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 --> 00:00:17,160 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,160 --> 00:00:18,370 at ocw.mit.edu. 8 00:00:20,575 --> 00:00:22,700 IAIN STEWART: [INAUDIBLE] theory graphs calculating 9 00:00:22,700 --> 00:00:24,075 the effective theory graphs using 10 00:00:24,075 --> 00:00:27,290 the same infrared regulator, renormalizing each of them 11 00:00:27,290 --> 00:00:29,930 separately, and then subtracting them 12 00:00:29,930 --> 00:00:32,540 to figure out higher-order corrections to the Wilson 13 00:00:32,540 --> 00:00:34,340 coefficients. 14 00:00:34,340 --> 00:00:40,618 And as part of that discussion, we also 15 00:00:40,618 --> 00:00:42,410 talked about scheme dependence because when 16 00:00:42,410 --> 00:00:44,270 you do that procedure for normalization 17 00:00:44,270 --> 00:00:46,895 in the effective theory, you're making a choice for the scheme. 18 00:00:46,895 --> 00:00:49,975 We picked m s bar, but you could pick other choices. 19 00:00:49,975 --> 00:00:52,100 And at the end of lecture, we talked about the fact 20 00:00:52,100 --> 00:00:55,010 that when you make different scheme choices, 21 00:00:55,010 --> 00:00:56,610 it can affect things. 22 00:00:56,610 --> 00:00:59,180 So it can affect the Wilson coefficients. 23 00:00:59,180 --> 00:01:01,340 It'll affect the matrix elements of operators. 24 00:01:01,340 --> 00:01:03,380 It'll affect your matching coefficients 25 00:01:03,380 --> 00:01:07,010 at one loop and your anomalous dimensions at two loops, 26 00:01:07,010 --> 00:01:10,370 but those scheme dependencies cancel out in observables. 27 00:01:10,370 --> 00:01:14,598 So it's important to take the scheme into account and work 28 00:01:14,598 --> 00:01:15,390 in the same scheme. 29 00:01:15,390 --> 00:01:18,320 But if you consistently use the same scheme for everything, 30 00:01:18,320 --> 00:01:20,550 then you will be OK. 31 00:01:20,550 --> 00:01:23,265 Now, if you're just doing calculations on pen and paper, 32 00:01:23,265 --> 00:01:24,890 it's very easy that you just-- well, we 33 00:01:24,890 --> 00:01:25,890 love the m s bar scheme. 34 00:01:25,890 --> 00:01:27,080 Let's use that. 35 00:01:27,080 --> 00:01:29,360 But remember that sometimes, there 36 00:01:29,360 --> 00:01:31,288 will be things in your result that you 37 00:01:31,288 --> 00:01:32,580 may want to get from elsewhere. 38 00:01:32,580 --> 00:01:34,350 For example, maybe there's some matrix elements 39 00:01:34,350 --> 00:01:35,058 of the operators. 40 00:01:35,058 --> 00:01:36,760 You want to get them from the lattice. 41 00:01:36,760 --> 00:01:38,510 Well, if you look up results in a lattice, 42 00:01:38,510 --> 00:01:40,180 they're not using m s bar. 43 00:01:40,180 --> 00:01:43,130 There's no non-perturbative definition of m s bar. 44 00:01:43,130 --> 00:01:45,440 So they'll be using some other scheme. 45 00:01:45,440 --> 00:01:47,892 And those results will have to be converted 46 00:01:47,892 --> 00:01:49,850 to m s bar if you're going to put them together 47 00:01:49,850 --> 00:01:51,260 with your results. 48 00:01:51,260 --> 00:01:54,650 So that's just something to be aware of. 49 00:01:54,650 --> 00:01:57,140 Before we leave this topic and go on to something else, 50 00:01:57,140 --> 00:02:00,020 I thought I'd say a few words about phenomenology. 51 00:02:00,020 --> 00:02:03,470 What is all this technology good for? 52 00:02:03,470 --> 00:02:08,848 So a nice example of that is beta s gamma. 53 00:02:08,848 --> 00:02:11,120 So beta s gamma-- 54 00:02:11,120 --> 00:02:13,862 it's a neutral current process. 55 00:02:13,862 --> 00:02:16,070 It doesn't happen in the Steiner model at tree level. 56 00:02:30,060 --> 00:02:32,010 And therefore, what that means, if it doesn't 57 00:02:32,010 --> 00:02:33,427 happen at tree level, is that it's 58 00:02:33,427 --> 00:02:36,780 sensitive to loop corrections. 59 00:02:36,780 --> 00:02:37,830 And you could go-- 60 00:02:37,830 --> 00:02:43,050 since we talked about a channel with b to c, you bar d, 61 00:02:43,050 --> 00:02:44,250 but this is very analogous. 62 00:02:44,250 --> 00:02:46,950 We have a b meson in the initial-- b quark 63 00:02:46,950 --> 00:02:47,830 in the initial state. 64 00:02:47,830 --> 00:02:49,840 So it's the same scales in the problem. 65 00:02:49,840 --> 00:02:52,740 Effectively, the b quark scale is the light scale. 66 00:02:52,740 --> 00:02:54,720 We want to get rid of the things that 67 00:02:54,720 --> 00:02:57,390 would mediate this decay that are inside the loop, which 68 00:02:57,390 --> 00:03:02,530 will be w bosons and top quarks in this example. 69 00:03:02,530 --> 00:03:05,670 And so you draw this in the standard model. 70 00:03:05,670 --> 00:03:07,410 There's various diagrams that contribute. 71 00:03:07,410 --> 00:03:09,960 One is this one. 72 00:03:15,270 --> 00:03:18,900 So here's a b quark changing into a strange quark 73 00:03:18,900 --> 00:03:21,820 through a top quark and a w. 74 00:03:21,820 --> 00:03:25,560 And if we integrate out the w in the top, 75 00:03:25,560 --> 00:03:27,600 then we get some local operator, much as 76 00:03:27,600 --> 00:03:31,810 we were talking about in our examples up here. 77 00:03:31,810 --> 00:03:34,740 It's just that we have different diagrams. 78 00:03:34,740 --> 00:03:36,870 And we can also, from the low-energy point of view, 79 00:03:36,870 --> 00:03:40,590 enumerate the operators. 80 00:03:40,590 --> 00:03:48,780 For some reason, I started calling it Q instead of O. 81 00:03:48,780 --> 00:03:51,030 Let me give you an example of some of these operators. 82 00:04:00,180 --> 00:04:03,960 So here's what you would call a magnetic dipole 83 00:04:03,960 --> 00:04:07,020 operator, couples directly to the photon, which 84 00:04:07,020 --> 00:04:09,540 is in the F mu nu. 85 00:04:09,540 --> 00:04:12,720 And it takes the b to an s bar. 86 00:04:12,720 --> 00:04:17,310 And that's an example of a higher dimension operator. 87 00:04:17,310 --> 00:04:19,620 Remember, this is dimension 2 and then plus 3 88 00:04:19,620 --> 00:04:21,300 there, so that's 5. 89 00:04:21,300 --> 00:04:23,940 There's a factor of the b quark mass that just comes in 90 00:04:23,940 --> 00:04:26,670 because of the chiral structure of the operator, which 91 00:04:26,670 --> 00:04:29,640 you can think of as there needs to be a mass insertion here. 92 00:04:29,640 --> 00:04:32,280 You need one factor of that mass in order for the diagram 93 00:04:32,280 --> 00:04:33,120 not to give 0. 94 00:04:37,950 --> 00:04:40,800 If you really start to do beta s gamma, 95 00:04:40,800 --> 00:04:43,320 and you want to go through the whole story, 96 00:04:43,320 --> 00:04:45,600 then you have to actually think about other operators. 97 00:04:45,600 --> 00:04:48,305 And there's a whole basis of them. 98 00:04:48,305 --> 00:04:49,680 And one thing you could do is you 99 00:04:49,680 --> 00:04:52,290 could take the operator up there and just 100 00:04:52,290 --> 00:04:53,760 replace the photon by a gluon. 101 00:04:57,450 --> 00:05:00,510 That won't give a tree-level contribution to beta s gamma, 102 00:05:00,510 --> 00:05:03,043 but these guys are charged under electromagnetism. 103 00:05:03,043 --> 00:05:04,710 So you could just have a direct coupling 104 00:05:04,710 --> 00:05:06,930 to the b quark or the strange quark, 105 00:05:06,930 --> 00:05:09,270 and then loop up the gluon, and get a correction 106 00:05:09,270 --> 00:05:11,220 from an operator like this one. 107 00:05:11,220 --> 00:05:15,840 And then there's also four quark operators, 108 00:05:15,840 --> 00:05:22,950 like the ones we talked about before, 109 00:05:22,950 --> 00:05:24,480 but with different flavors. 110 00:05:32,550 --> 00:05:35,400 And if you enumerate all of them, 111 00:05:35,400 --> 00:05:43,470 there's nine more different ways of making four-quark operators. 112 00:05:43,470 --> 00:05:46,590 So you could build some basis using the equation of motion 113 00:05:46,590 --> 00:05:48,540 to simplify the operators as much as possible. 114 00:05:48,540 --> 00:05:52,190 And then you get down to these ones. 115 00:05:52,190 --> 00:05:53,630 And you can go through the program 116 00:05:53,630 --> 00:05:56,130 that we talked about, of doing matching and renormalizations 117 00:05:56,130 --> 00:05:57,472 with evolution. 118 00:05:57,472 --> 00:05:59,180 And really what I want to talk about here 119 00:05:59,180 --> 00:06:00,680 is a little bit about phenomenology, 120 00:06:00,680 --> 00:06:03,110 because the interesting thing about this loop 121 00:06:03,110 --> 00:06:06,470 is that if there was nu physics-- since 122 00:06:06,470 --> 00:06:08,900 in the standard model at tree level it doesn't happen, 123 00:06:08,900 --> 00:06:11,150 if there's nu physics, then you can be sensitive to it 124 00:06:11,150 --> 00:06:13,340 here, because you're sensitive to heavy particles 125 00:06:13,340 --> 00:06:14,460 in this loop. 126 00:06:14,460 --> 00:06:15,950 That's why beta s gamma is always 127 00:06:15,950 --> 00:06:19,520 used to constrain nu physics. 128 00:06:19,520 --> 00:06:23,510 So in our effective theory, we have just an operator that does 129 00:06:23,510 --> 00:06:30,800 this, which is this 07 gamma. 130 00:06:33,570 --> 00:06:38,670 And if you think about what C 7 gamma is at lowest order, 131 00:06:38,670 --> 00:06:41,050 let's just calculate that diagram over there. 132 00:06:41,050 --> 00:06:44,190 And the result from doing that will 133 00:06:44,190 --> 00:06:47,010 be some function of m w over m top, 134 00:06:47,010 --> 00:06:50,380 because those are the things that are appearing in the loop. 135 00:06:50,380 --> 00:06:56,220 And if you do that calculation, you get a number like that, 136 00:06:56,220 --> 00:07:00,180 if you stick values for the top m w. 137 00:07:00,180 --> 00:07:03,390 And then you can start doing loop corrections. 138 00:07:03,390 --> 00:07:05,350 The first thing you might think about, 139 00:07:05,350 --> 00:07:07,680 which is actually not even suppressed 140 00:07:07,680 --> 00:07:11,010 by any factors of the coupling alpha s strong, 141 00:07:11,010 --> 00:07:19,280 would be to just loop up the Q1 operator, which I guess I don't 142 00:07:19,280 --> 00:07:22,610 know why I wrote it this way. 143 00:07:22,610 --> 00:07:25,280 This should be b. 144 00:07:25,280 --> 00:07:29,000 You could just loop the Q1 operator, contract the u u var, 145 00:07:29,000 --> 00:07:32,090 and then you can get a beta s transition from this operator. 146 00:07:32,090 --> 00:07:36,770 So this is the u quark here, and then attach a photon. 147 00:07:36,770 --> 00:07:39,940 And there's no factor in that loop of the strong coupling, 148 00:07:39,940 --> 00:07:42,530 because this is an electromagnetic coupling, 149 00:07:42,530 --> 00:07:45,920 and this guy here is just this Q1 operator. 150 00:07:45,920 --> 00:07:48,470 So this doesn't look like it's loop suppressed relative 151 00:07:48,470 --> 00:07:51,080 to that. 152 00:07:51,080 --> 00:07:53,610 And this is a little subtle, but this guy actually is 0. 153 00:07:53,610 --> 00:08:01,860 So you have to use a good scheme for gamma 5, 154 00:08:01,860 --> 00:08:04,460 but it turns out to be 0. 155 00:08:04,460 --> 00:08:07,110 So the first type of loop corrections that you get, 156 00:08:07,110 --> 00:08:10,460 which are suppressed by a factor of alpha s, 157 00:08:10,460 --> 00:08:12,170 come from diagrams like-- 158 00:08:12,170 --> 00:08:14,600 well, there's various diagrams, but one of them 159 00:08:14,600 --> 00:08:17,360 is like this, where I take the same diagram there 160 00:08:17,360 --> 00:08:21,230 and I attach an extra gluon on top of it. 161 00:08:21,230 --> 00:08:21,995 This guy diverges. 162 00:08:24,500 --> 00:08:33,080 And this two-loop calculation is order alpha strong, 163 00:08:33,080 --> 00:08:36,530 and it gives the leading order anomalous dimension, 164 00:08:36,530 --> 00:08:38,299 what we were calling gamma 0. 165 00:08:41,000 --> 00:08:42,230 It's not the only diagram. 166 00:08:42,230 --> 00:08:43,429 There's other diagrams, too. 167 00:08:45,950 --> 00:08:49,070 So we could go through that and do the similar type of thing, 168 00:08:49,070 --> 00:08:52,290 just with more diagrams than we had in our example. 169 00:08:52,290 --> 00:08:54,620 In particular, we could construct 170 00:08:54,620 --> 00:08:56,990 from the tree level matching in the one-loop anomalous 171 00:08:56,990 --> 00:09:03,350 dimension, the leading log result. Let me 172 00:09:03,350 --> 00:09:04,610 just write that down for you. 173 00:09:17,810 --> 00:09:38,187 Putting in numbers for the anomalous dimensions, at least 174 00:09:38,187 --> 00:09:39,770 sometimes putting in numbers for them. 175 00:09:47,370 --> 00:09:52,460 So the eta factor here is the ratio of alphas. 176 00:09:52,460 --> 00:09:56,420 So this is similar to what we saw before in our example 177 00:09:56,420 --> 00:10:00,730 where we got a ratio of alphas raised to a power. 178 00:10:00,730 --> 00:10:03,770 And if you want to pick a mu for this process over here, 179 00:10:03,770 --> 00:10:08,150 beta s gamma, then the right mu to think about is m b. 180 00:10:08,150 --> 00:10:13,010 So we want to take mu to the m b. 181 00:10:24,650 --> 00:10:27,102 And so if I plug in numbers-- 182 00:10:27,102 --> 00:10:29,060 and this is really what I wanted to emphasize-- 183 00:10:29,060 --> 00:10:32,480 if I plug in numbers here for these various things, 184 00:10:32,480 --> 00:10:35,300 this guy here gives a factor of 0.7. 185 00:10:35,300 --> 00:10:41,030 This guy here is this minus 0.2 that we talked about up there. 186 00:10:41,030 --> 00:10:47,090 This factor here is a bit small, 0.085. 187 00:10:47,090 --> 00:10:50,690 0.96. 188 00:10:50,690 --> 00:10:52,360 That's right. 189 00:10:52,360 --> 00:10:59,130 And then this piece here is a substantial correction. 190 00:10:59,130 --> 00:11:02,150 And if I wrote down all these numbers correctly, 191 00:11:02,150 --> 00:11:06,440 then the final result comes out to be, 192 00:11:06,440 --> 00:11:11,180 if I keep three digits, minus 0.3, which you can see 193 00:11:11,180 --> 00:11:14,440 is a fairly substantial change from minus 0.2. 194 00:11:14,440 --> 00:11:16,670 It's a 50% change. 195 00:11:16,670 --> 00:11:20,810 So just taking into account the evolution 196 00:11:20,810 --> 00:11:23,215 gives a 50% correction. 197 00:11:23,215 --> 00:11:25,590 So if you didn't take it into account, and you just said, 198 00:11:25,590 --> 00:11:27,090 well, forget about effective theory. 199 00:11:27,090 --> 00:11:28,590 I just calculate this graph. 200 00:11:28,590 --> 00:11:32,940 That's the standard model, you'd think that there is nu physics. 201 00:11:32,940 --> 00:11:36,710 We've certainly tested beta s gamma at better than 50%, 202 00:11:36,710 --> 00:11:38,582 more like the 10% level. 203 00:11:38,582 --> 00:11:40,790 So you really have to take into account these effects 204 00:11:40,790 --> 00:11:42,920 that we've been talking about, like this leading log 205 00:11:42,920 --> 00:11:45,462 evolution, if you want to look for nu physics, because you've 206 00:11:45,462 --> 00:11:52,745 got to get the right standard model result. 50% larger. 207 00:11:56,860 --> 00:12:01,240 And actually, people go two orders beyond what 208 00:12:01,240 --> 00:12:02,340 I'm talking about here. 209 00:12:02,340 --> 00:12:04,690 They go to the next, the next leading log order, 210 00:12:04,690 --> 00:12:07,130 when they really do precision beta s gamma physics. 211 00:12:07,130 --> 00:12:10,870 So some of the state of the art calculations of multiloop 212 00:12:10,870 --> 00:12:13,150 diagrams have been done exactly for beta s gamma 213 00:12:13,150 --> 00:12:15,250 because these effects are so important for looking 214 00:12:15,250 --> 00:12:15,875 for nu physics. 215 00:12:42,850 --> 00:12:44,600 And it's even worse when you get it put it 216 00:12:44,600 --> 00:12:47,895 in the branching ratio, because a 50% enhancement in a coupling 217 00:12:47,895 --> 00:12:49,520 when you put it in the branching ratio, 218 00:12:49,520 --> 00:12:50,790 you're squaring the amplitude. 219 00:12:50,790 --> 00:12:54,950 So that's a factor of 2.3. 220 00:12:54,950 --> 00:12:57,315 So these are really crucial corrections 221 00:12:57,315 --> 00:12:58,190 to take into account. 222 00:13:00,490 --> 00:13:02,990 So that's what this electroweak Hamiltonian is actually used 223 00:13:02,990 --> 00:13:06,620 for when you do phenomenology. 224 00:13:06,620 --> 00:13:10,370 So that's what I wanted to say about the electroweak 225 00:13:10,370 --> 00:13:12,317 Hamiltonian, just to give you a flavor for it. 226 00:13:12,317 --> 00:13:14,150 There's lots more that you could do with it. 227 00:13:14,150 --> 00:13:16,550 We could talk about more phenomenology, 228 00:13:16,550 --> 00:13:18,440 but let me stop there, since the idea is 229 00:13:18,440 --> 00:13:21,080 to give you an introduction to the concepts 230 00:13:21,080 --> 00:13:22,740 and we've done that. 231 00:13:22,740 --> 00:13:24,510 So now we'll move on to something else, 232 00:13:24,510 --> 00:13:28,308 which is a different concept, unless there's any questions. 233 00:13:36,930 --> 00:13:38,915 So all this business of schemes and stuff 234 00:13:38,915 --> 00:13:41,040 comes in when people are talking about beta s gamma 235 00:13:41,040 --> 00:13:44,738 and making this kind of model prediction. 236 00:13:44,738 --> 00:13:47,280 And there's actually schemes you can pick where you mess this 237 00:13:47,280 --> 00:13:49,760 up, but then, if you're careful, you 238 00:13:49,760 --> 00:13:51,690 get the same answer in the very end. 239 00:14:04,140 --> 00:14:07,020 So the next topic that I want to talk about 240 00:14:07,020 --> 00:14:08,832 is an example of something that's bottom up 241 00:14:08,832 --> 00:14:09,790 effective field theory. 242 00:14:09,790 --> 00:14:12,240 We've talked about top down with this example 243 00:14:12,240 --> 00:14:14,340 of removing heavy particles. 244 00:14:14,340 --> 00:14:16,620 And the classic example of bottom 245 00:14:16,620 --> 00:14:29,557 up is chiral perturbation theory or chiral Lagrangians. 246 00:14:34,030 --> 00:14:36,370 So our purposes here are not-- 247 00:14:36,370 --> 00:14:38,050 again, they're not a full exploration 248 00:14:38,050 --> 00:14:41,380 of this topic, which is a very large topic. 249 00:14:45,890 --> 00:14:48,260 So what are our goals? 250 00:14:48,260 --> 00:14:49,720 Bottom up effective theory example. 251 00:15:00,540 --> 00:15:03,860 We will also see in this example the utility 252 00:15:03,860 --> 00:15:06,260 of using something that's called the nonlinear 253 00:15:06,260 --> 00:15:09,620 realization of symmetry, non-linear symmetry 254 00:15:09,620 --> 00:15:14,780 representations, and the kind of connection 255 00:15:14,780 --> 00:15:16,160 to field redefinitions. 256 00:15:27,907 --> 00:15:29,490 Since field redefinitions is something 257 00:15:29,490 --> 00:15:31,490 we've been talking about, we'll talk about that. 258 00:15:34,220 --> 00:15:36,120 Another thing that this is an example of 259 00:15:36,120 --> 00:15:40,740 is an example where loops are not suppressed by the coupling 260 00:15:40,740 --> 00:15:43,470 constant. 261 00:15:43,470 --> 00:15:47,040 Instead, they're actually suppressed by powers 262 00:15:47,040 --> 00:15:48,744 in the power expansion. 263 00:16:02,300 --> 00:16:04,480 So that's kind of totally different than what 264 00:16:04,480 --> 00:16:07,690 we saw when we were integrating all the heavy particles, 265 00:16:07,690 --> 00:16:09,760 where you just put a loop. 266 00:16:09,760 --> 00:16:11,655 You're down by some factor of alpha s, 267 00:16:11,655 --> 00:16:13,780 but you're at the same order in the power expansion 268 00:16:13,780 --> 00:16:15,960 and 1 over large scale. 269 00:16:15,960 --> 00:16:18,810 This is going to be different. 270 00:16:18,810 --> 00:16:21,930 And therefore, in some ways this has a non-trivial power 271 00:16:21,930 --> 00:16:24,870 accounting-- 272 00:16:24,870 --> 00:16:27,630 more non-trivial, anyway, than what 273 00:16:27,630 --> 00:16:28,810 we were just talking about. 274 00:16:28,810 --> 00:16:31,960 So we'd like to give this as an example of non-trivial power 275 00:16:31,960 --> 00:16:34,680 accounting, and in fact, prove something that's called 276 00:16:34,680 --> 00:16:36,905 a "power accounting theorem." 277 00:16:36,905 --> 00:16:38,280 What the power accounting theorem 278 00:16:38,280 --> 00:16:40,230 means is that ahead of time, you should 279 00:16:40,230 --> 00:16:42,690 be able to figure out from your effective theory what 280 00:16:42,690 --> 00:16:44,070 order various things are. 281 00:16:44,070 --> 00:16:45,540 If you draw a diagram, you should 282 00:16:45,540 --> 00:16:49,950 know even before you calculate it how many powers in the power 283 00:16:49,950 --> 00:16:51,528 accounting expansion you have. 284 00:16:51,528 --> 00:16:53,070 If you didn't know that, you wouldn't 285 00:16:53,070 --> 00:16:55,660 know which diagrams to compute and which ones not to compute. 286 00:16:55,660 --> 00:16:57,160 You'd just have to compute them all. 287 00:16:57,160 --> 00:16:58,930 That's, of course, way too much work. 288 00:16:58,930 --> 00:17:01,900 So in order to formulate the effective theory-- 289 00:17:01,900 --> 00:17:04,773 especially since there's an infinite number of diagrams. 290 00:17:04,773 --> 00:17:06,690 So in order to formulate the theory, remember, 291 00:17:06,690 --> 00:17:09,630 you really have to have power accounting under control. 292 00:17:09,630 --> 00:17:12,420 And that means you need things like this in order 293 00:17:12,420 --> 00:17:15,210 to identify what's leading order. 294 00:17:15,210 --> 00:17:16,950 So we'll talk about that. 295 00:17:16,950 --> 00:17:20,550 So I'm imagining that maybe 50% of the class 296 00:17:20,550 --> 00:17:23,217 has seen chiral perturbation theory before in some form. 297 00:17:23,217 --> 00:17:25,050 I know I teach it in Quantum Field Theory 3, 298 00:17:25,050 --> 00:17:28,517 so if you took QFT 3 with me, you saw a glimpse into it. 299 00:17:28,517 --> 00:17:30,600 The things we're going to emphasize here are a bit 300 00:17:30,600 --> 00:17:34,230 different, but I am going to assume that you have knowledge 301 00:17:34,230 --> 00:17:38,310 of it at some level, and I'll assign you reading if you 302 00:17:38,310 --> 00:17:39,702 don't. 303 00:17:39,702 --> 00:17:41,160 The other thing I'm going to assume 304 00:17:41,160 --> 00:17:43,452 that you have some knowledge of is spontaneous symmetry 305 00:17:43,452 --> 00:17:45,125 breaking, because that's not the topic. 306 00:17:45,125 --> 00:17:47,250 That's not one of the things that I've listed here. 307 00:17:47,250 --> 00:17:50,130 That's not something that I really want to delve into, 308 00:17:50,130 --> 00:17:55,110 but of course, if we talk about the chiral Lagrangian, that's 309 00:17:55,110 --> 00:17:57,990 something that comes in, in particular, when we're 310 00:17:57,990 --> 00:17:59,580 talking about it from QCD. 311 00:17:59,580 --> 00:18:07,830 So I'll remind you of some things that are hopefully 312 00:18:07,830 --> 00:18:09,810 familiar, and anything that's unfamiliar, 313 00:18:09,810 --> 00:18:11,310 you should do additional reading on. 314 00:18:17,472 --> 00:18:19,180 So I know that there are some people that 315 00:18:19,180 --> 00:18:21,990 are taking QFT 3 right now, so this may be something 316 00:18:21,990 --> 00:18:22,990 they haven't got to yet. 317 00:18:22,990 --> 00:18:27,040 So I will do some review, but for further reading, 318 00:18:27,040 --> 00:18:29,720 you should see QFT 3 or some other source. 319 00:18:29,720 --> 00:18:33,700 I've posted some readings on the website. 320 00:18:40,590 --> 00:18:43,190 So if we start with QCD, massless QCD, 321 00:18:43,190 --> 00:18:44,970 we can divide it into a left-handed part 322 00:18:44,970 --> 00:18:46,040 and a right-handed part. 323 00:18:54,980 --> 00:18:57,200 And then we can talk about the transformation 324 00:18:57,200 --> 00:18:59,780 where we take the left-handed field 325 00:18:59,780 --> 00:19:02,750 and transform it by a separate amount 326 00:19:02,750 --> 00:19:09,680 than the right-handed field under a unitary transformation. 327 00:19:09,680 --> 00:19:13,440 And that's a symmetry of the theory. 328 00:19:13,440 --> 00:19:15,620 Now, exactly what symmetry you have 329 00:19:15,620 --> 00:19:18,498 depends on how many components you're 330 00:19:18,498 --> 00:19:19,790 talking about in the psi field. 331 00:19:19,790 --> 00:19:22,610 If I say the psi field is light, then you 332 00:19:22,610 --> 00:19:26,553 might think of up and down, and that's one possibility. 333 00:19:26,553 --> 00:19:28,220 Or you could throw the strange in there, 334 00:19:28,220 --> 00:19:29,570 and say, well, the strange is light, too, 335 00:19:29,570 --> 00:19:31,237 and then you have up, down, and strange. 336 00:19:33,480 --> 00:19:35,820 And that just the difference between SU2 and SU3, 337 00:19:35,820 --> 00:19:41,590 and both of these are viable choices. 338 00:19:41,590 --> 00:19:45,090 So let's make a little table. 339 00:19:45,090 --> 00:19:49,350 We have a group, which is the transformations given 340 00:19:49,350 --> 00:19:51,510 by the left and the right. 341 00:19:51,510 --> 00:19:54,247 And it's going to be broken spontaneously 342 00:19:54,247 --> 00:19:55,080 to some other group. 343 00:19:59,138 --> 00:19:59,930 So it could be SU3. 344 00:20:03,880 --> 00:20:06,040 So there's saying that the left and the right 345 00:20:06,040 --> 00:20:08,470 are in SU3, each of them. 346 00:20:08,470 --> 00:20:12,235 And in QCD, that's broken to the vector subgroup. 347 00:20:15,010 --> 00:20:17,623 So in this case, the psi would be u d s. 348 00:20:20,800 --> 00:20:24,070 And you get Goldstone bosons from that. 349 00:20:24,070 --> 00:20:26,700 And there's eight of them. 350 00:20:26,700 --> 00:20:28,796 And that's the pions, the kaon, and the eta. 351 00:20:32,388 --> 00:20:34,180 And keeping thing from our perspective what 352 00:20:34,180 --> 00:20:44,290 is the expansion parameter, and it's not so great, 353 00:20:44,290 --> 00:20:47,260 because the strange quark mass is not that light. 354 00:20:47,260 --> 00:20:50,740 So you can think that the strange quark mass over lambda 355 00:20:50,740 --> 00:20:55,240 QCD is maybe something like 1/3-ish, 356 00:20:55,240 --> 00:20:58,910 but you're not really getting better than that. 357 00:20:58,910 --> 00:21:03,790 So there was 8 generations here, 8 generators here, 358 00:21:03,790 --> 00:21:05,860 broken to 8 here. 359 00:21:05,860 --> 00:21:09,560 And then you have 8 Goldstones. 360 00:21:09,560 --> 00:21:14,250 You could do better, but make less predictions, 361 00:21:14,250 --> 00:21:20,360 if you just considered SU2 and just maybe up and down light. 362 00:21:20,360 --> 00:21:23,090 Then you just have the pions. 363 00:21:23,090 --> 00:21:28,350 And then you just have MU and MD over lambda QCD, 364 00:21:28,350 --> 00:21:33,380 which is more like 1/50, so a much better expansion. 365 00:21:33,380 --> 00:21:35,270 But then you can't do kaon physics 366 00:21:35,270 --> 00:21:38,390 with this, because the kaon is not 367 00:21:38,390 --> 00:21:43,020 part of the effective theory, and that can be the case. 368 00:21:43,020 --> 00:21:46,663 So in order to construct a theory for the Goldstones, 369 00:21:46,663 --> 00:21:48,330 which are the light degrees of freedom-- 370 00:21:48,330 --> 00:21:52,730 so this is we've identified the light degrees of freedom-- 371 00:21:52,730 --> 00:21:55,860 you'd like to construct an effective theory for them. 372 00:21:55,860 --> 00:21:57,620 Those are bound states of the particles 373 00:21:57,620 --> 00:22:01,590 in your original theory, and you don't know, on pen and paper, 374 00:22:01,590 --> 00:22:05,120 how to calculate the matching. 375 00:22:05,120 --> 00:22:08,090 And that's characteristic of a bottom up effective theory-- 376 00:22:08,090 --> 00:22:12,230 that you don't do the matching, that you just 377 00:22:12,230 --> 00:22:15,250 start from the bottom up. 378 00:22:15,250 --> 00:22:17,000 So in this case, you don't do the matching 379 00:22:17,000 --> 00:22:21,339 because it's non-perturbative. 380 00:22:42,320 --> 00:22:44,150 So what you do is you say, let's construct 381 00:22:44,150 --> 00:22:46,190 a type of field theory based on the fact 382 00:22:46,190 --> 00:22:48,140 that I know what the degrees of freedom are 383 00:22:48,140 --> 00:22:50,612 and I know something about symmetry. 384 00:22:50,612 --> 00:22:52,070 And in particular in this case, you 385 00:22:52,070 --> 00:22:53,695 know something about symmetry breaking. 386 00:23:05,050 --> 00:23:08,080 So one kind of logic is, then what you'll get 387 00:23:08,080 --> 00:23:10,330 is you'll get coefficients times operators again. 388 00:23:12,880 --> 00:23:17,650 The operators will be built out of your pion/kaon/eta fields. 389 00:23:17,650 --> 00:23:20,320 And you'll be able to calculate matrix elements of these. 390 00:23:26,477 --> 00:23:29,060 But you'll get some coefficients and you don't know the value. 391 00:23:29,060 --> 00:23:33,592 So these coefficients you could fit to data, 392 00:23:33,592 --> 00:23:35,050 because you haven't determined them 393 00:23:35,050 --> 00:23:36,910 from some high-level theory. 394 00:23:36,910 --> 00:23:41,290 You just could fit them to the data and do phenomenology. 395 00:23:41,290 --> 00:23:42,820 So it's kind of exactly the opposite 396 00:23:42,820 --> 00:23:44,290 from our high-energy point of view, 397 00:23:44,290 --> 00:23:46,762 from the theory with the electroweak Hamiltonian, where 398 00:23:46,762 --> 00:23:48,220 we thought the matrix elements were 399 00:23:48,220 --> 00:23:49,532 going to be non-perturbative. 400 00:23:49,532 --> 00:23:50,740 Here are the matrix elements. 401 00:23:50,740 --> 00:23:51,760 They're something you calculate. 402 00:23:51,760 --> 00:23:53,177 And the coefficients are including 403 00:23:53,177 --> 00:23:54,940 the non-perturbative physics. 404 00:23:57,810 --> 00:23:59,560 You could also get these from the lattice. 405 00:24:03,560 --> 00:24:06,890 So a lattice QCD calculation can tell you 406 00:24:06,890 --> 00:24:08,570 the values for the C's to use. 407 00:24:08,570 --> 00:24:10,612 And then you could just use the chiral Lagrangian 408 00:24:10,612 --> 00:24:13,670 to do phenomenology. 409 00:24:13,670 --> 00:24:17,180 Now, because we have this bottom-up point of view, 410 00:24:17,180 --> 00:24:20,400 there's something that we don't know. 411 00:24:20,400 --> 00:24:24,320 And that is, we actually don't know precisely what 412 00:24:24,320 --> 00:24:29,420 theory we started with, because all we're 413 00:24:29,420 --> 00:24:34,220 encoding about that theory is the symmetry breaking pattern. 414 00:24:34,220 --> 00:24:37,670 And so if there's a bunch of upper-level theories 415 00:24:37,670 --> 00:24:41,364 that have the same symmetry breaking pattern, 416 00:24:41,364 --> 00:24:42,810 they all look the same. 417 00:24:42,810 --> 00:24:45,337 They'll look like they have the same chiral Lagrangian. 418 00:24:55,370 --> 00:24:58,782 And the thing that distinguishes them 419 00:24:58,782 --> 00:25:00,740 is that they would have different coefficients. 420 00:25:00,740 --> 00:25:02,782 And you wouldn't know that unless you figured out 421 00:25:02,782 --> 00:25:06,080 what the coefficients are. 422 00:25:06,080 --> 00:25:07,640 So that's another way of just saying 423 00:25:07,640 --> 00:25:09,440 that the high-energy physics is being 424 00:25:09,440 --> 00:25:13,670 encoded in the coefficients. 425 00:25:13,670 --> 00:25:16,280 So the same chiral Lagrangian would show up 426 00:25:16,280 --> 00:25:19,265 for different high-level theories 427 00:25:19,265 --> 00:25:21,140 that have the same symmetry breaking pattern. 428 00:25:30,580 --> 00:25:33,160 So this is just one example of chiral Lagrangians. 429 00:25:33,160 --> 00:25:35,260 And we're going to use it as an example, which 430 00:25:35,260 --> 00:25:38,920 is perhaps the most familiar, to illustrate our bullets. 431 00:25:50,990 --> 00:25:53,210 So one of our bullets was related 432 00:25:53,210 --> 00:25:55,632 to non-linear representations in field redefinition, 433 00:25:55,632 --> 00:25:56,840 so let's start with that one. 434 00:26:12,960 --> 00:26:14,345 Problems set's due today. 435 00:26:14,345 --> 00:26:15,345 Problem set 2 is posted. 436 00:26:22,468 --> 00:26:24,760 So let me talk about something called the "linear sigma 437 00:26:24,760 --> 00:26:26,650 model." 438 00:26:26,650 --> 00:26:28,690 And we'll use this as an example of something 439 00:26:28,690 --> 00:26:30,482 that has a symmetry breaking pattern that's 440 00:26:30,482 --> 00:26:33,610 similar to the one we want, same as the one we want. 441 00:26:36,920 --> 00:26:40,450 And we'll construct, from this, the chiral Lagrangian. 442 00:26:40,450 --> 00:26:43,040 And since it's the same symmetry breaking pattern, 443 00:26:43,040 --> 00:26:49,390 it's also a viable chiral Lagrangian directly for QCD. 444 00:26:49,390 --> 00:26:51,280 So what is the linear sigma model? 445 00:26:51,280 --> 00:26:53,410 So I'll talk about fields that I call pi 446 00:26:53,410 --> 00:26:55,540 without a vector symbol on top. 447 00:26:55,540 --> 00:26:56,840 They have two components. 448 00:26:56,840 --> 00:27:00,370 One component that's in the diagonal, then one component-- 449 00:27:00,370 --> 00:27:06,210 these are matrices in the off diagonal, 450 00:27:06,210 --> 00:27:10,620 and different entries in the sigma 3. 451 00:27:10,620 --> 00:27:13,740 So I want to think of this is a kind of full theory. 452 00:27:18,452 --> 00:27:25,500 So the full theory for the sigma model is the following-- 453 00:27:25,500 --> 00:27:38,970 kinetic term, mass term, interaction term. 454 00:27:45,040 --> 00:27:46,720 And I can couple it to something else 455 00:27:46,720 --> 00:27:58,320 like a fermion to make it more interesting, 456 00:27:58,320 --> 00:27:59,998 with Yukawa couplings. 457 00:28:11,460 --> 00:28:24,140 And this theory has an SU2 left cross SU2 right symmetry, where 458 00:28:24,140 --> 00:28:32,090 I take psi left to L psi left, psi right to R psi right. 459 00:28:32,090 --> 00:28:36,920 And I also transform pi to L pi R dagger. 460 00:28:42,720 --> 00:28:45,120 And if you think of these L's and these R's, you 461 00:28:45,120 --> 00:28:50,590 should think of them as something with some generators. 462 00:28:50,590 --> 00:28:58,640 So L would have something like that, with tau left generators, 463 00:28:58,640 --> 00:29:01,910 and then some parameters alpha L. If I worked infinitesimally 464 00:29:01,910 --> 00:29:04,785 in those parameters, then if I expanded this out, 465 00:29:04,785 --> 00:29:06,410 this would give a linear transformation 466 00:29:06,410 --> 00:29:08,300 to the sigma and the pi vector. 467 00:29:10,905 --> 00:29:12,655 That's what it's meant, that it's linear-- 468 00:29:18,470 --> 00:29:22,830 so a linear infinitesimal transformation to pi and sigma. 469 00:29:26,410 --> 00:29:29,050 So this theory has spontaneous symmetry breaking. 470 00:29:39,880 --> 00:29:56,410 And if we write out the potential by taking the traces, 471 00:29:56,410 --> 00:29:57,910 and we can write in a way that makes 472 00:29:57,910 --> 00:30:01,810 that obvious by completing the square. 473 00:30:09,220 --> 00:30:11,500 And we can always throw away irrelevant constants 474 00:30:11,500 --> 00:30:13,420 when we do that. 475 00:30:13,420 --> 00:30:15,390 So we get something like that. 476 00:30:15,390 --> 00:30:16,600 So the minimum is shifted. 477 00:30:21,790 --> 00:30:24,340 And so if we want to shift ourselves over 478 00:30:24,340 --> 00:30:26,830 to describe perturbations around that vacuum, 479 00:30:26,830 --> 00:30:28,970 then we can do that. 480 00:30:28,970 --> 00:30:35,970 So let this guy have a VEV, which I'll call curly 481 00:30:35,970 --> 00:30:39,705 V, square root mu squared over lambda. 482 00:30:42,840 --> 00:30:45,420 Pi vector field has no VEV, because we 483 00:30:45,420 --> 00:30:48,120 want to maintain the vector symmetry. 484 00:30:48,120 --> 00:30:49,900 And we talk about a shifted field, 485 00:30:49,900 --> 00:30:53,190 which describes perturbations around the VEV 486 00:30:53,190 --> 00:30:57,450 when we do quantum field theory in the sigma twiddle. 487 00:30:57,450 --> 00:31:00,750 And then we write our Lagrangian in terms of the sigma twiddle, 488 00:31:00,750 --> 00:31:02,250 just by making a change of variable. 489 00:31:12,380 --> 00:31:14,690 And that makes clear who's getting masses 490 00:31:14,690 --> 00:31:17,240 when we expand around that vacuum 491 00:31:17,240 --> 00:31:18,500 and who's remaining massless. 492 00:31:18,500 --> 00:31:20,708 And of course, the Goldstones are remaining massless, 493 00:31:20,708 --> 00:31:23,060 and the Goldstones, by clever choice of notation, 494 00:31:23,060 --> 00:31:23,810 are called Pi. 495 00:31:47,762 --> 00:31:50,800 Let's see if I can squeeze it all in here. 496 00:31:53,570 --> 00:31:56,120 This is something they tell you in Professor 101 never to do, 497 00:31:56,120 --> 00:31:57,370 but I'm going to do it anyway. 498 00:32:08,190 --> 00:32:10,617 Since if you can't read something, 499 00:32:10,617 --> 00:32:12,450 you can always look at my notes when I them. 500 00:32:15,480 --> 00:32:19,770 So I didn't write all the fermion notes down again, 501 00:32:19,770 --> 00:32:23,550 but I could do the field redefinition in those modes 502 00:32:23,550 --> 00:32:31,340 as well, the shift in those terms. 503 00:32:36,360 --> 00:32:40,980 And the vector subgroup, which remains unbroken, 504 00:32:40,980 --> 00:32:47,400 is no transformation for sigma and the Pi field 505 00:32:47,400 --> 00:32:52,260 has left equal to right, so we just get some matrix V 506 00:32:52,260 --> 00:32:54,940 on both sides. 507 00:32:54,940 --> 00:32:58,048 And if we look at the Lagrangian that we have, 508 00:32:58,048 --> 00:33:00,090 then we can identify the masses of various things 509 00:33:00,090 --> 00:33:00,720 at tree level. 510 00:33:09,430 --> 00:33:11,170 And I didn't write down the fermion part, 511 00:33:11,170 --> 00:33:14,710 but the fermion gets mass 2 from the [INAUDIBLE] coupling, 512 00:33:14,710 --> 00:33:18,070 just like in the standard model. 513 00:33:18,070 --> 00:33:20,453 And the Pi, of course, remain massless. 514 00:33:20,453 --> 00:33:22,870 And the idea of thinking about this is an effective theory 515 00:33:22,870 --> 00:33:24,453 is that we can take these to be large. 516 00:33:28,360 --> 00:33:29,920 And then there's a clear separation 517 00:33:29,920 --> 00:33:33,010 between low-energy degrees of freedom 518 00:33:33,010 --> 00:33:36,827 and heavy degrees of freedom. 519 00:33:36,827 --> 00:33:39,160 So if we want to describe this with an effective theory, 520 00:33:39,160 --> 00:33:41,243 we would like to describe the physics of the pions 521 00:33:41,243 --> 00:33:44,466 without worrying too much about the sigma and the psi. 522 00:33:48,040 --> 00:33:50,830 So we could do that just by thinking about this Lagrangian, 523 00:33:50,830 --> 00:33:53,590 but it turns out that you can make field redefinitions 524 00:33:53,590 --> 00:33:55,930 and think about using different formulations which 525 00:33:55,930 --> 00:33:57,700 are entirely equivalent. 526 00:33:57,700 --> 00:34:00,560 And some of those are more useful than this linear sigma 527 00:34:00,560 --> 00:34:01,060 model. 528 00:34:03,690 --> 00:34:05,680 So we're thinking here of field redefinitions 529 00:34:05,680 --> 00:34:07,170 as an organizational tool. 530 00:34:22,159 --> 00:34:24,960 So you'll see what I mean when we go through this. 531 00:34:24,960 --> 00:34:30,239 So let's consider some different choices. 532 00:34:30,239 --> 00:34:33,087 So there's something called the "square root representation." 533 00:34:36,500 --> 00:34:53,989 So I just make a field redefinition like that-- 534 00:34:53,989 --> 00:34:56,280 involves a square root. 535 00:34:56,280 --> 00:34:59,645 And if I expand it, it starts a sigma twiddle, 536 00:34:59,645 --> 00:35:01,933 and then it goes a bunch of other terms. 537 00:35:01,933 --> 00:35:03,350 So it's certainly within the realm 538 00:35:03,350 --> 00:35:06,590 of the things that are allowed by our field redefinition 539 00:35:06,590 --> 00:35:08,060 theorem. 540 00:35:08,060 --> 00:35:14,270 And then I also talk about making a field redefinition 541 00:35:14,270 --> 00:35:15,320 for the Pi as well. 542 00:35:30,870 --> 00:35:37,527 And again, this is Pi plus other terms. 543 00:35:37,527 --> 00:35:39,110 So in this square root representation, 544 00:35:39,110 --> 00:35:42,270 we're going over to these fields S and psi. 545 00:35:42,270 --> 00:35:45,000 So these are our new fields. 546 00:35:45,000 --> 00:35:47,420 So we can make that change of variable 547 00:35:47,420 --> 00:35:50,350 and just write out our linear sigma model again. 548 00:35:55,790 --> 00:35:58,220 And it's the same theory, it's just field redefined. 549 00:36:08,800 --> 00:36:13,257 And it looks kind of Godawful, but it's more beautiful 550 00:36:13,257 --> 00:36:13,840 than it looks. 551 00:37:13,017 --> 00:37:15,350 So that's something you don't want to do more than once. 552 00:37:19,520 --> 00:37:20,960 So that's one possible way that we 553 00:37:20,960 --> 00:37:22,502 could deal with the effective theory, 554 00:37:22,502 --> 00:37:25,570 is in terms of these variables, S and phi. 555 00:37:25,570 --> 00:37:27,200 Let me write down a couple other ways, 556 00:37:27,200 --> 00:37:31,010 and then we'll talk about which we might pick. 557 00:37:31,010 --> 00:37:33,980 So there's something called the "exponential representation," 558 00:37:33,980 --> 00:37:35,550 also very common. 559 00:37:35,550 --> 00:37:37,820 These are very common in the literature, 560 00:37:37,820 --> 00:37:42,380 where instead of using S and phi, we use S and sigma. 561 00:37:42,380 --> 00:37:48,710 So we take our original fields, and we rewrite it 562 00:37:48,710 --> 00:37:54,470 as v plus S times sigma. 563 00:37:54,470 --> 00:37:58,370 And then we think of sigma as the exponential, and hence 564 00:37:58,370 --> 00:38:04,980 the name, of our fundamental Goldstone field. 565 00:38:04,980 --> 00:38:06,140 So this is not the same Pi. 566 00:38:06,140 --> 00:38:08,170 It's some Pi prime, if you like. 567 00:38:08,170 --> 00:38:09,840 But I'm going to drop the prime. 568 00:38:12,720 --> 00:38:15,260 So you write everything in terms of S and sigma, 569 00:38:15,260 --> 00:38:17,950 and you keep in mind that inside the sigma is the Pi. 570 00:38:21,530 --> 00:38:24,200 So the S part at the beginning is the same. 571 00:38:32,870 --> 00:38:34,870 And this guy's a little bit nicer to write down. 572 00:38:43,400 --> 00:38:49,040 So the terms that are pure S remain the same, 573 00:38:49,040 --> 00:38:50,000 so S cubed, S 4th. 574 00:39:06,560 --> 00:39:08,890 Since there's no funny square roots in our field 575 00:39:08,890 --> 00:39:10,640 redefinition, things are a little simpler. 576 00:39:14,120 --> 00:39:17,210 So that's, again, an equivalent version of the sigma model, 577 00:39:17,210 --> 00:39:19,928 just using different fields. 578 00:39:19,928 --> 00:39:22,220 Everything I can calculate in the original sigma model, 579 00:39:22,220 --> 00:39:25,690 I can calculate these formulations as well. 580 00:39:41,740 --> 00:39:47,270 And the final one is going to be different. 581 00:39:47,270 --> 00:39:53,860 And that's the non-linearity chiral Lagrangian. 582 00:39:53,860 --> 00:39:55,780 And what I'm going to do to get there is I'm 583 00:39:55,780 --> 00:39:58,000 just going to drop the things that 584 00:39:58,000 --> 00:40:02,050 have mass from my previous theory, which are S and psi. 585 00:40:02,050 --> 00:40:03,820 So I take this exponential representation 586 00:40:03,820 --> 00:40:05,620 of the sigma model. 587 00:40:05,620 --> 00:40:08,800 These guys are massive. 588 00:40:08,800 --> 00:40:12,130 I think about-- well, I integrate them out explicitly. 589 00:40:12,130 --> 00:40:15,850 In this case, what we're doing here, which is not QCD, 590 00:40:15,850 --> 00:40:17,980 we can do that, just remove them. 591 00:40:17,980 --> 00:40:21,010 That amounts to the lowest order, just dropping them. 592 00:40:21,010 --> 00:40:31,000 And then we have just the thing involving the pion, 593 00:40:31,000 --> 00:40:32,930 which is just very simple-- 594 00:40:32,930 --> 00:40:33,430 that. 595 00:40:37,460 --> 00:40:40,870 So this is a viable thing for low energies. 596 00:40:40,870 --> 00:40:48,560 The first three actions that we wrote down are identical. 597 00:40:48,560 --> 00:40:51,937 They're just field redefined versions of each other. 598 00:40:51,937 --> 00:40:53,395 They're really equivalent theories. 599 00:40:59,860 --> 00:41:04,000 And this final one, L chi, is equivalent 600 00:41:04,000 --> 00:41:10,750 for low-energy phenomenology of the pions. 601 00:41:14,770 --> 00:41:18,040 So the first three are actually equivalent to the last one 602 00:41:18,040 --> 00:41:21,540 if we restrict ourselves to low-energy interactions. 603 00:41:25,290 --> 00:41:27,800 So in order to see that, let's do an example 604 00:41:27,800 --> 00:41:30,080 where we calculate something. 605 00:41:30,080 --> 00:41:35,150 Let's think about calculating Pi plus Pi 0 goes to Pi plus Pi 606 00:41:35,150 --> 00:41:39,680 0, so scattering of Goldstone bosons. 607 00:41:39,680 --> 00:41:42,320 And q will be the momentum transfer. 608 00:41:42,320 --> 00:41:43,910 So in kind of an obvious notation, 609 00:41:43,910 --> 00:41:51,149 it's the difference between the final and the initial, 610 00:41:51,149 --> 00:41:53,600 or the initial and the final for the charge guy 611 00:41:53,600 --> 00:41:56,275 or the neutral guy. 612 00:41:56,275 --> 00:41:58,400 And there's two types of diagrams that can come in. 613 00:41:58,400 --> 00:42:08,362 You could have a direct scattering diagram 614 00:42:08,362 --> 00:42:09,945 or you could have an exchange diagram. 615 00:42:14,950 --> 00:42:21,530 so this is Pi plus coupling to either the sigma or the S, 616 00:42:21,530 --> 00:42:25,190 depending on which representation we're using. 617 00:42:25,190 --> 00:42:29,000 And then the Pi 0 is down here. 618 00:42:29,000 --> 00:42:32,990 And I want to take these, and I want to expand. 619 00:42:32,990 --> 00:42:36,710 And I'm going to expand in q squared over v squared. 620 00:42:36,710 --> 00:42:38,727 That's what I'm going to mean by "low energy". 621 00:42:38,727 --> 00:42:40,310 v was setting the scale of the masses. 622 00:42:44,318 --> 00:42:46,610 So let's see how that works out in the different cases. 623 00:42:46,610 --> 00:42:49,090 If I do it in the linear case, and I 624 00:42:49,090 --> 00:42:50,840 have both of these types of contributions, 625 00:42:50,840 --> 00:42:53,900 this guy just gives me some symmetry factor, 626 00:42:53,900 --> 00:42:58,950 which is 2 minus 2i lambda. 627 00:42:58,950 --> 00:43:20,742 This guy as a propagator, square root looks very different. 628 00:43:23,440 --> 00:43:24,940 And it turns out in the square root, 629 00:43:24,940 --> 00:43:27,500 that you can find out that this guy is order q to the 4th, 630 00:43:27,500 --> 00:43:29,810 so we won't even write it down. 631 00:43:36,180 --> 00:43:43,650 Exponential-- this guy is again ditto and ditto, 632 00:43:43,650 --> 00:43:49,800 and then I don't have room, but I'll put it up here. 633 00:43:49,800 --> 00:43:58,125 In the nonlinear, we don't even have that diagram 634 00:43:58,125 --> 00:43:59,250 because we just dropped it. 635 00:44:02,110 --> 00:44:04,300 And if I take this line, which is the only one that 636 00:44:04,300 --> 00:44:08,540 looks different, I can combine these two terms together. 637 00:44:08,540 --> 00:44:11,020 And once I do that, then I realize that this is also 638 00:44:11,020 --> 00:44:14,040 giving the same answer. 639 00:44:14,040 --> 00:44:20,380 So just rearrange it, put in the relation between mass 640 00:44:20,380 --> 00:44:27,430 and coupling and VEV, which looks like that, and expand. 641 00:44:27,430 --> 00:44:32,230 And you get i q squared over v squared as well. 642 00:44:32,230 --> 00:44:34,880 So if I stop at order q squared, all versions 643 00:44:34,880 --> 00:44:36,790 give the same thing. 644 00:44:36,790 --> 00:44:38,800 The linear and the square root were equivalent. 645 00:44:38,800 --> 00:44:40,000 The linear, the square root, and the exponential 646 00:44:40,000 --> 00:44:42,238 are just different versions of the same thing. 647 00:44:42,238 --> 00:44:44,530 And all that's changed by making the field redefinition 648 00:44:44,530 --> 00:44:46,390 is how I think about these two diagrams. 649 00:44:46,390 --> 00:44:48,940 In the linear, the leading order term is not 650 00:44:48,940 --> 00:44:50,590 coming from just this diagram, it's 651 00:44:50,590 --> 00:44:52,237 also coming from that diagram. 652 00:44:52,237 --> 00:44:54,070 And it's actually coming from a cancellation 653 00:44:54,070 --> 00:44:56,037 between those diagrams. 654 00:44:56,037 --> 00:44:58,120 In the square root and exponential representation, 655 00:44:58,120 --> 00:45:00,862 this diagram alone gives the leading order term 656 00:45:00,862 --> 00:45:02,320 and this diagram gives higher order 657 00:45:02,320 --> 00:45:04,910 terms, which is kind of what you want 658 00:45:04,910 --> 00:45:06,910 if you want to think about these heavy particles 659 00:45:06,910 --> 00:45:10,490 as something that is giving high-order corrections. 660 00:45:10,490 --> 00:45:15,500 So that can depend on exactly how you formulate it. 661 00:45:15,500 --> 00:45:18,580 And from the point of view of just keeping 662 00:45:18,580 --> 00:45:21,950 the low-energy degree of freedom, 663 00:45:21,950 --> 00:45:24,280 the nonlinear just gives you immediately the answer 664 00:45:24,280 --> 00:45:27,482 from very simple Lagrangian. 665 00:45:27,482 --> 00:45:29,690 Of course, it was just the exponential with something 666 00:45:29,690 --> 00:45:32,530 dropped. 667 00:45:32,530 --> 00:45:35,170 So from the point of view of doing calculations, 668 00:45:35,170 --> 00:45:39,220 the linear version here is the one that you don't want to use, 669 00:45:39,220 --> 00:45:41,410 because there's cancellations between diagrams 670 00:45:41,410 --> 00:45:43,360 that you have to figure out. 671 00:45:43,360 --> 00:45:46,180 And those cancellations actually affect 672 00:45:46,180 --> 00:45:48,940 what you call "leading order," because leading 673 00:45:48,940 --> 00:45:51,370 order for this calculation is order q squared, 674 00:45:51,370 --> 00:45:53,620 and you don't see that until there's that cancellation 675 00:45:53,620 --> 00:45:54,505 that's taken effect. 676 00:45:59,688 --> 00:46:01,730 So it's very hard to formulate a power accounting 677 00:46:01,730 --> 00:46:05,767 for the linear theory for that reason. 678 00:46:05,767 --> 00:46:07,350 But if any of these nonlinear theories 679 00:46:07,350 --> 00:46:10,170 we could formulate a power accounting, 680 00:46:10,170 --> 00:46:14,771 then everything is nice and beautiful. 681 00:46:41,300 --> 00:46:44,150 So what happens is that because of chiral symmetry, 682 00:46:44,150 --> 00:46:46,340 we have derivative couplings. 683 00:46:46,340 --> 00:46:48,350 And in the linear version, we don't 684 00:46:48,350 --> 00:46:50,240 see that until we cancel terms. 685 00:46:54,600 --> 00:46:56,240 So if you like, if you think about that 686 00:46:56,240 --> 00:46:59,270 as a property of the symmetry, you'd like to make it explicit, 687 00:46:59,270 --> 00:47:01,910 and the other representations do that. 688 00:47:17,323 --> 00:47:18,740 The version that's most convenient 689 00:47:18,740 --> 00:47:21,380 is the non-linear guy, since it's the simplest. 690 00:47:21,380 --> 00:47:24,650 And we just think about it, forgetting 691 00:47:24,650 --> 00:47:27,170 about the heavy stuff. 692 00:47:27,170 --> 00:47:30,590 And it's what we really want for our bottom-up effective theory 693 00:47:30,590 --> 00:47:36,630 because it only has the low energy sigma 694 00:47:36,630 --> 00:47:41,790 field, which has a pion in it, and it 695 00:47:41,790 --> 00:47:43,450 has the derivative couplings. 696 00:47:53,250 --> 00:47:56,228 If you look at how the symmetry works, 697 00:47:56,228 --> 00:47:58,020 which I didn't talk about when I wrote down 698 00:47:58,020 --> 00:48:04,290 all the different versions, S is for singlet. 699 00:48:04,290 --> 00:48:08,370 That's what's behind the name S, so it doesn't transform. 700 00:48:08,370 --> 00:48:12,540 Sigma transforms on both sides with an L 701 00:48:12,540 --> 00:48:15,090 on the left and an R dagger on the right. 702 00:48:15,090 --> 00:48:17,940 And that comes from looking back, noting 703 00:48:17,940 --> 00:48:20,430 that S doesn't transform, and looking back 704 00:48:20,430 --> 00:48:23,400 at how the Pi field transformed. 705 00:48:23,400 --> 00:48:24,590 It's just exactly that way. 706 00:48:28,480 --> 00:48:33,450 Now, if you write out the sigma, which is transforming linearly, 707 00:48:33,450 --> 00:48:38,250 as the exponential of i tau dot Pi, 708 00:48:38,250 --> 00:48:41,340 and if I put it in a convenient normalization, which 709 00:48:41,340 --> 00:48:49,030 is the VEV, or the v, then you can work out 710 00:48:49,030 --> 00:48:51,490 from this transformation how the Pi field transforms. 711 00:48:51,490 --> 00:48:54,580 And it transforms nonlinearly, and hence, that's 712 00:48:54,580 --> 00:48:55,270 the name for-- 713 00:49:01,750 --> 00:49:05,140 that's where the name comes from. 714 00:49:05,140 --> 00:49:11,350 If you do the infinitesimal version of the transformation, 715 00:49:11,350 --> 00:49:13,630 and you work it out for the Pi a field, which 716 00:49:13,630 --> 00:49:23,670 is a-th component of the vector, a being 1, 2, or 3, 717 00:49:23,670 --> 00:49:26,760 part of the transformation is simply a shift. 718 00:49:26,760 --> 00:49:29,790 And then there are additional pieces, 719 00:49:29,790 --> 00:49:31,840 which are things that you keep, that are 720 00:49:31,840 --> 00:49:33,510 order pi squared and higher. 721 00:49:33,510 --> 00:49:37,980 And that's those pi squared and higher terms that they're 722 00:49:37,980 --> 00:49:39,297 saying it's nonlinear. 723 00:49:39,297 --> 00:49:41,880 This term here is what's telling you it better be derivatively 724 00:49:41,880 --> 00:49:43,797 coupled, because if it's derivatively coupled, 725 00:49:43,797 --> 00:49:47,960 if you have a derivative, that kills this constant, 726 00:49:47,960 --> 00:49:50,460 and that was something that was hidden in our linear version 727 00:49:50,460 --> 00:49:51,043 of the theory. 728 00:49:53,870 --> 00:49:57,370 Now, we've constructed it here from this kind of point 729 00:49:57,370 --> 00:49:59,020 of view of integrating out. 730 00:49:59,020 --> 00:50:01,750 So I said QCD, we can integrate it out, 731 00:50:01,750 --> 00:50:04,890 but all we care about is the symmetry-breaking pattern. 732 00:50:04,890 --> 00:50:07,390 So we wrote down a theory, which is this linear sigma model, 733 00:50:07,390 --> 00:50:09,460 had the same symmetry-breaking pattern. 734 00:50:09,460 --> 00:50:11,800 And we could remove the heavy fields in that, 735 00:50:11,800 --> 00:50:13,870 and then the low-energy theory, I claim, 736 00:50:13,870 --> 00:50:16,270 is the same one that you would use to describe QCD, 737 00:50:16,270 --> 00:50:19,330 because it's the same symmetry-breaking pattern. 738 00:50:19,330 --> 00:50:22,360 But that's kind of a roundabout way of getting 739 00:50:22,360 --> 00:50:24,160 to where we wanted to go. 740 00:50:24,160 --> 00:50:26,350 And you'd really like to just get 741 00:50:26,350 --> 00:50:31,150 right to the chiral Lagrangian from the start, 742 00:50:31,150 --> 00:50:32,170 and you can do that. 743 00:50:46,963 --> 00:50:49,130 So rather than going through the linear sigma model, 744 00:50:49,130 --> 00:50:51,755 we could also just directly get to where we want to go. 745 00:50:51,755 --> 00:50:52,880 And here's how you do that. 746 00:50:55,490 --> 00:51:02,250 So go back to what the symmetry-breaking pattern was 747 00:51:02,250 --> 00:51:04,027 and figure out what is it that we're 748 00:51:04,027 --> 00:51:06,110 trying to do with the low-energy effective theory. 749 00:51:06,110 --> 00:51:14,250 So we have G that's broken to H. The Goldstones are 750 00:51:14,250 --> 00:51:21,690 transformations in the coset, which is G/H. 751 00:51:21,690 --> 00:51:23,900 And we'd like to parameterize them. 752 00:51:23,900 --> 00:51:25,650 And what we're doing with this sigma field 753 00:51:25,650 --> 00:51:27,255 is we're parameterizing fluctuations 754 00:51:27,255 --> 00:51:29,790 to take us around the coset. 755 00:51:29,790 --> 00:51:34,080 AUDIENCE: I'm a little confused, because if you don't expand 756 00:51:34,080 --> 00:51:37,920 sigma, that Lagrangian has still the full symmetry as you 757 00:51:37,920 --> 00:51:40,740 do left, as you do right. 758 00:51:40,740 --> 00:51:43,480 But it contains the v, [INAUDIBLE] 759 00:51:43,480 --> 00:51:45,105 parameter for the rest of the symmetry. 760 00:51:47,790 --> 00:51:51,007 That Lagrangian is presenting the unbroken phase or-- 761 00:51:51,007 --> 00:51:52,590 IAIN STEWART: So remember, I went over 762 00:51:52,590 --> 00:51:55,740 to the shifted fields, the sigma twiddle, 763 00:51:55,740 --> 00:51:58,163 and then I was expanding around the correct vacuum-- 764 00:51:58,163 --> 00:52:00,330 I mean, the one that's the lowest energy vacuum, not 765 00:52:00,330 --> 00:52:02,850 the unbroken. 766 00:52:02,850 --> 00:52:03,870 I went from sigma-- 767 00:52:03,870 --> 00:52:05,950 I originally had sigma-- 768 00:52:05,950 --> 00:52:07,980 sorry. 769 00:52:07,980 --> 00:52:10,930 Thank you for that. 770 00:52:10,930 --> 00:52:13,515 So really, when I talked about the linear version, 771 00:52:13,515 --> 00:52:15,390 there was two versions of the linear version. 772 00:52:15,390 --> 00:52:17,010 There was the original version where I wrote it down 773 00:52:17,010 --> 00:52:19,218 in terms of sigma and Pi vector, and then I went over 774 00:52:19,218 --> 00:52:21,337 to sigma twiddle and Pi vector. 775 00:52:21,337 --> 00:52:22,920 And when I went over to sigma twiddle, 776 00:52:22,920 --> 00:52:25,830 it was just a shift where I shifted myself 777 00:52:25,830 --> 00:52:27,128 to the proper vacuum. 778 00:52:27,128 --> 00:52:28,920 So I'm really doing the perturbation theory 779 00:52:28,920 --> 00:52:32,100 in the linear here about the proper vacuum. 780 00:52:32,100 --> 00:52:33,630 And all the other versions, I'm also 781 00:52:33,630 --> 00:52:37,905 doing about that proper vacuum, the lowest energy vacuum. 782 00:52:40,818 --> 00:52:42,610 So there was a step at the beginning, which 783 00:52:42,610 --> 00:52:44,980 is the classic step for spontaneous symmetry 784 00:52:44,980 --> 00:52:46,180 breaking, which is-- 785 00:52:46,180 --> 00:52:47,980 in some sense, I had to do that to get 786 00:52:47,980 --> 00:52:49,420 started when I went from the sigma 787 00:52:49,420 --> 00:52:50,710 to the sigma twiddle field. 788 00:52:50,710 --> 00:52:52,210 But you should really think about it 789 00:52:52,210 --> 00:52:53,720 as the sigma twiddle field as being 790 00:52:53,720 --> 00:52:55,700 field in this linear case. 791 00:52:58,510 --> 00:53:01,870 That's a sigma twiddle there. 792 00:53:01,870 --> 00:53:02,800 Other questions? 793 00:53:06,210 --> 00:53:08,880 So how would we do this construction of sigma 794 00:53:08,880 --> 00:53:11,935 if we didn't know about this linear sigma model 795 00:53:11,935 --> 00:53:12,810 way of getting there? 796 00:53:15,990 --> 00:53:18,710 So we have a generator, G, which is 797 00:53:18,710 --> 00:53:25,220 in L comma R. That's my notation for the combined SU2 left 798 00:53:25,220 --> 00:53:27,552 cross SU2 right, for example. 799 00:53:27,552 --> 00:53:29,510 And that's going to be broken down to something 800 00:53:29,510 --> 00:53:32,600 that's in the vector sub-group, which I'll denote like this-- 801 00:53:32,600 --> 00:53:38,090 V comma V. And then an element of that we can call h. 802 00:53:41,390 --> 00:53:43,220 And if we want to parameterize the coset, 803 00:53:43,220 --> 00:53:46,520 you can think that you have a G left and a G right 804 00:53:46,520 --> 00:53:50,870 in a kind of obvious notation. 805 00:53:50,870 --> 00:53:55,400 And you want some kind of field to parameterize fluctuations, 806 00:53:55,400 --> 00:53:57,140 where we pull out the h. 807 00:53:57,140 --> 00:53:58,640 So I'll call that "cascade." 808 00:54:19,170 --> 00:54:21,912 So let's just think about an example. 809 00:54:21,912 --> 00:54:23,370 So say we had something that looked 810 00:54:23,370 --> 00:54:30,700 like this, which is some set of generators that 811 00:54:30,700 --> 00:54:35,380 is in the original L comma R. And let 812 00:54:35,380 --> 00:54:38,830 me insert in here 1 in the form of gR, 813 00:54:38,830 --> 00:54:49,610 gR dagger, and then write it as a product of generators 814 00:54:49,610 --> 00:54:52,640 that looks like this. 815 00:54:59,542 --> 00:55:01,250 So I just multiply the various-- this guy 816 00:55:01,250 --> 00:55:04,610 multiplies that guy, this guy multiplies that guy. 817 00:55:04,610 --> 00:55:06,980 So this guy here has the same entries in both, 818 00:55:06,980 --> 00:55:10,150 so that's in h. 819 00:55:10,150 --> 00:55:16,520 This is in the v comma v, which is h. 820 00:55:16,520 --> 00:55:19,970 And this guy here, this gR dagger, 821 00:55:19,970 --> 00:55:22,430 is then the thing that's parameterizing the cascade 822 00:55:22,430 --> 00:55:24,780 field. 823 00:55:24,780 --> 00:55:45,040 So this is a matrix that's parameterizing the coset, 824 00:55:45,040 --> 00:55:48,160 and transforms in the way that we said. 825 00:55:51,930 --> 00:55:55,170 So that's the idea behind what the sigma field is doing. 826 00:55:55,170 --> 00:56:00,480 Now, you can ask, well, that just seems like one choice. 827 00:56:00,480 --> 00:56:02,880 And I could think about parameterizing the coset 828 00:56:02,880 --> 00:56:04,720 in many different ways. 829 00:56:04,720 --> 00:56:05,580 And that's true. 830 00:56:10,860 --> 00:56:12,480 And there's actually a nice formalism 831 00:56:12,480 --> 00:56:15,790 that takes that into account. 832 00:56:15,790 --> 00:56:20,970 So if you just talk about broken generators, which I can call X, 833 00:56:20,970 --> 00:56:26,460 then a very general definition of what this cascade field is 834 00:56:26,460 --> 00:56:32,580 is just an exponential involving those broken generators, 835 00:56:32,580 --> 00:56:35,850 and some fields to describe the fluctuations, 836 00:56:35,850 --> 00:56:37,410 and normalized in some way. 837 00:56:41,950 --> 00:56:45,660 And this is due to Callan, Coleman, Wess and Zamino. 838 00:56:49,740 --> 00:56:59,910 So it's the CCWZ prescription for parameterizing a coset. 839 00:56:59,910 --> 00:57:01,530 And the point of their prescription 840 00:57:01,530 --> 00:57:05,070 is that you have choices here for how you represent 841 00:57:05,070 --> 00:57:06,300 the broken generators. 842 00:57:14,305 --> 00:57:15,680 We start out with left generators 843 00:57:15,680 --> 00:57:18,200 and right generators, and then you 844 00:57:18,200 --> 00:57:19,940 go over to the vector generators, 845 00:57:19,940 --> 00:57:21,110 but what ones are broken? 846 00:57:21,110 --> 00:57:23,193 Well, you could take different linear combinations 847 00:57:23,193 --> 00:57:24,080 that are broken. 848 00:57:24,080 --> 00:57:27,980 And those would be equally viable choices. 849 00:57:27,980 --> 00:57:31,890 Doesn't have to be a group, the broken generators, 850 00:57:31,890 --> 00:57:47,020 so you can pick different choices of this X. 851 00:57:47,020 --> 00:57:54,520 And if we pick X, which is the left generators, 852 00:57:54,520 --> 00:57:58,832 that's actually what gives us what 853 00:57:58,832 --> 00:58:00,040 we were talking about before. 854 00:58:00,040 --> 00:58:11,200 Because then we end up parameterizing the cosets 855 00:58:11,200 --> 00:58:15,425 by something as 1 in the second entry, because we 856 00:58:15,425 --> 00:58:16,300 don't have the right. 857 00:58:16,300 --> 00:58:19,120 We're just saying pick the left, the left are broken. 858 00:58:19,120 --> 00:58:23,217 One possible choice, left plus right is unbroken. 859 00:58:23,217 --> 00:58:25,300 Then we end up with an entry here, which is sigma. 860 00:58:28,040 --> 00:58:29,960 That's our sigma. 861 00:58:29,960 --> 00:58:33,220 And also, if you work out the transformation, 862 00:58:33,220 --> 00:58:37,180 how the sigma transforms, you could reproduce-- 863 00:58:37,180 --> 00:58:45,820 you can also derive that sigma goes to L, sigma R dagger. 864 00:58:45,820 --> 00:58:47,110 But that's just one choice. 865 00:58:47,110 --> 00:58:48,290 You could do other choices. 866 00:58:48,290 --> 00:58:52,030 So for example, you could pick X a to be tau left 867 00:58:52,030 --> 00:58:54,640 a minus tau right a. 868 00:58:54,640 --> 00:58:56,500 That's another possible choice. 869 00:58:56,500 --> 00:59:09,520 And that would lead to a different field, 870 00:59:09,520 --> 00:59:12,630 but an equally valid parameterization of the coset. 871 00:59:12,630 --> 00:59:14,240 And this was actually also popular. 872 00:59:14,240 --> 00:59:18,800 It's something that people usually denote by C. 873 00:59:18,800 --> 00:59:22,280 I'm going to leave further discussion of that 874 00:59:22,280 --> 00:59:23,540 to your reading. 875 00:59:23,540 --> 00:59:26,360 There's a very nice review by Aneesh Manohar, where 876 00:59:26,360 --> 00:59:29,840 he goes through some of these things for CCSW, 877 00:59:29,840 --> 00:59:32,600 and does a nice job of talking about both of these in more 878 00:59:32,600 --> 00:59:35,850 detail than I've done here. 879 00:59:35,850 --> 00:59:39,753 So further discussion of this point will be in your reading. 880 00:59:39,753 --> 00:59:42,170 You can also look back at the original paper, if you like. 881 00:59:42,170 --> 00:59:44,360 But Aneesh has distilled it nicely. 882 00:59:51,610 --> 00:59:53,130 So any questions about that? 883 00:59:53,130 --> 00:59:55,210 There's a way of thinking-- 884 00:59:55,210 --> 00:59:57,335 I'll just keep talking until someone raises their-- 885 00:59:57,335 --> 00:59:59,502 there's a way of thinking about field redefinitions, 886 00:59:59,502 --> 01:00:01,300 even from this low-energy point of view. 887 01:00:01,300 --> 01:00:02,800 That's the key here. 888 01:00:02,800 --> 01:00:04,810 And it has to do with the freedom that you have 889 01:00:04,810 --> 01:00:06,268 and how you parameterize the coset. 890 01:00:15,910 --> 01:00:18,210 So even if we're constructing it from the bottom up, 891 01:00:18,210 --> 01:00:20,380 there's different representations. 892 01:00:20,380 --> 01:00:23,760 In fact, Weinberg likes the square root version 893 01:00:23,760 --> 01:00:27,030 of parameterizing the coset. 894 01:00:27,030 --> 01:00:30,630 Maybe he's the only person. 895 01:00:30,630 --> 01:00:33,360 He has very good reasons for being allowed 896 01:00:33,360 --> 01:00:37,830 to stick with his original results, 897 01:00:37,830 --> 01:00:39,900 rather than the exponential representation, 898 01:00:39,900 --> 01:00:42,850 which everyone else seems to favor, including me. 899 01:00:46,630 --> 01:00:50,610 So if we go back to talking about QCD, 900 01:00:50,610 --> 01:00:54,180 then it's common to call this curly V f over root 2. 901 01:00:54,180 --> 01:00:55,290 1/2 is a decay constant. 902 01:00:58,715 --> 01:01:00,090 And so the conventions, then, are 903 01:01:00,090 --> 01:01:03,322 the following-- at least one possible choice of conventions. 904 01:01:08,050 --> 01:01:12,930 There's some choice about where you put the 2's, and then 905 01:01:12,930 --> 01:01:16,000 everything else is fixed. 906 01:01:16,000 --> 01:01:17,880 So this is one possible common choice 907 01:01:17,880 --> 01:01:20,350 for the exponential representation. 908 01:01:20,350 --> 01:01:24,834 And then that gives you the chiral Lagrangian-- 909 01:01:30,270 --> 01:01:32,110 looks like that. 910 01:01:32,110 --> 01:01:35,550 And if you expand this out, you get the standard kinetic term 911 01:01:35,550 --> 01:01:37,928 for the phi field. 912 01:01:37,928 --> 01:01:39,720 That's what this normalization factor does. 913 01:01:51,500 --> 01:01:57,600 And then if you keep expanding, you get some interaction terms 914 01:01:57,600 --> 01:02:02,945 like this one, et cetera. 915 01:02:02,945 --> 01:02:05,300 So there's four point interactions in that Lagrangian 916 01:02:05,300 --> 01:02:05,800 as well. 917 01:02:09,170 --> 01:02:11,680 And that's because we had a curved-- 918 01:02:11,680 --> 01:02:13,870 our coset is nontrivial. 919 01:02:23,130 --> 01:02:26,760 So part of this story here when you talk about symmetry 920 01:02:26,760 --> 01:02:28,230 breaking has to be about the fact 921 01:02:28,230 --> 01:02:30,960 that symmetry is also broken explicitly 922 01:02:30,960 --> 01:02:33,740 in the standard model. 923 01:02:33,740 --> 01:02:36,280 And so you have to add-- 924 01:02:36,280 --> 01:02:38,480 the Goldstones are only pseudogoldstones. 925 01:02:38,480 --> 01:02:40,310 You have to add a mass. 926 01:02:40,310 --> 01:02:44,210 And the way you do that is by doing a spurion analysis. 927 01:02:49,480 --> 01:02:50,700 Remind you how that goes. 928 01:02:57,990 --> 01:03:00,900 You write a mass term like this, and to left and right. 929 01:03:03,630 --> 01:03:06,420 The mass, you could say, is just the diagonal matrix 930 01:03:06,420 --> 01:03:09,370 of up and down for SU2. 931 01:03:09,370 --> 01:03:12,330 And you could pretend, rather than talking about things that 932 01:03:12,330 --> 01:03:14,970 are violating the symmetry like these guys, 933 01:03:14,970 --> 01:03:19,590 you let the mass transform and pretend 934 01:03:19,590 --> 01:03:25,080 that you actually have something that's invariant. 935 01:03:25,080 --> 01:03:26,940 By pretending that the mass transforms, 936 01:03:26,940 --> 01:03:28,950 you're able-- if you pretend the mass transforms 937 01:03:28,950 --> 01:03:30,600 in the chiral theory-- of constructing 938 01:03:30,600 --> 01:03:33,300 things that violate the symmetry in the same way. 939 01:03:33,300 --> 01:03:36,150 You construct invariance in both theories with this kind 940 01:03:36,150 --> 01:03:37,740 of transformation law. 941 01:03:37,740 --> 01:03:40,650 And then when you go back and fix them, and say no, no, 942 01:03:40,650 --> 01:03:43,080 it doesn't transform, it's really a fixed thing, 943 01:03:43,080 --> 01:03:45,570 not a field that could transform-- it's fixed, 944 01:03:45,570 --> 01:03:46,680 the number-- 945 01:03:46,680 --> 01:03:49,600 then you violate the symmetry explicitly in the same way 946 01:03:49,600 --> 01:03:50,100 as up here. 947 01:03:50,100 --> 01:03:54,210 It's a trick for how to build things that are 948 01:03:54,210 --> 01:03:56,370 covariant in a certain form. 949 01:04:02,220 --> 01:04:05,550 So you could add the variant operators 950 01:04:05,550 --> 01:04:07,078 of the chiral Lagrangian. 951 01:04:10,910 --> 01:04:14,490 This is a different v, some parameter, 952 01:04:14,490 --> 01:04:16,650 different than the curly one we were talking about 953 01:04:16,650 --> 01:04:22,140 earlier, which is now called f. 954 01:04:36,080 --> 01:04:38,790 So this is the spurion story-- 955 01:04:38,790 --> 01:04:43,640 how to break symmetry explicitly in a different theory, 956 01:04:43,640 --> 01:04:44,360 the same way. 957 01:04:49,420 --> 01:04:51,750 And if you expand a quadratic order, 958 01:04:51,750 --> 01:04:53,325 you get mass terms for the pions. 959 01:04:57,020 --> 01:04:58,020 I won't go through that. 960 01:05:11,420 --> 01:05:13,490 But one kind of an important thing about it 961 01:05:13,490 --> 01:05:19,130 is that the mass is proportional to v times the quark 962 01:05:19,130 --> 01:05:27,500 mass, so v0, so the parameter in the Lagrangian, linear 963 01:05:27,500 --> 01:05:30,560 in the quark masses, quadratic in the Goldstone mass. 964 01:05:40,830 --> 01:05:43,110 You can also a couple of currents. 965 01:05:43,110 --> 01:05:45,210 And you can do a similar type of analysis 966 01:05:45,210 --> 01:05:49,230 with spurion-type transformation analysis for the currents. 967 01:05:52,170 --> 01:05:53,850 And I'm not going to go through this 968 01:05:53,850 --> 01:05:56,430 in any gory detail, but just enough 969 01:05:56,430 --> 01:05:58,780 to do what we need to do here. 970 01:05:58,780 --> 01:06:02,340 So here's a left-handed fermion current, standard model. 971 01:06:02,340 --> 01:06:05,490 Could be coupling to the W boson, for example. 972 01:06:05,490 --> 01:06:11,640 And you could think of getting J by taking a functional 973 01:06:11,640 --> 01:06:15,270 derivative with respect to a field that's left handed. 974 01:06:19,932 --> 01:06:21,640 And then I can do the same kind of thing, 975 01:06:21,640 --> 01:06:25,300 where I think about this L and let it transform, just like I 976 01:06:25,300 --> 01:06:27,070 was letting the M transform. 977 01:06:27,070 --> 01:06:34,220 So think about J as this capital J times minus L. 978 01:06:34,220 --> 01:06:37,012 And if I then think about this L mu transforming, 979 01:06:37,012 --> 01:06:38,720 I can build a chiral Lagrangian, and then 980 01:06:38,720 --> 01:06:41,030 use this formula to construct the current 981 01:06:41,030 --> 01:06:44,000 in the chiral Lagrangian. 982 01:06:44,000 --> 01:06:59,130 So you couple a spurion current, L mu a tau a, 983 01:06:59,130 --> 01:07:03,370 and you, in this case, in order to couple it, 984 01:07:03,370 --> 01:07:06,100 you can let it transform like a left-handed gauge field. 985 01:07:13,450 --> 01:07:15,487 And then you get something that's invariant. 986 01:07:21,700 --> 01:07:31,385 So the transformation that will make it invariant 987 01:07:31,385 --> 01:07:33,260 is to think of it as a left-hand gauge field. 988 01:07:33,260 --> 01:07:41,500 So L mu goes to L, L mu, L dagger. 989 01:07:53,680 --> 01:08:02,900 And then that gives you an invariant 990 01:08:02,900 --> 01:08:05,380 in the original theory. 991 01:08:05,380 --> 01:08:08,230 And then you build in a variant of the x theory. 992 01:08:12,570 --> 01:08:14,500 And so if you do this, and you're 993 01:08:14,500 --> 01:08:16,540 replacing, in your chiral Lagrangian, 994 01:08:16,540 --> 01:08:19,223 your partial derivative by a covariate one, 995 01:08:19,223 --> 01:08:20,890 it acts on the left because that's where 996 01:08:20,890 --> 01:08:22,899 the sigma is transforming. 997 01:08:22,899 --> 01:08:25,450 And so it's an easy way of building that in. 998 01:08:29,074 --> 01:08:31,399 And you just get an i L times the sigma on the left. 999 01:08:31,399 --> 01:08:33,899 So you take everywhere you have a derivative, you replace it 1000 01:08:33,899 --> 01:08:35,585 by this combination. 1001 01:08:35,585 --> 01:08:37,710 So we had two derivatives in our chiral Lagrangian. 1002 01:08:37,710 --> 01:08:40,770 Expand it out, and we can figure out 1003 01:08:40,770 --> 01:08:44,770 how to put it in the left-handed spurion. 1004 01:08:44,770 --> 01:08:47,082 So the spurion is just kind of a general way 1005 01:08:47,082 --> 01:08:48,540 of going from one to the other just 1006 01:08:48,540 --> 01:08:50,130 by tracking symmetry breaking. 1007 01:09:07,515 --> 01:09:08,890 And you'll get some more practice 1008 01:09:08,890 --> 01:09:12,220 with some of this on your problem set. 1009 01:09:15,944 --> 01:09:17,319 So on your problem set, what I've 1010 01:09:17,319 --> 01:09:22,330 asked you to do on problem set 2 is to do a one-loop calculation 1011 01:09:22,330 --> 01:09:24,640 in this chiral theory. 1012 01:09:24,640 --> 01:09:28,652 So you can really see that this is a quantum field 1013 01:09:28,652 --> 01:09:30,069 theory, an effective field theory, 1014 01:09:30,069 --> 01:09:33,790 not only a tree-level mnemonic theory, but really 1015 01:09:33,790 --> 01:09:35,590 a field theory in its own right where 1016 01:09:35,590 --> 01:09:39,850 you have to consider loops, contractions, Wick's theorem. 1017 01:09:39,850 --> 01:09:43,210 Everything is the same as a real quantum field theory should be. 1018 01:09:43,210 --> 01:09:45,880 It's an effective field theory of certain degrees of freedom. 1019 01:09:53,180 --> 01:09:56,320 So you do a one-loop calculation for the decay constant. 1020 01:09:56,320 --> 01:09:59,710 And there's two problems in the problem set. 1021 01:09:59,710 --> 01:10:02,002 The first one is fairly straightforward. 1022 01:10:02,002 --> 01:10:03,460 And then that's the second problem. 1023 01:10:03,460 --> 01:10:06,002 That's all I'm asking you to do because it's fairly involved. 1024 01:10:09,190 --> 01:10:11,090 It's like maybe two normal problems. 1025 01:10:14,280 --> 01:10:17,785 So here's our chiral Lagrangian. 1026 01:10:17,785 --> 01:10:19,910 And we're going to count in this chiral Lagrangian, 1027 01:10:19,910 --> 01:10:22,680 in terms of power counting, partial squared as being order 1028 01:10:22,680 --> 01:10:23,272 v0 ab q. 1029 01:10:23,272 --> 01:10:24,980 That means we're going to count p squared 1030 01:10:24,980 --> 01:10:26,210 as being order M pi squared. 1031 01:10:29,720 --> 01:10:32,400 That's going to be our accounting. 1032 01:10:32,400 --> 01:10:36,740 And so we're going to expand in these things as small 1033 01:10:36,740 --> 01:10:38,455 and something else that is large. 1034 01:10:38,455 --> 01:10:40,830 And we'll figure out what the large thing is in a minute. 1035 01:10:46,200 --> 01:10:48,410 So we do the expansion like that, where something 1036 01:10:48,410 --> 01:10:51,290 is downstairs that has mass dimensions and the things that 1037 01:10:51,290 --> 01:10:54,330 are upstairs are p squared and then Pi squared. 1038 01:10:54,330 --> 01:10:57,140 So the type of expansion we have here 1039 01:10:57,140 --> 01:11:02,360 is a combination of a derivative expansion, although it's 1040 01:11:02,360 --> 01:11:05,540 a bit of a different derivative expansion than before, 1041 01:11:05,540 --> 01:11:09,110 and an M q expansion, simultaneously driven 1042 01:11:09,110 --> 01:11:10,400 expansion in these two things. 1043 01:11:15,090 --> 01:11:17,520 If we look at L0, then it has things in it 1044 01:11:17,520 --> 01:11:19,800 like a propagator-- 1045 01:11:19,800 --> 01:11:22,350 dashed line for the scalar plan propagator. 1046 01:11:26,580 --> 01:11:29,730 It has four point interactions. 1047 01:11:29,730 --> 01:11:32,067 And if you look at how those scale, 1048 01:11:32,067 --> 01:11:33,900 we could write out what the Feynman rule is, 1049 01:11:33,900 --> 01:11:35,317 but if you look at how they scale, 1050 01:11:35,317 --> 01:11:41,700 they go like p squared over f squared, or M Pi squared 1051 01:11:41,700 --> 01:11:43,848 over f squared. 1052 01:11:43,848 --> 01:11:45,390 The p squared you could put on shell, 1053 01:11:45,390 --> 01:11:47,430 and it would be M Pi over f squared. 1054 01:11:52,890 --> 01:11:56,510 And you will actually need this Lagrangian 1055 01:11:56,510 --> 01:11:58,500 and this Feynman rule for your homework, 1056 01:11:58,500 --> 01:11:59,750 so let me tell you what it is. 1057 01:12:03,860 --> 01:12:06,800 And I'll leave it to you to work out the Feynman rule. 1058 01:12:22,490 --> 01:12:28,424 So commutators, there's a factor of 6. 1059 01:12:28,424 --> 01:12:40,550 And then there's also a term that 1060 01:12:40,550 --> 01:12:43,250 would give four-point scattering from the M2 term 1061 01:12:43,250 --> 01:12:44,810 in the Lagrangian. 1062 01:12:44,810 --> 01:12:47,938 So that also gives an M Pi squared over f squared term, 1063 01:12:47,938 --> 01:12:48,980 and that looks like that. 1064 01:12:53,190 --> 01:12:55,530 So you could build up a higher point function, 1065 01:12:55,530 --> 01:12:58,760 six-point function, et cetera from 1066 01:12:58,760 --> 01:13:01,290 the leading-order Lagrangian that has all this in it. 1067 01:13:01,290 --> 01:13:04,100 It has two parameters, f and v0. 1068 01:13:07,130 --> 01:13:08,187 So what about loops? 1069 01:13:08,187 --> 01:13:09,020 And what about this? 1070 01:13:09,020 --> 01:13:11,647 What is this scale lambda chi? 1071 01:13:11,647 --> 01:13:13,730 So the type of loops that you can have if you have 1072 01:13:13,730 --> 01:13:17,360 a four-point interaction, and just draw a solid line since 1073 01:13:17,360 --> 01:13:20,210 it's easier than drawing dashed lines-- 1074 01:13:20,210 --> 01:13:21,440 think of all these as dashed. 1075 01:13:31,170 --> 01:13:33,810 So there's cross diagrams as well as the original diagram. 1076 01:13:36,547 --> 01:13:38,130 This is not the calculation I'm asking 1077 01:13:38,130 --> 01:13:39,213 you to do in the homework. 1078 01:13:42,810 --> 01:13:49,920 If we thought about our example of Pi plus Pi 0 scattering, 1079 01:13:49,920 --> 01:13:52,800 then we could contract up Pi plus and Pi 0 fields, 1080 01:13:52,800 --> 01:13:56,250 and we could have a loop that looks like that. 1081 01:13:56,250 --> 01:13:59,280 And in order for our theory to be unitary, 1082 01:13:59,280 --> 01:14:01,890 we have to think that these loops make sense, 1083 01:14:01,890 --> 01:14:04,380 because there's cutting rules, and all that story 1084 01:14:04,380 --> 01:14:06,330 should go over for this quantum field theory. 1085 01:14:06,330 --> 01:14:08,100 And it does. 1086 01:14:08,100 --> 01:14:12,870 And that means that if you like the imaginary part of this loop 1087 01:14:12,870 --> 01:14:15,600 is needed, and you can't get the imaginary part 1088 01:14:15,600 --> 01:14:17,280 without the real part, so we have 1089 01:14:17,280 --> 01:14:19,200 to also think that the real part is 1090 01:14:19,200 --> 01:14:20,690 a physically meaningful thing. 1091 01:14:23,137 --> 01:14:24,720 And so we can go ahead and compute it, 1092 01:14:24,720 --> 01:14:26,012 and figure out what's going on. 1093 01:14:30,370 --> 01:14:35,160 So there's derivative couplings that go like f squared, 1 1094 01:14:35,160 --> 01:14:36,600 over p squared over f squared. 1095 01:14:43,930 --> 01:14:46,700 L is the loop momenta. 1096 01:14:46,700 --> 01:14:48,360 Primes are the final. 1097 01:14:48,360 --> 01:14:49,745 Unprimes are the initial. 1098 01:14:56,892 --> 01:14:59,970 You can parameterize it however you like. 1099 01:14:59,970 --> 01:15:02,070 I'm using here dimensional regularization. 1100 01:15:02,070 --> 01:15:05,700 I like to dimensional regularization, and so do you. 1101 01:15:05,700 --> 01:15:09,820 Dimensional regularization here preserves chiral symmetry, 1102 01:15:09,820 --> 01:15:11,860 which is something that we want to preserve. 1103 01:15:11,860 --> 01:15:14,070 And I'll tell you a little later what 1104 01:15:14,070 --> 01:15:16,960 would have happened if we'd done this with a cut-off. 1105 01:15:16,960 --> 01:15:20,670 So if you do this calculation, just calculate those guys, 1106 01:15:20,670 --> 01:15:23,460 you get terms that go like p to the 4th. 1107 01:15:23,460 --> 01:15:26,550 You get terms that go like p squared, M Pi squared, 1108 01:15:26,550 --> 01:15:28,200 terms that go like M Pi to the 4th. 1109 01:15:30,780 --> 01:15:33,490 You get a 4 Pi, because it's a loop, 1110 01:15:33,490 --> 01:15:34,920 and then you get an f to the 4th, 1111 01:15:34,920 --> 01:15:38,970 because there's four factors of f. 1112 01:15:38,970 --> 01:15:41,070 So what this behaves like, it's p squared 1113 01:15:41,070 --> 01:15:44,070 or M pi squared over f squared, which 1114 01:15:44,070 --> 01:15:51,930 was our tree-level, four-point function, times something 1115 01:15:51,930 --> 01:15:54,750 that's causing suppression, which is p squared 1116 01:15:54,750 --> 01:16:04,090 or M Pi squared over 4 Pi f squared. 1117 01:16:04,090 --> 01:16:07,350 So the loop is actually down by a factor of p 1118 01:16:07,350 --> 01:16:09,420 squared over 4 Pi f, all squared. 1119 01:16:13,200 --> 01:16:15,920 So just by explicit calculation and not breaking the symmetry, 1120 01:16:15,920 --> 01:16:22,100 we see that loops are suppressed by p squared over lambda chi 1121 01:16:22,100 --> 01:16:28,370 squared, where lambda chi here is defined to be 4 Pi f. 1122 01:16:31,200 --> 01:16:35,980 So the 4 Pi is important because f is like 130 GeV, 1123 01:16:35,980 --> 01:16:40,290 and you need this 4 Pi in order to have some range 1124 01:16:40,290 --> 01:16:42,370 for your chiral expansion. 1125 01:16:42,370 --> 01:16:44,640 So if you plug in numbers, depending 1126 01:16:44,640 --> 01:16:46,738 on your conventions for f-- and there's 1127 01:16:46,738 --> 01:16:48,030 a couple of common conventions. 1128 01:16:48,030 --> 01:16:53,010 You could get numbers that are like 1.6 or 1.2 GeV, so 1129 01:16:53,010 --> 01:16:55,523 some scale of order a GeV. 1130 01:16:55,523 --> 01:16:56,940 On general grounds, you could also 1131 01:16:56,940 --> 01:16:58,953 ask the question, if I just think about it 1132 01:16:58,953 --> 01:17:00,370 from the effective theory, what is 1133 01:17:00,370 --> 01:17:03,480 a reasonable scale for the denominator, 1134 01:17:03,480 --> 01:17:05,227 for this lambda chi? 1135 01:17:05,227 --> 01:17:07,560 Remember, you're going to have to construct higher order 1136 01:17:07,560 --> 01:17:09,130 operators as well. 1137 01:17:09,130 --> 01:17:11,225 And so you need to figure out what's 1138 01:17:11,225 --> 01:17:13,350 the right scale for the suppression of those higher 1139 01:17:13,350 --> 01:17:14,820 order operators. 1140 01:17:14,820 --> 01:17:16,470 And if you did it on physical grounds, 1141 01:17:16,470 --> 01:17:18,465 than what you'd expect the lambda chi to be 1142 01:17:18,465 --> 01:17:20,400 is about the mass of the rho. 1143 01:17:20,400 --> 01:17:23,010 Because that's the lowest energy hadron that you've 1144 01:17:23,010 --> 01:17:24,360 left out of this chiral theory. 1145 01:17:27,870 --> 01:17:31,830 You might expect lambda chi's of order M 1146 01:17:31,830 --> 01:17:38,130 rho, which is sort of 770 MeV. 1147 01:17:38,130 --> 01:17:41,910 And so you can't really say whether it's 770 or 1.2, 1148 01:17:41,910 --> 01:17:44,370 because the difference between those, 1149 01:17:44,370 --> 01:17:46,500 although important numerically, is not 1150 01:17:46,500 --> 01:17:52,110 within the realm of the kind of factor 2-ish meaning 1151 01:17:52,110 --> 01:17:55,500 of these twiddle symbols. 1152 01:17:55,500 --> 01:17:59,760 So these twiddle symbols mean "parametrically of order." 1153 01:17:59,760 --> 01:18:02,560 Then there's two symbols-- there's 1154 01:18:02,560 --> 01:18:03,930 much greater than or twiddle. 1155 01:18:03,930 --> 01:18:05,762 "Twiddle" means factor of 2-ish. 1156 01:18:05,762 --> 01:18:07,720 "Much greater" means parametrically [INAUDIBLE] 1157 01:18:07,720 --> 01:18:10,910 or much less than. 1158 01:18:10,910 --> 01:18:15,200 And just with those two symbols, there's some freedom. 1159 01:18:15,200 --> 01:18:17,450 These are different ways of doing an estimate for what 1160 01:18:17,450 --> 01:18:20,480 lambda chi are, just saying that there is that freedom, 1161 01:18:20,480 --> 01:18:22,520 and you have to do the full calculation 1162 01:18:22,520 --> 01:18:24,890 in a particular channel to really figure out-- 1163 01:18:24,890 --> 01:18:27,380 or a particular piece of phenomenology-- 1164 01:18:27,380 --> 01:18:29,330 what the lambda is. 1165 01:18:29,330 --> 01:18:31,040 Extract it from data if you like. 1166 01:18:37,040 --> 01:18:42,838 So that is likely a good place to stop. 1167 01:18:42,838 --> 01:18:44,630 And next time, we'll talk a little bit more 1168 01:18:44,630 --> 01:18:46,400 about these loops. 1169 01:18:46,400 --> 01:18:48,560 And we'll talk about what happens 1170 01:18:48,560 --> 01:18:51,350 to these loops in the power counting theorem that you have, 1171 01:18:51,350 --> 01:18:53,450 that organizes all diagrams in this theory, 1172 01:18:53,450 --> 01:18:57,030 and tells you what you have to do if you want to construct 1173 01:18:57,030 --> 01:18:59,008 the Lagrangian order by order-- 1174 01:18:59,008 --> 01:19:00,800 how you deal with when to put in the loops, 1175 01:19:00,800 --> 01:19:02,960 when to put in the counter terms. 1176 01:19:02,960 --> 01:19:05,500 We'll talk about those things next time.