUDIBLE] does not change a one-particle state. 260 00:18:51,315 --> 00:18:52,398 IAN STEWART: That's right. 261 00:18:52,398 --> 00:18:53,650 That was an assumption. 262 00:18:53,650 --> 00:18:57,220 AUDIENCE: I mean, is that the same as what you're-- 263 00:18:57,220 --> 00:18:59,297 IAN STEWART: It's the same, yeah. 264 00:18:59,297 --> 00:19:03,083 It's effectively the same, though it's 265 00:19:03,083 --> 00:19:05,500 being more careful about what the statement of that means, 266 00:19:05,500 --> 00:19:09,400 right, because I mean, you're changing the residue 267 00:19:09,400 --> 00:19:13,360 but you're not changing the S-matrix. 268 00:19:13,360 --> 00:19:14,162 AUDIENCE: Right. 269 00:19:14,162 --> 00:19:16,120 This is what I was thinking when you said that. 270 00:19:16,120 --> 00:19:18,070 IAN STEWART: Yeah, all right. 271 00:19:25,470 --> 00:19:27,390 So really, this statement of-- 272 00:19:27,390 --> 00:19:30,708 yeah, I could have been a little more careful. 273 00:19:30,708 --> 00:19:33,000 So instead of saying that the hypothesis of the theorem 274 00:19:33,000 --> 00:19:35,580 is that it wouldn't change one-particle states, what 275 00:19:35,580 --> 00:19:38,550 I should have really said is that there's a linear term 276 00:19:38,550 --> 00:19:40,880 in the field redefinition. 277 00:19:40,880 --> 00:19:46,160 Yeah, which now I'm showing you is equivalent to not changing 278 00:19:46,160 --> 00:19:48,740 the one-particle states. 279 00:19:48,740 --> 00:19:54,560 And let's do one final example just 280 00:19:54,560 --> 00:19:56,510 to rule out all possible things we could 281 00:19:56,510 --> 00:19:58,530 think of that are different. 282 00:19:58,530 --> 00:20:03,860 So what if we had a derivative [INAUDIBLE] squared phi prime? 283 00:20:03,860 --> 00:20:05,660 So then we would do the following. 284 00:20:16,230 --> 00:20:19,337 We could add and subtract a mass term. 285 00:20:19,337 --> 00:20:21,920 The reason we would want to do that is, if we looked at a term 286 00:20:21,920 --> 00:20:25,643 like this, it's, again, less singular than this term. 287 00:20:25,643 --> 00:20:27,560 If this term is giving the one particle state, 288 00:20:27,560 --> 00:20:30,860 this term has got a factor of the propagator upstairs. 289 00:20:30,860 --> 00:20:33,800 So if you look for a pole that would come from that term, 290 00:20:33,800 --> 00:20:34,700 it's canceling out. 291 00:20:34,700 --> 00:20:36,260 You get p squared minus m squared 292 00:20:36,260 --> 00:20:37,760 over p squared minus m squared. 293 00:20:41,130 --> 00:20:43,980 So there's no pole from this term. 294 00:20:43,980 --> 00:20:47,030 And this term is, again, just of the type 295 00:20:47,030 --> 00:20:48,840 from our first example. 296 00:20:48,840 --> 00:20:51,520 It's just shifting 5 by some constant. 297 00:20:51,520 --> 00:20:54,650 So probably to get dimensions right, 298 00:20:54,650 --> 00:20:56,760 I should put some factors of eta in here. 299 00:21:03,450 --> 00:21:06,268 So again, something like that doesn't change anything, 300 00:21:06,268 --> 00:21:08,060 because it can be decomposed into something 301 00:21:08,060 --> 00:21:10,070 that has no pole and then something 302 00:21:10,070 --> 00:21:14,020 that's just a shift, OK? 303 00:21:14,020 --> 00:21:16,540 So as long as you have this linear term in your field 304 00:21:16,540 --> 00:21:21,608 redefinition, and it doesn't have to even have a trivial-- 305 00:21:21,608 --> 00:21:23,650 it doesn't have to even have a trivial prefactor. 306 00:21:23,650 --> 00:21:27,310 You could have 2 times phi or whatever you like, 307 00:21:27,310 --> 00:21:30,400 because that'll cancel out when you normalize things correctly. 308 00:21:30,400 --> 00:21:33,550 As long as you have that linear term, you're fine. 309 00:21:33,550 --> 00:21:37,570 And you don't have to worry about changes to the Jacobian 310 00:21:37,570 --> 00:21:41,287 or changes from the source term. 311 00:21:41,287 --> 00:21:43,870 So you just have the changes to the Lagrangian to worry about. 312 00:21:43,870 --> 00:21:45,870 You don't have to think about the path integral. 313 00:21:45,870 --> 00:21:48,893 Just make field redefinitions on the Lagrangian. 314 00:21:48,893 --> 00:21:50,560 And that's what you'll get some practice 315 00:21:50,560 --> 00:21:54,170 with on the problem set. 316 00:21:54,170 --> 00:21:57,400 So as I said at the beginning before everybody was here, 317 00:21:57,400 --> 00:21:59,362 there's a problem set number 1 that's posted. 318 00:21:59,362 --> 00:22:01,820 Everyone should make sure that they can actually access it. 319 00:22:01,820 --> 00:22:04,480 And if they can't access it, they should let me know-- 320 00:22:04,480 --> 00:22:08,110 if you can't get access to the web page. 321 00:22:08,110 --> 00:22:11,600 Any final questions about this before we move on? 322 00:22:11,600 --> 00:22:13,660 Yeah? 323 00:22:13,660 --> 00:22:15,160 AUDIENCE: I get nervous about trying 324 00:22:15,160 --> 00:22:18,490 to cancel a pole by adding an m squared term, because then, I 325 00:22:18,490 --> 00:22:20,220 guess, at some point, you start to worry 326 00:22:20,220 --> 00:22:22,210 that masses get shifted when you start [? normalizing these. ?] 327 00:22:22,210 --> 00:22:23,127 IAN STEWART: Oh, yeah. 328 00:22:23,127 --> 00:22:24,640 Yeah. 329 00:22:24,640 --> 00:22:27,790 Yeah, you should be careful about that, too. 330 00:22:27,790 --> 00:22:29,830 But you don't have to worry about it. 331 00:22:36,210 --> 00:22:36,710 Yeah. 332 00:22:43,300 --> 00:22:46,630 You could think of it as doing the mass renormalization first 333 00:22:46,630 --> 00:22:48,940 and then worry about this, or-- 334 00:22:58,373 --> 00:23:00,290 so that you're using a renormalized mass here. 335 00:23:06,310 --> 00:23:06,810 All right. 336 00:23:12,720 --> 00:23:14,160 So we'll start a new section. 337 00:23:35,052 --> 00:23:36,760 And the goal of this section is basically 338 00:23:36,760 --> 00:23:40,240 to be more careful about loop diagrams 339 00:23:40,240 --> 00:23:41,740 and, in particular, to show you what 340 00:23:41,740 --> 00:23:44,590 matching is between two effective field theories. 341 00:23:44,590 --> 00:23:48,190 We'll still be thinking in the context of mass of particles. 342 00:23:48,190 --> 00:23:51,130 That's the simplest place to describe this 343 00:23:51,130 --> 00:23:55,510 that we'll probably do an example later on of a case that 344 00:23:55,510 --> 00:23:57,310 doesn't just involve mass of particles. 345 00:24:00,590 --> 00:24:02,800 So let's start out with a very trivial example just 346 00:24:02,800 --> 00:24:05,140 to see what we're talking about. 347 00:24:07,970 --> 00:24:09,155 So we'll take two particles. 348 00:24:09,155 --> 00:24:10,405 One of them is a heavy scalar. 349 00:24:17,660 --> 00:24:20,865 It has mass capital M. Actually, I 350 00:24:20,865 --> 00:24:22,240 put a little underline so you can 351 00:24:22,240 --> 00:24:26,230 tell my capitals from my lowercase, 352 00:24:26,230 --> 00:24:28,150 because the light fermion that we're also 353 00:24:28,150 --> 00:24:37,560 going to have has a lowercase mass, little m. 354 00:24:37,560 --> 00:24:38,060 OK. 355 00:24:38,060 --> 00:24:41,450 So heavy scalar, light fermion. 356 00:24:41,450 --> 00:24:43,450 And so what we want to talk about here 357 00:24:43,450 --> 00:24:47,920 is really this picture that we described to you earlier 358 00:24:47,920 --> 00:24:51,700 where we've got two theories, 1 and 2, 359 00:24:51,700 --> 00:24:53,920 and we want to pass from theory 1 to 2 360 00:24:53,920 --> 00:24:56,800 by removing something from theory 1. 361 00:24:56,800 --> 00:25:00,590 So theory 1 here will be just the theory of these things. 362 00:25:00,590 --> 00:25:03,670 And we'll make it a renormalizable theory, 363 00:25:03,670 --> 00:25:07,150 in the traditional sense, just so we know where to stop. 364 00:25:18,320 --> 00:25:19,700 And this guy should be capital. 365 00:25:23,825 --> 00:25:25,700 And then there's some [INAUDIBLE] interaction 366 00:25:25,700 --> 00:25:27,910 between them. 367 00:25:27,910 --> 00:25:32,390 So we'll think of this as being the theory 1, 368 00:25:32,390 --> 00:25:36,110 where we're in a situation where capital M is 369 00:25:36,110 --> 00:25:37,413 much bigger than little m. 370 00:25:41,577 --> 00:25:44,160 So then we want to think about describing psi at low energies. 371 00:25:47,220 --> 00:25:49,140 And that means we can get rid of the scalar 372 00:25:49,140 --> 00:25:50,820 as an explicit degree of freedom. 373 00:25:58,900 --> 00:26:01,510 So low energy is relative to this mass scale, 374 00:26:01,510 --> 00:26:03,495 which is heavy-- 375 00:26:03,495 --> 00:26:17,370 M, capital M. We're going to remove 376 00:26:17,370 --> 00:26:19,440 the scalar from the theory. 377 00:26:19,440 --> 00:26:21,375 So you could just say remove it, or you 378 00:26:21,375 --> 00:26:22,500 could say integrate it out. 379 00:26:22,500 --> 00:26:25,920 When you say integrate out, the words you're using 380 00:26:25,920 --> 00:26:27,630 have a path-integral connotation, 381 00:26:27,630 --> 00:26:30,120 where you had this [INAUDIBLE] in the path integral, 382 00:26:30,120 --> 00:26:32,820 and you think about just doing the path 383 00:26:32,820 --> 00:26:35,565 integral over that field and removing it. 384 00:26:35,565 --> 00:26:37,815 But the words of "removing it" or "integrating it out" 385 00:26:37,815 --> 00:26:40,230 are synonymous. 386 00:26:40,230 --> 00:26:42,000 So what is theory 2 going to look like? 387 00:26:45,007 --> 00:26:47,340 Well, we'll still have the kinetic term for our fermion. 388 00:26:52,800 --> 00:26:56,370 And then removing the scalar will 389 00:26:56,370 --> 00:26:57,840 generate some new operators. 390 00:26:57,840 --> 00:27:01,260 In particular, there'll be a dimension-six operator 391 00:27:01,260 --> 00:27:03,220 like this. 392 00:27:03,220 --> 00:27:07,295 And then there could well be other terms. 393 00:27:07,295 --> 00:27:08,670 And so if we wanted to figure out 394 00:27:08,670 --> 00:27:11,520 what this dimension-six operator is at tree level, 395 00:27:11,520 --> 00:27:14,580 that's a pretty straightforward exercise. 396 00:27:14,580 --> 00:27:21,000 We would simply think about the Feynman diagram in theory 1. 397 00:27:21,000 --> 00:27:25,780 So here's a Feynman diagram in theory 1 with [INAUDIBLE] 398 00:27:25,780 --> 00:27:28,933 couplings here. 399 00:27:28,933 --> 00:27:30,600 We would calculate this Feynman diagram. 400 00:27:40,202 --> 00:27:41,910 And we would assume that all the momentum 401 00:27:41,910 --> 00:27:44,520 of the external particles are small. 402 00:27:44,520 --> 00:27:46,628 And that means that this is small. 403 00:27:46,628 --> 00:27:47,670 And we would just expand. 404 00:28:02,130 --> 00:28:05,040 And then we would just ask that, when I take the Feynman 405 00:28:05,040 --> 00:28:06,150 rule from this-- 406 00:28:06,150 --> 00:28:08,940 whoops, should have been a square there-- 407 00:28:08,940 --> 00:28:10,830 when I take the Feynman rule from that, 408 00:28:10,830 --> 00:28:14,970 then I should get the same thing as the Feynman rule from that. 409 00:28:14,970 --> 00:28:16,980 And that would fix this coefficient a, 410 00:28:16,980 --> 00:28:18,990 which is, so far, arbitrary. 411 00:28:18,990 --> 00:28:21,450 But we can determine it by using theory 1. 412 00:28:21,450 --> 00:28:24,720 And that's the idea of doing a matching calculation that you 413 00:28:24,720 --> 00:28:26,550 have introduced the theory 2. 414 00:28:26,550 --> 00:28:28,650 It has some parameters in front of operators. 415 00:28:28,650 --> 00:28:31,420 In this case, I called it a. 416 00:28:31,420 --> 00:28:33,730 And we want to determine those parameters by doing 417 00:28:33,730 --> 00:28:35,890 calculations in theory 1. 418 00:28:35,890 --> 00:28:37,330 And in particular, what you do is 419 00:28:37,330 --> 00:28:40,690 you make sure that S-matrix elements in the two theories 420 00:28:40,690 --> 00:28:42,430 agree. 421 00:28:42,430 --> 00:28:44,605 But at tree level, that's just matching up diagrams. 422 00:29:16,840 --> 00:29:18,460 So a is simply g squared. 423 00:29:29,310 --> 00:29:31,470 So we just have to match up this guy with that guy. 424 00:29:31,470 --> 00:29:36,770 And so taking the leading-order term, a is just g squared. 425 00:29:36,770 --> 00:29:39,392 Very simple. 426 00:29:39,392 --> 00:29:41,600 So the place where this gets more involved and more-- 427 00:29:41,600 --> 00:29:43,790 where you have to think a little bit more-- this 428 00:29:43,790 --> 00:29:45,650 seems almost automatic, that you just do 429 00:29:45,650 --> 00:29:48,740 calculations in here and here, and just match them up. 430 00:29:48,740 --> 00:29:49,982 And that's the basic idea. 431 00:29:49,982 --> 00:29:51,440 It's only a little more complicated 432 00:29:51,440 --> 00:29:53,510 when you have to take into account loops. 433 00:29:53,510 --> 00:29:56,600 So that's what we'll spend most of this section discussing, 434 00:29:56,600 --> 00:29:59,790 since this part is easier. 435 00:29:59,790 --> 00:30:00,635 So what about loops? 436 00:30:03,612 --> 00:30:05,570 What are some of the issues that come up there? 437 00:30:09,260 --> 00:30:12,222 Well, the first one is that the Feynman diagrams diverge, 438 00:30:12,222 --> 00:30:13,430 so you have to regulate them. 439 00:30:24,810 --> 00:30:27,520 And the thing that can be confusing about thinking 440 00:30:27,520 --> 00:30:30,020 about effective field theories and thinking about divergence 441 00:30:30,020 --> 00:30:33,660 diagrams is separating the ideas of regularization 442 00:30:33,660 --> 00:30:38,280 and the mass scale lambda nu that we've been talking about. 443 00:30:38,280 --> 00:30:41,660 So that's something I want to talk 444 00:30:41,660 --> 00:30:49,910 about in some detail, because they're not the same thing. 445 00:30:52,675 --> 00:30:54,800 So we would need to cut off ultraviolet divergences 446 00:30:54,800 --> 00:30:57,020 to obtain finite results. 447 00:30:57,020 --> 00:31:01,970 And that means we introduce cutoff parameters 448 00:31:01,970 --> 00:31:03,185 into our results. 449 00:31:12,480 --> 00:31:20,580 So examples would be taking the Minkowski momentum, 450 00:31:20,580 --> 00:31:23,070 continuing it to be Euclidean, and then just putting 451 00:31:23,070 --> 00:31:26,970 a hard cutoff on it-- that's one example. 452 00:31:26,970 --> 00:31:29,520 Or you could use dim reg. 453 00:31:29,520 --> 00:31:30,870 That's another example. 454 00:31:30,870 --> 00:31:32,340 Or you could use a lattice spacing. 455 00:31:32,340 --> 00:31:34,540 That's another example. 456 00:31:34,540 --> 00:31:36,390 So these are all examples of how you 457 00:31:36,390 --> 00:31:39,537 might cut off the theory to remove ultraviolet divergences. 458 00:31:43,120 --> 00:31:51,340 And then there's a second step, renormalization, 459 00:31:51,340 --> 00:31:53,110 distinct from regularization. 460 00:31:56,140 --> 00:31:58,030 So some of the things I'm teaching you here 461 00:31:58,030 --> 00:31:59,650 should be familiar. 462 00:31:59,650 --> 00:32:02,170 And the only real generalization that we're going to do 463 00:32:02,170 --> 00:32:04,390 is we're going to be able to apply some of the things 464 00:32:04,390 --> 00:32:05,973 that you learned about renormalization 465 00:32:05,973 --> 00:32:07,600 and regularization to any operators 466 00:32:07,600 --> 00:32:09,017 that you might have in the theory, 467 00:32:09,017 --> 00:32:13,030 whether they're renormalizable in the dimension-four 468 00:32:13,030 --> 00:32:16,850 traditional sense or in a higher-dimensional sense. 469 00:32:16,850 --> 00:32:19,660 So here, what we're doing when we talk about renormalization 470 00:32:19,660 --> 00:32:22,930 is we're picking a scheme that gives 471 00:32:22,930 --> 00:32:27,870 precise or definite meaning to the parameters 472 00:32:27,870 --> 00:32:28,870 in the effective theory. 473 00:32:33,220 --> 00:32:38,530 Or, even more explicitly, each coefficient in the Lagrangian, 474 00:32:38,530 --> 00:32:44,230 as well as the operators in the Lagrangian, 475 00:32:44,230 --> 00:32:49,450 are given a meaning by this procedure. 476 00:32:49,450 --> 00:32:51,850 If we didn't have a renormalization procedure, 477 00:32:51,850 --> 00:32:54,640 then, because of the ultraviolet divergences, 478 00:32:54,640 --> 00:32:57,190 there'd be ambiguities in how we define the coefficients 479 00:32:57,190 --> 00:32:59,410 and the operators. 480 00:32:59,410 --> 00:33:01,720 Or, even if you don't have ultraviolet divergences, 481 00:33:01,720 --> 00:33:05,290 you still have freedom to pick different schemes 482 00:33:05,290 --> 00:33:09,760 for the definitions of the coefficients. 483 00:33:09,760 --> 00:33:14,080 And when you do this, you also can introduce parameters. 484 00:33:18,933 --> 00:33:20,850 So you can get parameters from regularization. 485 00:33:20,850 --> 00:33:24,130 You can also get parameters from this. 486 00:33:24,130 --> 00:33:26,670 So some examples that are familiar-- 487 00:33:26,670 --> 00:33:29,205 there's this scale mu that shows up when you do MS bar. 488 00:33:32,485 --> 00:33:34,110 If you do something that's a little bit 489 00:33:34,110 --> 00:33:36,870 different, if you take Green's functions, 490 00:33:36,870 --> 00:33:40,290 and you go to some offshell point, 491 00:33:40,290 --> 00:33:45,300 that's another renormalization scheme called offshell momentum 492 00:33:45,300 --> 00:33:47,460 subtraction. 493 00:33:47,460 --> 00:33:54,100 And that scheme also has a parameter that shows up. 494 00:33:54,100 --> 00:33:58,820 And if you did a Wilsonian renormalization, 495 00:33:58,820 --> 00:34:02,580 there would also be a cutoff associated to that. 496 00:34:02,580 --> 00:34:05,115 And this Wilsonian cutoff here doesn't 497 00:34:05,115 --> 00:34:06,990 have to be the same as this lambda uv cutoff. 498 00:34:14,040 --> 00:34:14,540 OK. 499 00:34:14,540 --> 00:34:16,130 So every coefficient, every operator, 500 00:34:16,130 --> 00:34:18,409 no matter the dimension, no matter 501 00:34:18,409 --> 00:34:21,650 where it turns up in the series, we're 502 00:34:21,650 --> 00:34:23,900 going to have to think about whether there's 503 00:34:23,900 --> 00:34:26,750 diagrams that generate ultraviolet divergences. 504 00:34:26,750 --> 00:34:31,363 And if those diagrams look like these operators, 505 00:34:31,363 --> 00:34:33,030 then you're going to get renormalization 506 00:34:33,030 --> 00:34:33,822 of those operators. 507 00:34:33,822 --> 00:34:37,350 And you're going to have to be careful about taking care 508 00:34:37,350 --> 00:34:39,725 of those divergences and also the scheme 509 00:34:39,725 --> 00:34:41,100 that you define the operators in. 510 00:34:46,020 --> 00:34:49,409 So you can think about this, just as far 511 00:34:49,409 --> 00:34:59,061 as the coefficients are concerned, 512 00:34:59,061 --> 00:35:04,710 as starting out with some bare coefficients that 513 00:35:04,710 --> 00:35:07,860 depend on your ultraviolet regulator, 514 00:35:07,860 --> 00:35:11,282 and switching to some renormalized coefficients which 515 00:35:11,282 --> 00:35:12,990 don't depend on the ultraviolet regulator 516 00:35:12,990 --> 00:35:15,810 but do depend on the scheme. 517 00:35:15,810 --> 00:35:18,855 And then, also, you have some counterterms 518 00:35:18,855 --> 00:35:19,980 that depend on both things. 519 00:35:23,230 --> 00:35:24,960 So that's how it would look if you 520 00:35:24,960 --> 00:35:30,090 wrote that down for a cutoff, in a Wilsonian sense. 521 00:35:30,090 --> 00:35:34,140 If you wrote it down in dimensional regularization, 522 00:35:34,140 --> 00:35:35,190 it would look like this. 523 00:35:39,630 --> 00:35:43,102 Same idea, different names for the parameters. 524 00:35:46,760 --> 00:35:47,260 OK. 525 00:35:47,260 --> 00:35:48,635 So in dimensional regularization, 526 00:35:48,635 --> 00:35:50,620 epsilon is the ultraviolet regulator. 527 00:35:50,620 --> 00:35:53,320 It shows up in the bare coefficients. 528 00:35:53,320 --> 00:35:57,400 The renormalized ones don't depend on epsilon anymore. 529 00:35:57,400 --> 00:35:59,140 And these are your 1-over-epsilon poles. 530 00:36:05,340 --> 00:36:06,900 All right. 531 00:36:06,900 --> 00:36:09,480 So one of the things that makes this tricky 532 00:36:09,480 --> 00:36:11,970 is when you start thinking about your power counting. 533 00:36:11,970 --> 00:36:17,700 And so I want to spend a few minutes talking about that. 534 00:36:27,850 --> 00:36:30,240 So let's consider this example that I wrote down 535 00:36:30,240 --> 00:36:33,370 with the four-fermion operator. 536 00:36:33,370 --> 00:36:37,290 And if we just loop up the four-fermion operator, 537 00:36:37,290 --> 00:36:39,552 then we can get mass renormalization. 538 00:36:42,510 --> 00:36:45,600 So this is psi dagger psi squared. 539 00:36:45,600 --> 00:36:53,920 And I'm sticking it in and looping it up. 540 00:36:59,470 --> 00:37:02,265 So we start out with a tree-level mass, little m. 541 00:37:02,265 --> 00:37:03,640 But there's a one-loop correction 542 00:37:03,640 --> 00:37:05,807 that involves one of our higher-dimension operators. 543 00:37:08,343 --> 00:37:10,010 And there's some connection to the mass, 544 00:37:10,010 --> 00:37:12,240 which I'll call delta m. 545 00:37:12,240 --> 00:37:14,740 I'm not going to worry too much about the overall prefactor. 546 00:37:17,980 --> 00:37:21,260 This operator came with an a over M squared. 547 00:37:21,260 --> 00:37:24,370 And then there's a loop integral. 548 00:37:24,370 --> 00:37:30,014 There's a fermionic propagator, so that's a k slash plus m 549 00:37:30,014 --> 00:37:31,690 over k squared minus m squared. 550 00:37:35,290 --> 00:37:38,980 This guy here drops away. 551 00:37:38,980 --> 00:37:41,230 And really, we just have a correction that's a scalar. 552 00:37:41,230 --> 00:37:47,060 And that's a correction to the mass scalar in the spin space. 553 00:37:47,060 --> 00:37:51,100 So it's a, little m, over capital M, and then 554 00:37:51,100 --> 00:37:53,840 just this integral. 555 00:37:53,840 --> 00:37:59,950 And if we continue it to be Euclidean, 556 00:37:59,950 --> 00:38:04,510 then it looks like that if I continue from Minkowski 557 00:38:04,510 --> 00:38:05,860 to Euclidean. 558 00:38:05,860 --> 00:38:09,440 That's why I put the i there. 559 00:38:09,440 --> 00:38:12,340 So if we just assume that this integral that's 560 00:38:12,340 --> 00:38:15,730 sitting there-- there's only one mass scale that seems 561 00:38:15,730 --> 00:38:17,740 explicit there-- the little m. 562 00:38:17,740 --> 00:38:20,110 So if we just assume that that integral is 563 00:38:20,110 --> 00:38:21,460 dominated by the scale m-- 564 00:38:30,040 --> 00:38:38,360 little m-- then you could do a power counting 565 00:38:38,360 --> 00:38:39,860 for this loop integral. 566 00:38:39,860 --> 00:38:41,930 You just assume that all the factors of k 567 00:38:41,930 --> 00:38:42,970 are of order little m. 568 00:38:42,970 --> 00:38:46,140 You've got an explicit little m, four powers in the numerator, 569 00:38:46,140 --> 00:38:48,740 and two powers downstairs. 570 00:38:48,740 --> 00:38:55,610 So you would say that this integral scales like m squared. 571 00:38:58,310 --> 00:39:02,240 And then you would put that together with your delta m 572 00:39:02,240 --> 00:39:08,090 and say that delta m scales like a, little m cubed, capital M 573 00:39:08,090 --> 00:39:09,330 squared. 574 00:39:09,330 --> 00:39:11,780 And so this is something that would then be suppressed. 575 00:39:11,780 --> 00:39:14,030 Relative to the original tree-level contribution, 576 00:39:14,030 --> 00:39:16,028 it would be suppressed by, perhaps, 577 00:39:16,028 --> 00:39:17,570 some [? 4 pi ?] [? is in ?] the loop, 578 00:39:17,570 --> 00:39:20,570 but also little m squared over big M squared. 579 00:39:24,690 --> 00:39:29,490 So it would be a small correction, which is really 580 00:39:29,490 --> 00:39:31,952 what we would like to be the case when we're talking 581 00:39:31,952 --> 00:39:33,660 about some higher-dimension operator that 582 00:39:33,660 --> 00:39:34,910 was supposed to be suppressed. 583 00:39:34,910 --> 00:39:39,120 We'd like it to be a small correction. 584 00:39:39,120 --> 00:39:40,500 All right. 585 00:39:40,500 --> 00:39:43,128 So does anyone have any idea what can go wrong 586 00:39:43,128 --> 00:39:43,920 with this argument? 587 00:39:47,970 --> 00:39:49,180 I've been glib about it. 588 00:39:49,180 --> 00:39:52,200 I just said, if the integral is dominated by that. 589 00:39:56,270 --> 00:39:58,020 So the thing is that we have to be careful 590 00:39:58,020 --> 00:40:00,970 about our choice of regulator, because regulators also 591 00:40:00,970 --> 00:40:01,845 can introduce scales. 592 00:40:13,910 --> 00:40:15,759 So let's do this more carefully. 593 00:40:33,564 --> 00:40:35,490 Well, let's consider two different choices 594 00:40:35,490 --> 00:40:37,110 for the regulator. 595 00:40:37,110 --> 00:40:40,967 So we'll start out with just a cutoff, which 596 00:40:40,967 --> 00:40:43,050 seems very natural, from an effective field theory 597 00:40:43,050 --> 00:40:46,200 point of view-- that your theory is supposed to be only valid up 598 00:40:46,200 --> 00:40:48,030 to some scale. 599 00:40:48,030 --> 00:40:52,080 So why not just take and cut off the momentum explicitly 600 00:40:52,080 --> 00:40:53,100 above some scale? 601 00:40:59,060 --> 00:41:02,210 And you can think of that scale as being of order big M. 602 00:41:02,210 --> 00:41:04,520 So we just don't include any momentum 603 00:41:04,520 --> 00:41:07,050 that are higher than that scale in our loop integrals. 604 00:41:07,050 --> 00:41:09,772 So we're only integrating over the region in momentum space 605 00:41:09,772 --> 00:41:11,480 where the theory is supposed to be valid. 606 00:41:11,480 --> 00:41:13,820 It seems like a perfectly reasonable thing to do. 607 00:41:27,260 --> 00:41:30,070 So in this case, we can take this integral that 608 00:41:30,070 --> 00:41:33,250 has some angular parts to it as well as a radial part, 609 00:41:33,250 --> 00:41:36,610 decompose it into the radial piece and the angular piece. 610 00:41:40,480 --> 00:41:44,830 I'll give you a handout, or I'll post a page of lecture notes 611 00:41:44,830 --> 00:41:49,090 that have all those fun formulas that are useful to remember 612 00:41:49,090 --> 00:41:54,040 but not fun to talk about for decomposing integrals 613 00:41:54,040 --> 00:42:00,302 in arbitrary dimensions into radial and angular pieces. 614 00:42:00,302 --> 00:42:02,260 So if we do that, in this case, then the radial 615 00:42:02,260 --> 00:42:04,390 integral-- we can cut off the radial integral 616 00:42:04,390 --> 00:42:07,210 with a hard cutoff lambda uv. 617 00:42:07,210 --> 00:42:12,520 And just by dimensions, the d 4 k becomes kE times kE cubed. 618 00:42:12,520 --> 00:42:15,673 So this is a radial k now. 619 00:42:15,673 --> 00:42:17,215 And this integral, we can do exactly. 620 00:42:24,340 --> 00:42:27,460 There's a loop factor, 4 pi squared. 621 00:42:27,460 --> 00:42:30,340 There's a lambda uv squared. 622 00:42:30,340 --> 00:42:33,091 And then there's a logarithm. 623 00:42:33,091 --> 00:42:36,910 It goes like, lambda uv squared over little m squared. 624 00:42:42,443 --> 00:42:44,110 There's two scales that are showing up-- 625 00:42:44,110 --> 00:42:45,760 lambda uv and little m. 626 00:42:45,760 --> 00:42:47,708 The answer depends on both of them. 627 00:42:47,708 --> 00:42:49,750 And then we can start expanding, because little m 628 00:42:49,750 --> 00:42:51,667 is supposed to be much smaller than lambda uv. 629 00:43:00,890 --> 00:43:04,970 Let me pull out an a m over 4 pi squared. 630 00:43:15,270 --> 00:43:18,530 So there would be a logarithmic term. 631 00:43:18,530 --> 00:43:20,420 There's this lambda uv squared over capital 632 00:43:20,420 --> 00:43:24,401 M squared if I put that factor in. 633 00:43:24,401 --> 00:43:25,707 And then there's some-- 634 00:43:31,920 --> 00:43:33,920 so everything I'm writing in the square brackets 635 00:43:33,920 --> 00:43:36,200 here is dimensionless. 636 00:43:36,200 --> 00:43:37,880 This has the right dimensions of a mass. 637 00:43:40,950 --> 00:43:41,880 OK. 638 00:43:41,880 --> 00:43:45,090 So you can see that, if I do this, 639 00:43:45,090 --> 00:43:47,460 it's not satisfying what I said over here. 640 00:43:47,460 --> 00:43:50,910 I'd like the correction to be m cubed over M squared. 641 00:43:50,910 --> 00:43:54,360 There are corrections that go like m cubed over M squared, 642 00:43:54,360 --> 00:43:55,600 like this one. 643 00:43:55,600 --> 00:43:57,130 These ones are even higher. 644 00:43:57,130 --> 00:43:59,370 But there's this term. 645 00:43:59,370 --> 00:44:01,392 And that's not a small correction. 646 00:44:13,200 --> 00:44:17,460 So for that particular piece, what you're finding 647 00:44:17,460 --> 00:44:19,410 is that your power counting-- your naive power 648 00:44:19,410 --> 00:44:20,785 counting-- of the loop was wrong, 649 00:44:20,785 --> 00:44:23,020 because there's a piece from k of order-- 650 00:44:23,020 --> 00:44:28,760 the cutoff that's contributing for that part of it. 651 00:44:31,330 --> 00:44:35,100 And that's why your naive scaling argument didn't work. 652 00:44:35,100 --> 00:44:38,400 Now, this is the bare result. And we 653 00:44:38,400 --> 00:44:40,690 have to go through the renormalization procedure. 654 00:44:45,760 --> 00:44:48,030 And so if we go through the renormalization procedure, 655 00:44:48,030 --> 00:44:50,857 you can think of that as taking a piece of the integral-- 656 00:44:55,250 --> 00:44:58,770 so in a Wilsonian sense, you would 657 00:44:58,770 --> 00:45:02,440 take a piece of the integral and absorb it into the counterterm. 658 00:45:07,840 --> 00:45:10,020 So as promised, the counterterm depends 659 00:45:10,020 --> 00:45:11,430 on both lambda uv and lambda. 660 00:45:11,430 --> 00:45:14,432 And those are just explicitly-- 661 00:45:14,432 --> 00:45:16,140 I'm cutting off the radial integral here. 662 00:45:21,210 --> 00:45:23,670 And that does improve things, because then, I can lower 663 00:45:23,670 --> 00:45:27,610 this cutoff, make it smaller. 664 00:45:27,610 --> 00:45:31,920 And what is left would just be lambda squared over capital M 665 00:45:31,920 --> 00:45:39,880 squared and log of m squared over capital lambda squared. 666 00:45:39,880 --> 00:45:42,780 So I change the lambda uv's up here into lambdas 667 00:45:42,780 --> 00:45:44,231 by that procedure. 668 00:45:51,780 --> 00:45:53,580 So when I renormalize, the psi bar psi 669 00:45:53,580 --> 00:45:56,570 squared matrix element-- there's a correction from that 670 00:45:56,570 --> 00:45:57,260 to the mass. 671 00:45:57,260 --> 00:46:01,520 And the renormalized thing would depend on this Wilsonian cutoff 672 00:46:01,520 --> 00:46:04,040 lambda, OK? 673 00:46:04,040 --> 00:46:06,230 But I'm not fully getting around the issue 674 00:46:06,230 --> 00:46:07,880 that I'm generating this type of term. 675 00:46:15,020 --> 00:46:21,770 So let's do it with a different regulator, which 676 00:46:21,770 --> 00:46:23,420 is dimensional regularization. 677 00:46:23,420 --> 00:46:41,840 Let's see what happens using the MS-bar scheme. 678 00:46:41,840 --> 00:46:42,920 Same calculation. 679 00:46:57,170 --> 00:47:04,100 So again, split it into radial and angular parts. 680 00:47:04,100 --> 00:47:06,150 And again, I'll give you formulas 681 00:47:06,150 --> 00:47:07,400 for doing something like that. 682 00:47:09,992 --> 00:47:11,690 But I won't write them down in lecture. 683 00:47:18,160 --> 00:47:21,280 Actually, I think I will write one down a little later. 684 00:47:21,280 --> 00:47:24,690 But I'll give you a more complete set as a handout. 685 00:47:34,532 --> 00:47:35,990 So leaving over the radial integral 686 00:47:35,990 --> 00:47:37,520 but regulating it dimensionally just 687 00:47:37,520 --> 00:47:40,290 means that, instead of having this guy to the cubed power, 688 00:47:40,290 --> 00:47:41,930 it's to the d minus 1. 689 00:47:41,930 --> 00:47:45,290 And there's some pieces here that depend on epsilon. 690 00:47:45,290 --> 00:47:47,400 And my convention for the entire course 691 00:47:47,400 --> 00:47:50,850 is that d is 4 minus 2 epsilon, not d minus epsilon. 692 00:47:50,850 --> 00:47:55,390 So again, this is an integral we can do exactly. 693 00:48:19,540 --> 00:48:21,942 We get something like that. 694 00:48:21,942 --> 00:48:22,900 And then we can expand. 695 00:48:37,176 --> 00:48:40,330 We get a 1-over-epsilon pole. 696 00:48:40,330 --> 00:48:41,340 We get a logarithm. 697 00:48:43,930 --> 00:48:46,850 There is some constant as well. 698 00:48:46,850 --> 00:48:49,180 And then there's order-epsilon pieces. 699 00:48:53,240 --> 00:48:57,435 So you should contrast this result here 700 00:48:57,435 --> 00:48:59,730 with the result we had over here. 701 00:48:59,730 --> 00:49:03,017 The order-epsilon pieces are like these terms. 702 00:49:03,017 --> 00:49:05,600 They're the terms that would go away if I took the cutoff here 703 00:49:05,600 --> 00:49:09,350 to infinity-- lambda uv to infinity or epsilon to 0. 704 00:49:09,350 --> 00:49:12,660 There's terms here that go like m cubed over M squared. 705 00:49:12,660 --> 00:49:14,485 That's like this term. 706 00:49:14,485 --> 00:49:15,610 And this term is not there. 707 00:49:22,750 --> 00:49:26,200 This 1 over epsilon is related to the log divergence. 708 00:49:26,200 --> 00:49:28,960 And this thing here is a power divergence. 709 00:49:28,960 --> 00:49:32,080 So the 1 over epsilon is related to this log of lambda uv. 710 00:49:32,080 --> 00:49:33,760 You should think of log of lambda uv 711 00:49:33,760 --> 00:49:38,288 as going to 1 over epsilon plus log of mu squared. 712 00:49:38,288 --> 00:49:39,830 But we don't see the power divergence 713 00:49:39,830 --> 00:49:46,140 in dimensional regularization in MS bar. 714 00:49:46,140 --> 00:49:46,640 OK. 715 00:49:46,640 --> 00:49:50,750 So what's happening is, from this result, 716 00:49:50,750 --> 00:50:02,450 we are getting something that's the size that we expected 717 00:50:02,450 --> 00:50:04,070 by our power-counting argument. 718 00:50:04,070 --> 00:50:09,560 The regularized result is the right size by power counting. 719 00:50:09,560 --> 00:50:21,300 The logarithm here is the same log as in a with, if you like, 720 00:50:21,300 --> 00:50:24,660 a correspondence between mu and lambda. 721 00:50:24,660 --> 00:50:26,927 So we're seeing that logarithm there. 722 00:50:26,927 --> 00:50:28,260 We're just not seeing this term. 723 00:50:37,070 --> 00:50:41,440 And if we wanted to write down the MS-bar counterterm. 724 00:50:41,440 --> 00:50:47,935 Some correction to the mass, it would go like, a, m squared. 725 00:50:53,610 --> 00:50:56,775 The diagram would look like just the 1-over-epsilon pole 726 00:50:56,775 --> 00:50:59,710 in MS bar. 727 00:50:59,710 --> 00:51:00,210 OK. 728 00:51:00,210 --> 00:51:04,290 So the two regulators seem to give us similar results but not 729 00:51:04,290 --> 00:51:06,420 exactly-identical results. 730 00:51:06,420 --> 00:51:09,340 And the question is, how should I think about this? 731 00:51:09,340 --> 00:51:11,670 What's the right way of thinking about this extra term? 732 00:51:14,990 --> 00:51:16,540 So there's a language that people 733 00:51:16,540 --> 00:51:19,510 use in effective field theory and dimensional 734 00:51:19,510 --> 00:51:21,670 counting for the difference between thinking 735 00:51:21,670 --> 00:51:28,110 about doing a regularization of type A and type B. 736 00:51:28,110 --> 00:51:31,290 And what you say is that, in type B, 737 00:51:31,290 --> 00:51:35,670 your regularization of the problem 738 00:51:35,670 --> 00:51:52,330 does not break the power counting, which 739 00:51:52,330 --> 00:51:57,400 means that you can do power counting for regulated graphs 740 00:51:57,400 --> 00:51:58,750 prior to renormalization them. 741 00:52:08,250 --> 00:52:13,978 But you can see that, in case A, that would be problematic, 742 00:52:13,978 --> 00:52:16,270 because our naive power counting wouldn't have given us 743 00:52:16,270 --> 00:52:20,770 the right result. 744 00:52:20,770 --> 00:52:22,424 So what do we say about case A? 745 00:52:28,635 --> 00:52:30,690 Well, if case B didn't break the power counting, 746 00:52:30,690 --> 00:52:31,590 then case A does. 747 00:52:35,970 --> 00:52:38,820 But you can say something a little bit more positive 748 00:52:38,820 --> 00:52:45,600 about case A. And that is that, when you think about case A, 749 00:52:45,600 --> 00:52:51,390 you can set up the theory with renormalization conditions 750 00:52:51,390 --> 00:52:57,390 such that you can restore power counting 751 00:52:57,390 --> 00:53:00,720 in the renormalized graphs and renormalized couplings 752 00:53:00,720 --> 00:53:02,697 and renormalized operators. 753 00:53:02,697 --> 00:53:04,780 But you don't have power counting-- explicit power 754 00:53:04,780 --> 00:53:07,715 counting-- in just the regularized results. 755 00:53:07,715 --> 00:53:09,090 So you can think that I just take 756 00:53:09,090 --> 00:53:10,890 the Wilsonian cutoff small enough that I 757 00:53:10,890 --> 00:53:15,990 make that annoying term small. 758 00:53:15,990 --> 00:53:18,630 And order by order, I can do that-- 759 00:53:21,180 --> 00:53:22,860 order by order in my loop expansion, 760 00:53:22,860 --> 00:53:24,480 order by order in my calculations. 761 00:53:31,730 --> 00:53:32,340 OK. 762 00:53:32,340 --> 00:53:34,130 So in some sense, there's nothing wrong 763 00:53:34,130 --> 00:53:35,960 with doing A. It's just that you have 764 00:53:35,960 --> 00:53:38,060 to work a little harder to think about it. 765 00:53:38,060 --> 00:53:40,213 And when you think about the power counting, 766 00:53:40,213 --> 00:53:42,380 you have to do that for the renormalized quantities, 767 00:53:42,380 --> 00:53:45,707 not for the bare quantities. 768 00:53:45,707 --> 00:53:47,290 So if you like, what you're doing here 769 00:53:47,290 --> 00:53:49,540 is you're adding counterterms. 770 00:53:49,540 --> 00:53:55,670 Another way you can say this is that you're adding counterterms 771 00:53:55,670 --> 00:53:59,060 to restore the power counting that you want. 772 00:54:02,940 --> 00:54:05,142 And that's not too different than-- 773 00:54:05,142 --> 00:54:06,600 you could always do that, actually. 774 00:54:06,600 --> 00:54:08,340 I made this analogy that you should 775 00:54:08,340 --> 00:54:14,310 think about power counting as being like gauge symmetry. 776 00:54:14,310 --> 00:54:17,550 So let's say you had a theory, and it had nice gauge symmetry, 777 00:54:17,550 --> 00:54:19,560 but you picked some crazy regulator 778 00:54:19,560 --> 00:54:21,805 that broke gauge symmetry. 779 00:54:21,805 --> 00:54:23,430 Well, you could always put counterterms 780 00:54:23,430 --> 00:54:24,847 in that would break gauge symmetry 781 00:54:24,847 --> 00:54:26,925 and restore it in the renormalized quantity. 782 00:54:26,925 --> 00:54:28,440 That would be OK. 783 00:54:28,440 --> 00:54:29,850 It would be more work. 784 00:54:29,850 --> 00:54:31,590 We don't like to do that. 785 00:54:31,590 --> 00:54:33,810 We avoid it at all costs. 786 00:54:33,810 --> 00:54:36,600 But if we had to, we could do it. 787 00:54:36,600 --> 00:54:38,748 If you do supersymmetry, you might 788 00:54:38,748 --> 00:54:40,290 try to use dimensional regularization 789 00:54:40,290 --> 00:54:42,540 and supersymmetry, standard dimensional regularization 790 00:54:42,540 --> 00:54:44,880 and supersymmetry, break supersymmetry. 791 00:54:44,880 --> 00:54:47,340 If you do that, you have to introduce counterterms 792 00:54:47,340 --> 00:54:49,140 that restore supersymmetry. 793 00:54:49,140 --> 00:54:52,240 So the same language of symmetry is being applied here, 794 00:54:52,240 --> 00:54:54,720 except now to power counting, where 795 00:54:54,720 --> 00:54:57,780 we say that if your regulator messes up your power counting, 796 00:54:57,780 --> 00:55:00,420 you can restore it in the renormalized couplings. 797 00:55:00,420 --> 00:55:02,790 But you may be smart enough to think up a regulator-- 798 00:55:02,790 --> 00:55:05,130 in this case, dimensional regularization-- 799 00:55:05,130 --> 00:55:08,210 where you don't have to deal with that complication. 800 00:55:08,210 --> 00:55:08,710 OK. 801 00:55:20,656 --> 00:55:21,820 So let me write that. 802 00:56:15,070 --> 00:56:16,740 So in an effective theory, you should 803 00:56:16,740 --> 00:56:19,860 think about regulating to preserve symmetries 804 00:56:19,860 --> 00:56:23,130 as well as to preserve power counting, if you can. 805 00:56:36,660 --> 00:56:40,620 And one way, in a more formal language, 806 00:56:40,620 --> 00:56:43,350 that you could say what happens with the generation 807 00:56:43,350 --> 00:56:45,960 of that term that we talked about with the cutoff 808 00:56:45,960 --> 00:56:50,310 is that you'd mix up different orders in the expansion, 809 00:56:50,310 --> 00:56:54,840 and it looks like your naively higher-order term 810 00:56:54,840 --> 00:56:58,325 is mixing back to a term of lower dimension. 811 00:57:02,880 --> 00:57:06,060 And so if you can get away with taking a regulator that doesn't 812 00:57:06,060 --> 00:57:09,510 have that mixing back to more relevant operators, 813 00:57:09,510 --> 00:57:12,720 then you could preserve your power counting, 814 00:57:12,720 --> 00:57:15,730 make it simpler. 815 00:57:15,730 --> 00:57:20,950 This is true irrespective of what the power counting is in. 816 00:57:20,950 --> 00:57:23,800 This is a general statement. 817 00:57:23,800 --> 00:57:25,940 In the context of what we've been talking about, 818 00:57:25,940 --> 00:57:29,230 which is dimensional power counting, 819 00:57:29,230 --> 00:57:31,480 there's a particular phrase that goes along with this. 820 00:57:39,980 --> 00:57:42,820 And that is that we talk about using a mass independent 821 00:57:42,820 --> 00:57:50,415 regulator, like dim reg. 822 00:57:56,845 --> 00:58:00,070 If you like, it has a mass scale, mu, 823 00:58:00,070 --> 00:58:02,260 but it's put in softly in a way that 824 00:58:02,260 --> 00:58:04,360 doesn't mess up power counting. 825 00:58:04,360 --> 00:58:07,780 We call that using a mass independent regulator. 826 00:58:07,780 --> 00:58:11,110 So we want to avoid having different orders 827 00:58:11,110 --> 00:58:13,780 in the expansion mix up with each other. 828 00:58:13,780 --> 00:58:16,810 In general, I should comment, then, 829 00:58:16,810 --> 00:58:18,700 that terms that are the same order 830 00:58:18,700 --> 00:58:23,530 will definitely typically mix up with each other 831 00:58:23,530 --> 00:58:25,990 under renormalization. 832 00:58:25,990 --> 00:58:27,760 So even if you thought you were smart, 833 00:58:27,760 --> 00:58:30,895 and you enumerated all the operators, but you missed one, 834 00:58:30,895 --> 00:58:33,520 and then you started calculating, 835 00:58:33,520 --> 00:58:35,740 that operator might just pop out at you, 836 00:58:35,740 --> 00:58:37,900 because you could have some calculation 837 00:58:37,900 --> 00:58:41,260 with another operator mix into that operator. 838 00:58:41,260 --> 00:58:44,680 Or you could have an operator that you did a matching at tree 839 00:58:44,680 --> 00:58:47,080 level, and you didn't generate, but then you 840 00:58:47,080 --> 00:58:50,060 start renormalizing that tree-level operator, 841 00:58:50,060 --> 00:58:52,925 and another operator pops out at you. 842 00:58:52,925 --> 00:58:55,300 So it's important that you include all the operators that 843 00:58:55,300 --> 00:58:58,893 have the same dimension and the same quantum numbers, 844 00:58:58,893 --> 00:59:00,310 because if you don't include them, 845 00:59:00,310 --> 00:59:04,963 you're bound to get them from loops anyway. 846 00:59:04,963 --> 00:59:06,130 And you want to be complete. 847 00:59:11,140 --> 00:59:18,690 So if you like, in a matrix [INAUDIBLE] notation, 848 00:59:18,690 --> 00:59:22,530 you could say that the bare operators mix up with the set 849 00:59:22,530 --> 00:59:24,420 of possible renormalized operators, 850 00:59:24,420 --> 00:59:27,420 and there's some matrix of counterterms that would correct 851 00:59:27,420 --> 00:59:28,020 them-- 852 00:59:28,020 --> 00:59:31,030 connect them. 853 00:59:31,030 --> 00:59:31,570 OK. 854 00:59:31,570 --> 00:59:33,120 So any questions so far? 855 00:59:37,030 --> 00:59:40,480 So sometimes, in the literature, you'll see that these kinds 856 00:59:40,480 --> 00:59:44,050 of things-- regulator discussions in effective field 857 00:59:44,050 --> 00:59:44,710 theory-- 858 00:59:44,710 --> 00:59:47,860 generate all sorts of papers. 859 00:59:47,860 --> 00:59:50,410 Keep this in mind if you ever run into that. 860 01:00:01,045 --> 01:00:02,920 You should be able to think about the physics 861 01:00:02,920 --> 01:00:04,712 that you're after with different regulators 862 01:00:04,712 --> 01:00:07,500 and come to the same conclusion. 863 01:00:07,500 --> 01:00:12,140 And it just may be easier with one regulator versus another. 864 01:00:12,140 --> 01:00:14,112 So we've said, in these kind of theories 865 01:00:14,112 --> 01:00:15,820 where we have dimensional power counting, 866 01:00:15,820 --> 01:00:21,040 that dimensional regularization is special. 867 01:00:21,040 --> 01:00:23,020 So I want to talk a bit more about 868 01:00:23,020 --> 01:00:24,739 dimensional regularization. 869 01:00:45,123 --> 01:00:47,540 So sometimes you'll hear people say that you should always 870 01:00:47,540 --> 01:00:51,130 use dimensional regularization for doing the power counting. 871 01:00:51,130 --> 01:00:55,250 But that's not quite true in the way that I've told you. 872 01:00:55,250 --> 01:00:56,750 It's not that you have to, it's just 873 01:00:56,750 --> 01:00:58,940 that it makes things simpler. 874 01:00:58,940 --> 01:01:01,650 But given that we want to make things as simple as possible, 875 01:01:01,650 --> 01:01:06,170 let's take dimensional regularization seriously. 876 01:01:06,170 --> 01:01:08,540 So you can actually derive dimensional regularization 877 01:01:08,540 --> 01:01:11,420 by just imposing axioms. 878 01:01:11,420 --> 01:01:16,445 If you say that you want a loop integration that's linear-- 879 01:01:22,260 --> 01:01:24,240 I should have said this earlier. 880 01:01:24,240 --> 01:01:26,190 So my notation with dimensional regularization 881 01:01:26,190 --> 01:01:27,607 is, I put a little cross on the d. 882 01:01:27,607 --> 01:01:30,640 And that means dividing by the 2 pi. 883 01:01:30,640 --> 01:01:36,720 So that means d d p over 2 pi to the d. 884 01:01:36,720 --> 01:01:39,720 So linearity means that if I'm integrating some function that 885 01:01:39,720 --> 01:01:42,360 can be decomposed into a sum of two pieces, a 886 01:01:42,360 --> 01:01:48,680 and b being constants, f and g being functions, 887 01:01:48,680 --> 01:01:55,030 then I can write that out as an integral over f 888 01:01:55,030 --> 01:01:59,590 plus an integral over g, which really is something 889 01:01:59,590 --> 01:02:05,948 that almost every reasonable definition of the integration 890 01:02:05,948 --> 01:02:06,490 will satisfy. 891 01:02:12,090 --> 01:02:18,660 The second one is translations, which is more restricting. 892 01:02:18,660 --> 01:02:21,240 So that says, if you have some integral over f, 893 01:02:21,240 --> 01:02:24,330 but it's a function of p plus q-- q is some external-- 894 01:02:24,330 --> 01:02:30,345 I can always shift away the q. 895 01:02:30,345 --> 01:02:32,205 It just goes-- p goes to p minus q. 896 01:02:32,205 --> 01:02:34,013 And then I just have an integral over p. 897 01:02:37,120 --> 01:02:38,800 And along with translations, you can 898 01:02:38,800 --> 01:02:40,990 think about having rotations. 899 01:02:40,990 --> 01:02:44,560 My whole notation is covariance, so we won't worry so much 900 01:02:44,560 --> 01:02:46,210 about rotations. 901 01:02:46,210 --> 01:02:47,340 and Lorentz group. 902 01:02:51,880 --> 01:02:56,560 And then the final one that's obviously 903 01:02:56,560 --> 01:03:00,370 a little bit special to dim reg is a scaling. 904 01:03:00,370 --> 01:03:04,150 So let's say we have a scalar s multiplying our momentum p. 905 01:03:07,610 --> 01:03:12,290 Then we can rescale the momentum p and get rid of-- 906 01:03:12,290 --> 01:03:18,410 pull the s outside by just taking p goes to p over s. 907 01:03:18,410 --> 01:03:20,000 So that changes this to a p. 908 01:03:20,000 --> 01:03:22,336 We get an s to the minus d. 909 01:03:22,336 --> 01:03:25,140 It pulls out front. 910 01:03:25,140 --> 01:03:29,210 And if we demand that, then that's 911 01:03:29,210 --> 01:03:30,830 special to dimensional regularization, 912 01:03:30,830 --> 01:03:34,760 because you can see that this depends on d. 913 01:03:34,760 --> 01:03:37,340 Even if I call this measure some abstract thing, 914 01:03:37,340 --> 01:03:38,910 now there's a d showing up. 915 01:03:38,910 --> 01:03:40,130 It's outside the measure. 916 01:03:45,600 --> 01:03:48,480 And these three together actually give 917 01:03:48,480 --> 01:03:52,410 a unique definition to the integration up 918 01:03:52,410 --> 01:03:54,285 to the overall normalization. 919 01:04:07,160 --> 01:04:09,263 And that unique thing is dim reg. 920 01:04:12,890 --> 01:04:17,150 So I'm going to refer you to reading, 921 01:04:17,150 --> 01:04:19,640 I have posted a chapter from Collins' book 922 01:04:19,640 --> 01:04:21,410 on regularization. 923 01:04:21,410 --> 01:04:24,860 And around page 65, he talks about how you prove that. 924 01:04:24,860 --> 01:04:25,730 It's not too hard. 925 01:04:34,310 --> 01:04:36,970 The standard definition of the normalization, 926 01:04:36,970 --> 01:04:41,184 which is something you have to specify, 927 01:04:41,184 --> 01:04:44,520 is that you let, say, this Gaussian 928 01:04:44,520 --> 01:04:46,140 integral be pi to the d over 2. 929 01:04:50,580 --> 01:04:55,770 So then, from that, you have some measure in some space 930 01:04:55,770 --> 01:04:56,895 that you can then just use. 931 01:05:01,220 --> 01:05:03,350 So one formula that I used earlier on 932 01:05:03,350 --> 01:05:08,720 was the ability to split that into pieces 933 01:05:08,720 --> 01:05:13,160 which were a radial piece and then an angular piece. 934 01:05:13,160 --> 01:05:15,310 And in general, this is a property 935 01:05:15,310 --> 01:05:17,060 that this integration [? measure obeys, ?] 936 01:05:17,060 --> 01:05:19,393 that you could split it, and you could split it further. 937 01:05:19,393 --> 01:05:28,670 You could pull out another angle, for example, 938 01:05:28,670 --> 01:05:34,810 and get one less dimension in the angular parts. 939 01:05:34,810 --> 01:05:38,110 And the uv divergences, if we're talking about us divergences-- 940 01:05:38,110 --> 01:05:40,500 they're occurring in this radial part-- 941 01:05:40,500 --> 01:05:41,920 Euclidean radial part. 942 01:05:48,070 --> 01:05:50,950 So by thinking about this kind of decomposition, 943 01:05:50,950 --> 01:05:53,620 you're moving the uv divergences to 944 01:05:53,620 --> 01:05:57,280 a one-dimensional integration, at least at one loop. 945 01:05:57,280 --> 01:05:59,635 And you can always do that. 946 01:05:59,635 --> 01:06:01,510 So in general, in dimensional regularization, 947 01:06:01,510 --> 01:06:03,220 there's many ways you could evaluate the integral. 948 01:06:03,220 --> 01:06:04,750 You're not used to using this one. 949 01:06:04,750 --> 01:06:07,030 You're used to keeping things covariant, 950 01:06:07,030 --> 01:06:11,770 using some Feynman parameters, combining propagators together, 951 01:06:11,770 --> 01:06:14,330 and then doing the integral. 952 01:06:14,330 --> 01:06:16,450 But you could also do it this way, 953 01:06:16,450 --> 01:06:19,070 and you get the same answer. 954 01:06:19,070 --> 01:06:22,355 So it's really a well-defined measure in the sense 955 01:06:22,355 --> 01:06:24,230 that you can manipulate it in different ways. 956 01:06:24,230 --> 01:06:26,860 And they should all lead to the same answer for your loop 957 01:06:26,860 --> 01:06:27,769 integrals. 958 01:06:31,763 --> 01:06:33,180 And that's part of what I'm trying 959 01:06:33,180 --> 01:06:37,110 to emphasize by saying that you could derive it 960 01:06:37,110 --> 01:06:38,295 by considering axioms. 961 01:06:46,740 --> 01:06:49,800 So d was equal to 4 minus 2 epsilon. 962 01:06:49,800 --> 01:06:53,070 Epsilon greater than 0 is what you 963 01:06:53,070 --> 01:06:57,554 need to lower the powers of p, and therefore tame the uv. 964 01:06:57,554 --> 01:07:03,630 Epsilon less than 0 can be used to regulate 965 01:07:03,630 --> 01:07:05,307 infrared singularities. 966 01:07:10,542 --> 01:07:12,000 There's some counterintuitive facts 967 01:07:12,000 --> 01:07:13,710 about dimensional regularization, 968 01:07:13,710 --> 01:07:15,730 and I want to mention a couple of them to you. 969 01:07:19,160 --> 01:07:23,280 One of them is that, if I have p to an arbitrary power-- 970 01:07:23,280 --> 01:07:24,650 think of it as Euclidean-- 971 01:07:27,270 --> 01:07:27,870 that's 0. 972 01:07:38,970 --> 01:07:43,950 So Collins constructs a proof of this 973 01:07:43,950 --> 01:07:47,970 on page 71, which is actually a little more involved, 974 01:07:47,970 --> 01:07:50,280 in general. 975 01:07:50,280 --> 01:07:54,270 I'll just give you an idea of how you can see that, 976 01:07:54,270 --> 01:07:56,850 from using our axioms, that something like this 977 01:07:56,850 --> 01:07:58,810 better be true. 978 01:07:58,810 --> 01:08:01,980 So let's consider a special example 979 01:08:01,980 --> 01:08:04,530 that won't be enough to prove it for arbitrary alpha. 980 01:08:04,530 --> 01:08:07,800 This is any alpha. 981 01:08:07,800 --> 01:08:09,450 Let's consider a special example that 982 01:08:09,450 --> 01:08:13,140 at least will be enough to prove it for integers. 983 01:08:16,359 --> 01:08:19,729 So we'll consider k's that are integers. 984 01:08:19,729 --> 01:08:23,550 And we'll think of k's that are greater than 0. 985 01:08:23,550 --> 01:08:32,240 So if I just expand out this p plus q squared, 986 01:08:32,240 --> 01:08:35,640 then the first term is p to the 2k. 987 01:08:35,640 --> 01:08:39,900 Then I get some coefficient, p to the 2k minus 2, 988 01:08:39,900 --> 01:08:46,500 q squared, some coefficient, p to the 2k minus 4, 989 01:08:46,500 --> 01:08:50,550 q to the fourth, et cetera. 990 01:08:50,550 --> 01:08:52,600 In general, there's p dot q terms as well. 991 01:08:52,600 --> 01:08:54,830 But then I could do integral-- 992 01:08:54,830 --> 01:08:56,880 angular average and combine those together 993 01:08:56,880 --> 01:08:58,840 with these terms. 994 01:08:58,840 --> 01:09:01,140 And that's why I'm not being very explicit about what 995 01:09:01,140 --> 01:09:03,069 the coefficients are. 996 01:09:03,069 --> 01:09:06,367 But they're some positive numbers. 997 01:09:06,367 --> 01:09:08,700 Now, I could also take this integral, and I could shift. 998 01:09:08,700 --> 01:09:09,825 That was one of our axioms. 999 01:09:18,192 --> 01:09:21,359 And that is p to the 2k. 1000 01:09:21,359 --> 01:09:25,200 So that means that all these terms here better be 0. 1001 01:09:25,200 --> 01:09:41,279 And they have to be 0 for arbitrary q and arbitrary k, 1002 01:09:41,279 --> 01:09:43,478 or any k under the assumptions that we used, 1003 01:09:43,478 --> 01:09:45,520 which are that it's an integer and it's positive. 1004 01:09:45,520 --> 01:09:47,979 So I could expand it in this way. 1005 01:09:47,979 --> 01:09:50,790 And so therefore, we have all these integrals over p 1006 01:09:50,790 --> 01:09:51,450 to the powers. 1007 01:09:51,450 --> 01:09:52,920 And they better be 0. 1008 01:09:52,920 --> 01:09:56,610 And that's enough to prove this for integer alphas-- 1009 01:09:56,610 --> 01:09:57,780 positive integer alphas. 1010 01:10:08,660 --> 01:10:10,280 OK. 1011 01:10:10,280 --> 01:10:12,380 And so if you want to fill in between the integers 1012 01:10:12,380 --> 01:10:14,030 and you want to do the negative cases, 1013 01:10:14,030 --> 01:10:15,405 then you have to look at Collins. 1014 01:10:15,405 --> 01:10:18,575 But you could do that, too. 1015 01:10:18,575 --> 01:10:20,450 Then it requires a little more heavy lifting. 1016 01:10:28,000 --> 01:10:31,000 There's one fact about this-- 1017 01:10:31,000 --> 01:10:33,490 fact number one-- which is a little bit 1018 01:10:33,490 --> 01:10:36,580 subtle, and you have to be careful. 1019 01:10:36,580 --> 01:10:39,500 And it's worth noting. 1020 01:10:39,500 --> 01:10:42,910 So let me do another example, which 1021 01:10:42,910 --> 01:10:46,120 is by way of warning you that this can sometimes 1022 01:10:46,120 --> 01:10:48,080 be dangerous. 1023 01:10:48,080 --> 01:10:50,740 So let's think of a scalar-field theory 1024 01:10:50,740 --> 01:10:52,810 and a simple loop diagram like this. 1025 01:10:52,810 --> 01:10:57,055 But let's take 0 momentum and 0 mass. 1026 01:11:00,032 --> 01:11:01,990 So if you do that, you'll encounter an integral 1027 01:11:01,990 --> 01:11:03,620 that looks like this. 1028 01:11:03,620 --> 01:11:05,290 There's two propagators, so I get a p 1029 01:11:05,290 --> 01:11:10,240 to the fourth downstairs, and I get d d p. 1030 01:11:10,240 --> 01:11:15,820 And that integral is 0, but it's 0 in a special way. 1031 01:11:15,820 --> 01:11:18,820 It's 0 due to a cancellation between ultraviolet and 1032 01:11:18,820 --> 01:11:21,850 infrared physics. 1033 01:11:21,850 --> 01:11:23,830 I said that epsilon could be regulating 1034 01:11:23,830 --> 01:11:25,450 both infrared divergences as well 1035 01:11:25,450 --> 01:11:27,790 as ultraviolet divergences. 1036 01:11:27,790 --> 01:11:29,620 If I only used epsilon to regulate 1037 01:11:29,620 --> 01:11:33,190 ultraviolet divergences, I'd get a 1-over-epsilon uv. 1038 01:11:33,190 --> 01:11:36,152 But in this integral, I'm actually using it to do both. 1039 01:11:36,152 --> 01:11:38,110 It's regulating an infrared divergence as well. 1040 01:11:38,110 --> 01:11:39,770 And it just comes in with the opposite sign. 1041 01:11:39,770 --> 01:11:41,645 And since epsilon uv is equal to epsilon IR-- 1042 01:11:44,950 --> 01:11:50,080 they're just notation to signify what region of physics 1043 01:11:50,080 --> 01:11:51,730 is giving the divergence-- 1044 01:11:51,730 --> 01:11:54,405 you get 0. 1045 01:11:54,405 --> 01:11:55,780 But even though that's true, that 1046 01:11:55,780 --> 01:11:59,140 doesn't mean you don't have to add a counterterm 1047 01:11:59,140 --> 01:12:01,690 for this diagram, because counterterms are supposed 1048 01:12:01,690 --> 01:12:06,550 to cancel ultraviolet divergences, not infrared ones, 1049 01:12:06,550 --> 01:12:08,800 OK? 1050 01:12:08,800 --> 01:12:16,870 So even though it's 0, you still need to add a counterterm, 1051 01:12:16,870 --> 01:12:18,910 because the 0 is actually a cancellation 1052 01:12:18,910 --> 01:12:22,760 between ultraviolet and infrared physics. 1053 01:12:22,760 --> 01:12:25,435 So there's some counterterm. 1054 01:12:28,230 --> 01:12:31,620 And it would be exactly of this sort, 1055 01:12:31,620 --> 01:12:34,670 because this is the epsilon uv. 1056 01:12:38,220 --> 01:12:44,160 And then if you add the bare diagram plus the counterterm, 1057 01:12:44,160 --> 01:12:47,460 the answer is non-zero. 1058 01:12:47,460 --> 01:12:50,040 You've canceled, if you like, the uv pole, 1059 01:12:50,040 --> 01:12:53,130 and you've left over the IR pole. 1060 01:12:53,130 --> 01:12:53,630 OK. 1061 01:12:53,630 --> 01:12:55,490 So you have to be a bit careful about using 1062 01:12:55,490 --> 01:12:57,140 dimensional regularization, because if you encounter 1063 01:12:57,140 --> 01:12:59,265 scaleless integrals, it could be that they actually 1064 01:12:59,265 --> 01:13:01,410 are affecting counterterm. 1065 01:13:01,410 --> 01:13:03,740 And if you want to do some renormalization-group 1066 01:13:03,740 --> 01:13:06,020 improvement of the theory or something like that, 1067 01:13:06,020 --> 01:13:08,970 you have to be aware of this. 1068 01:13:08,970 --> 01:13:11,420 If you know that all the infrared poles are going 1069 01:13:11,420 --> 01:13:14,555 to cancel because you're looking at some infrared safe quantity, 1070 01:13:14,555 --> 01:13:16,430 then you can be a little bit glib about this, 1071 01:13:16,430 --> 01:13:18,347 because if they're going to cancel in the end, 1072 01:13:18,347 --> 01:13:21,560 that means that any of these corresponding uv poles 1073 01:13:21,560 --> 01:13:23,570 will also cancel. 1074 01:13:23,570 --> 01:13:26,180 But it's not always the case that you're renormalizing 1075 01:13:26,180 --> 01:13:31,010 operators that have no 1-over-epsilon IR's [INAUDIBLE] 1076 01:13:31,010 --> 01:13:33,300 have to be careful about this. 1077 01:13:33,300 --> 01:13:35,990 So this is a subtlety that sometimes people get wrong 1078 01:13:35,990 --> 01:13:38,960 when they write papers. 1079 01:13:38,960 --> 01:13:41,450 So dim reg is beautiful, but there are some things about it 1080 01:13:41,450 --> 01:13:42,658 that are a little bit tricky. 1081 01:13:46,600 --> 01:13:51,070 Another thing that can be confusing about dim reg 1082 01:13:51,070 --> 01:13:58,053 is that it does this, that it regulates both uv and IR poles. 1083 01:13:58,053 --> 01:14:00,220 And even though it's doing that, and even though you 1084 01:14:00,220 --> 01:14:04,090 need different values of epsilon, 1085 01:14:04,090 --> 01:14:08,500 if you want to do that, it's actually 1086 01:14:08,500 --> 01:14:10,450 still a well-defined procedure, even 1087 01:14:10,450 --> 01:14:13,260 in the presence of uv and IR poles, 1088 01:14:13,260 --> 01:14:15,010 even if they're both in the same integral. 1089 01:14:17,680 --> 01:14:24,190 And basically, you're using analytic continuation here. 1090 01:14:28,930 --> 01:14:31,590 So let me give you a little example which 1091 01:14:31,590 --> 01:14:37,570 is not exactly related to this but will allow me to show you 1092 01:14:37,570 --> 01:14:39,670 both how you use analytic continuation 1093 01:14:39,670 --> 01:14:43,630 and how you could think about separating uv and IR poles. 1094 01:14:43,630 --> 01:14:45,520 So we'll start out just by thinking 1095 01:14:45,520 --> 01:14:47,710 about analytic continuation. 1096 01:14:47,710 --> 01:14:49,240 So suppose I had some integral. 1097 01:14:57,320 --> 01:14:59,620 So what does analytic continuation mean? 1098 01:14:59,620 --> 01:15:02,340 Or, how should I construct it? 1099 01:15:02,340 --> 01:15:04,850 So let me, again, write it in a way where I've separated 1100 01:15:04,850 --> 01:15:05,870 out the radial integral. 1101 01:15:10,250 --> 01:15:12,290 And let's suppose that this integral here 1102 01:15:12,290 --> 01:15:17,130 is perfectly well-defined for d in some range. 1103 01:15:17,130 --> 01:15:29,360 So this is well-defined for some range of positive d. 1104 01:15:29,360 --> 01:15:32,390 And then let's say that we wanted to continue 1105 01:15:32,390 --> 01:15:33,680 that integral to negative d. 1106 01:15:42,705 --> 01:15:44,450 Now, the problem with negative d may 1107 01:15:44,450 --> 01:15:46,160 be that, when you get to negative d, 1108 01:15:46,160 --> 01:15:48,438 you're getting some infrared divergences, 1109 01:15:48,438 --> 01:15:50,480 and you have to figure out how to deal with them. 1110 01:15:58,340 --> 01:16:01,600 And you can do this, if you'd like, step by step. 1111 01:16:04,340 --> 01:16:06,820 So if we wanted to extend the lower limit down to minus 2 1112 01:16:06,820 --> 01:16:09,959 from 0, then we would do the following. 1113 01:16:14,650 --> 01:16:17,740 We take our integral, write it out 1114 01:16:17,740 --> 01:16:27,730 in this angular/radial separation, 1115 01:16:27,730 --> 01:16:33,700 split up, in the radial variable, the piece that's 1116 01:16:33,700 --> 01:16:36,623 ultraviolet, which is the high-momentum piece, 1117 01:16:36,623 --> 01:16:37,915 from the piece that's infrared. 1118 01:16:50,975 --> 01:16:52,350 And in the piece that's infrared, 1119 01:16:52,350 --> 01:16:55,290 we could also just do some addition and some subtraction 1120 01:16:55,290 --> 01:16:57,840 to make it more convergent as p goes to 0. 1121 01:16:57,840 --> 01:17:00,600 So for example, we could subtract f at 0. 1122 01:17:00,600 --> 01:17:03,810 This thing would fall off faster and hence, give 1123 01:17:03,810 --> 01:17:05,110 more powers of p. 1124 01:17:05,110 --> 01:17:09,300 And we could make it more convergent at 0. 1125 01:17:09,300 --> 01:17:15,300 And then we just integrate the subtraction up to the cutoff c. 1126 01:17:17,950 --> 01:17:20,950 And the idea of introducing this cutoff c 1127 01:17:20,950 --> 01:17:23,680 is that we split the uv piece and the ultraviolet piece. 1128 01:17:27,470 --> 01:17:29,650 And so we can do one kind of continuation 1129 01:17:29,650 --> 01:17:32,860 for d up here, making epsilon positive, 1130 01:17:32,860 --> 01:17:34,700 one kind of continuation down here. 1131 01:17:34,700 --> 01:17:36,700 And the result, when we put these back together, 1132 01:17:36,700 --> 01:17:39,370 is independent of c, OK? 1133 01:17:39,370 --> 01:17:43,210 So that's the sense, actually, in which what I said up here 1134 01:17:43,210 --> 01:17:43,750 is-- 1135 01:17:43,750 --> 01:17:46,090 that you can use it for both uv and IR divergences, 1136 01:17:46,090 --> 01:17:48,132 because you could always introduce some parameter 1137 01:17:48,132 --> 01:17:48,932 c to split-- 1138 01:17:48,932 --> 01:17:51,140 they're occurring in different regions of space base, 1139 01:17:51,140 --> 01:17:56,740 so you could always split them up, regulate each one 1140 01:17:56,740 --> 01:18:00,680 with different values of d, and then put them back together. 1141 01:18:00,680 --> 01:18:03,370 And the answer, when you put them back together, 1142 01:18:03,370 --> 01:18:04,360 is independent of c. 1143 01:18:08,040 --> 01:18:08,970 OK. 1144 01:18:08,970 --> 01:18:11,100 Now, if you wanted to define-- 1145 01:18:11,100 --> 01:18:12,953 that was one of our goals. 1146 01:18:12,953 --> 01:18:14,370 The other one was just to show you 1147 01:18:14,370 --> 01:18:16,980 what we mean by analytic continuation. 1148 01:18:16,980 --> 01:18:19,290 So since it's independent of c, then you 1149 01:18:19,290 --> 01:18:21,970 could do the following. 1150 01:18:21,970 --> 01:18:27,390 So for minus 2 less than d less than 0, 1151 01:18:27,390 --> 01:18:30,410 let's take c goes to infinity. 1152 01:18:33,490 --> 01:18:35,640 And so for that particular range, what you 1153 01:18:35,640 --> 01:18:38,340 find, then, if you take that limit-- 1154 01:18:55,540 --> 01:18:57,430 Well, because of the c to the minus d, 1155 01:18:57,430 --> 01:18:59,320 the c is downstairs if d is negative. 1156 01:18:59,320 --> 01:19:03,550 So that term goes away, and you're just 1157 01:19:03,550 --> 01:19:06,860 left with this term. 1158 01:19:06,860 --> 01:19:08,870 This term here goes away, too, because c 1159 01:19:08,870 --> 01:19:12,080 approaches the upper limit, and everything is regulated. 1160 01:19:12,080 --> 01:19:15,750 And so I just would be left with that. 1161 01:19:15,750 --> 01:19:16,250 OK. 1162 01:19:16,250 --> 01:19:17,480 Making d negative is-- 1163 01:19:20,220 --> 01:19:20,720 OK. 1164 01:19:20,720 --> 01:19:22,480 So that would be the definition. 1165 01:19:22,480 --> 01:19:24,188 So you can see some of the kind of tricks 1166 01:19:24,188 --> 01:19:26,272 that you could use with dimensional regularization 1167 01:19:26,272 --> 01:19:27,680 or adding and subtracting terms. 1168 01:19:30,780 --> 01:19:33,050 And these are the things that are valid things to do. 1169 01:19:39,580 --> 01:19:40,800 Any questions about that? 1170 01:19:43,970 --> 01:19:45,410 OK. 1171 01:19:45,410 --> 01:19:48,740 So when we do dimensional regularization in MS bar, 1172 01:19:48,740 --> 01:19:50,720 you're used to doing that for a gauge theory. 1173 01:19:50,720 --> 01:19:53,420 That's what you've learned about. 1174 01:19:53,420 --> 01:19:55,670 But you can also do it for any effective field theory. 1175 01:19:55,670 --> 01:19:57,870 And the logic is the same. 1176 01:19:57,870 --> 01:20:00,770 So let me remind you of the gauge-theory logic 1177 01:20:00,770 --> 01:20:04,490 and then just tell you how you would define MS bar precisely 1178 01:20:04,490 --> 01:20:08,690 for the fermionic effective theory with the "psi bar psi 1179 01:20:08,690 --> 01:20:12,380 squared" operator that we had. 1180 01:20:12,380 --> 01:20:16,560 So we talked about dimensional regularization. 1181 01:20:16,560 --> 01:20:18,190 Let's talk about the MS scheme. 1182 01:20:30,530 --> 01:20:34,550 So if we've set up our effective theory in a way 1183 01:20:34,550 --> 01:20:39,510 where we've made the mass scale explicit, 1184 01:20:39,510 --> 01:20:41,135 which is often a nice thing to do-- 1185 01:20:41,135 --> 01:20:43,260 and we did that when we set up the effective theory 1186 01:20:43,260 --> 01:20:46,303 where you have the capital M showing up explicitly. 1187 01:20:46,303 --> 01:20:47,720 If you do that, then the couplings 1188 01:20:47,720 --> 01:20:49,370 start out dimensionless. 1189 01:20:49,370 --> 01:20:52,640 And that's a nice thing, to have dimensionless coupling. 1190 01:20:52,640 --> 01:20:54,290 And the MS scheme is simply the scheme 1191 01:20:54,290 --> 01:20:56,150 where you want to introduce a scale 1192 01:20:56,150 --> 01:20:59,520 to keep the renormalized couplings dimensionless. 1193 01:20:59,520 --> 01:21:04,490 So the example you're familiar with 1194 01:21:04,490 --> 01:21:06,370 is just having a gauge coupling. 1195 01:21:09,817 --> 01:21:11,900 And if you go through the dimensions of the fields 1196 01:21:11,900 --> 01:21:14,450 here, which I do in my notes, but I'm 1197 01:21:14,450 --> 01:21:17,900 going to assume that you've got some familiarity with this, 1198 01:21:17,900 --> 01:21:22,520 you find that the bare coupling has dimension epsilon. 1199 01:21:22,520 --> 01:21:25,618 And so you define a renormalized coupling as dimensionless. 1200 01:21:29,675 --> 01:21:31,675 And you introduce a factor of mu to the epsilon. 1201 01:21:35,367 --> 01:21:39,790 So you say g bare, which has dimensions, 1202 01:21:39,790 --> 01:21:43,540 some dimensionless z factor, some mu to the epsilon 1203 01:21:43,540 --> 01:21:44,750 to make up those dimensions. 1204 01:21:44,750 --> 01:21:47,710 And then left over is the renormalized coupling. 1205 01:21:54,210 --> 01:21:56,520 And the idea and the strategy for any other coupling 1206 01:21:56,520 --> 01:21:58,020 in the effective theory is the same, 1207 01:21:58,020 --> 01:22:00,400 so let's do one other example. 1208 01:22:00,400 --> 01:22:07,698 So take our [? dimension-six ?] a bare over capital M squared, 1209 01:22:07,698 --> 01:22:10,980 psi bar psi squared. 1210 01:22:10,980 --> 01:22:17,817 Do dimension counting on this guy, which 1211 01:22:17,817 --> 01:22:18,900 I do, again, in the notes. 1212 01:22:18,900 --> 01:22:21,600 But if you go through that dimension 1213 01:22:21,600 --> 01:22:24,270 counting, and you remember that the dimensions for the fermions 1214 01:22:24,270 --> 01:22:26,498 are assigned by the kinetic term-- 1215 01:22:26,498 --> 01:22:28,540 so we have a dimension counting for the fermions. 1216 01:22:28,540 --> 01:22:31,050 We know that this is dimension minus 2. 1217 01:22:31,050 --> 01:22:32,670 The whole thing has to add up to d. 1218 01:22:32,670 --> 01:22:34,950 So that tells us what the a bare is. 1219 01:22:34,950 --> 01:22:38,390 And we get 4 minus d, in this case. 1220 01:22:38,390 --> 01:22:45,000 And so then we can write down a formula analogous to that one 1221 01:22:45,000 --> 01:22:47,490 but for the a coefficient. 1222 01:22:47,490 --> 01:22:48,705 And it's mu to the 2 epsilon. 1223 01:22:53,540 --> 01:22:54,050 OK. 1224 01:22:54,050 --> 01:22:56,060 So it's as simple as that. 1225 01:23:14,010 --> 01:23:15,560 So in looking at the action where 1226 01:23:15,560 --> 01:23:17,180 this is a term in the Lagrangian, 1227 01:23:17,180 --> 01:23:19,520 we want to ensure that, when we go to the-- 1228 01:23:19,520 --> 01:23:21,440 that we figure out the dimensions of this guy 1229 01:23:21,440 --> 01:23:23,120 in dimensional regularization. 1230 01:23:23,120 --> 01:23:26,233 That's 4 minus d, which we determine from knowing 1231 01:23:26,233 --> 01:23:27,150 the other pieces here. 1232 01:23:27,150 --> 01:23:30,840 And then we just make a redefinition 1233 01:23:30,840 --> 01:23:33,810 to give a dimensionless coupling. 1234 01:23:33,810 --> 01:23:36,450 So that's how MS bar will work. 1235 01:23:36,450 --> 01:23:40,920 Well, this is MS, but this is how minimal subtraction 1236 01:23:40,920 --> 01:23:44,100 works for defining all the operators that you may have. 1237 01:23:49,990 --> 01:23:55,510 And then minimal subtraction is simply a rescaling. 1238 01:23:55,510 --> 01:24:00,070 And that's the same as it is in gauge theory, where we get rid 1239 01:24:00,070 --> 01:24:03,690 of some annoying factors, and [? g is ?] a slightly 1240 01:24:03,690 --> 01:24:05,930 different definition. 1241 01:24:05,930 --> 01:24:09,670 So this was MS, and this is MS bar. 1242 01:24:09,670 --> 01:24:10,290 OK. 1243 01:24:10,290 --> 01:24:13,410 So it really works in a very similar way to gauge theory. 1244 01:24:13,410 --> 01:24:15,450 And you figure out the factors of mu 1245 01:24:15,450 --> 01:24:18,270 to the epsilon to include in your calculation this way. 1246 01:24:18,270 --> 01:24:20,520 Sometimes you see books do it by saying 1247 01:24:20,520 --> 01:24:24,060 the loop measure is continued within mu to the epsilon. 1248 01:24:24,060 --> 01:24:25,650 That's not right. 1249 01:24:25,650 --> 01:24:27,810 This is right. 1250 01:24:27,810 --> 01:24:31,200 If you do that, you'll get into trouble-- 1251 01:24:31,200 --> 01:24:32,040 not always. 1252 01:24:32,040 --> 01:24:33,790 That's why the books can get away with it. 1253 01:24:33,790 --> 01:24:37,020 But in general, you'll get into trouble. 1254 01:24:37,020 --> 01:24:37,590 All right. 1255 01:24:37,590 --> 01:24:39,607 So we should stop there.