1 00:00:00,000 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high-quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,270 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,270 --> 00:00:18,276 at ocw.mit.edu. 8 00:00:22,192 --> 00:00:23,650 IAIN STEWART: So last time, we were 9 00:00:23,650 --> 00:00:26,260 talking about regularization and power counting. 10 00:00:26,260 --> 00:00:27,928 And in particular, we were-- 11 00:00:27,928 --> 00:00:29,470 we argued that there some things that 12 00:00:29,470 --> 00:00:31,852 are nice about dimensional regularization when you're 13 00:00:31,852 --> 00:00:34,060 doing dimensional power counting, which is what we've 14 00:00:34,060 --> 00:00:35,800 been discussing so far, power counting 15 00:00:35,800 --> 00:00:38,470 and ratios of mass scales. 16 00:00:38,470 --> 00:00:40,810 So I want to continue along that theme today, 17 00:00:40,810 --> 00:00:43,600 and in particular move towards really talking 18 00:00:43,600 --> 00:00:46,590 about matching calculations in the context of mass 19 00:00:46,590 --> 00:00:48,560 of particles. 20 00:00:48,560 --> 00:00:50,560 So just continuing with this discussion 21 00:00:50,560 --> 00:00:53,470 of dimensional regularization, we also have to pick a scheme. 22 00:00:53,470 --> 00:00:56,020 And the scheme that is a nice scheme 23 00:00:56,020 --> 00:00:59,275 for dimensional regularization is this MS bar scheme. 24 00:00:59,275 --> 00:01:01,150 There are some things that are good about it. 25 00:01:04,150 --> 00:01:06,580 Well, it's good because it works under the context 26 00:01:06,580 --> 00:01:09,295 of dimensional regularization and preserves-- 27 00:01:09,295 --> 00:01:11,170 and it doesn't mess up any of the nice things 28 00:01:11,170 --> 00:01:12,130 about that regulator. 29 00:01:12,130 --> 00:01:14,770 So it preserves symmetries-- 30 00:01:14,770 --> 00:01:16,390 gauge, symmetry, Lorentz symmetry, 31 00:01:16,390 --> 00:01:18,010 the things we mentioned last time. 32 00:01:20,650 --> 00:01:24,070 In terms of doing calculations, it 33 00:01:24,070 --> 00:01:30,500 makes them technically easy, or easier. 34 00:01:30,500 --> 00:01:31,618 So a lot of-- 35 00:01:31,618 --> 00:01:33,160 if you look at the literature and you 36 00:01:33,160 --> 00:01:34,840 look at multi-loop calculations, they're 37 00:01:34,840 --> 00:01:36,393 all done on the MS bar scheme. 38 00:01:36,393 --> 00:01:38,560 And there's a reason for that, because you've only-- 39 00:01:38,560 --> 00:01:40,300 you haven't introduced these extra scales 40 00:01:40,300 --> 00:01:43,138 that would make your loop calculations more complicated. 41 00:01:46,420 --> 00:01:53,730 And finally, in the context of our discussion 42 00:01:53,730 --> 00:01:57,580 of effective field theory, it's nice because it often 43 00:01:57,580 --> 00:02:00,040 gives what I would call manifest power counting, 44 00:02:00,040 --> 00:02:02,740 where we can power count both the regulator diagrams, 45 00:02:02,740 --> 00:02:03,900 the renormalized diagrams. 46 00:02:03,900 --> 00:02:05,650 We don't have to worry about whether we've 47 00:02:05,650 --> 00:02:07,840 added the counter terms or not. 48 00:02:07,840 --> 00:02:10,887 We can just do power counting. 49 00:02:10,887 --> 00:02:12,970 So if there's some good, there should be some bad. 50 00:02:12,970 --> 00:02:13,810 So what's the bad? 51 00:02:17,830 --> 00:02:18,970 Nothing is free in life. 52 00:02:23,330 --> 00:02:24,707 So one thing that we kind of lose 53 00:02:24,707 --> 00:02:27,040 with dimensional regularization is the physical picture. 54 00:02:30,520 --> 00:02:33,850 When we had this cutoff that we introduced, 55 00:02:33,850 --> 00:02:35,920 or if we have a Wilsonian picture, 56 00:02:35,920 --> 00:02:38,500 it's very clear what you're doing as far 57 00:02:38,500 --> 00:02:42,940 as removing degrees of freedom. 58 00:02:42,940 --> 00:02:45,640 And in MS bar, that's a little less clear. 59 00:02:48,440 --> 00:02:51,220 So if it's just less clear but it's still fine, 60 00:02:51,220 --> 00:02:53,140 that would be-- 61 00:02:53,140 --> 00:02:54,910 that would just be a technical aspect 62 00:02:54,910 --> 00:02:57,110 and we would just pretty much ignore it. 63 00:02:57,110 --> 00:02:57,790 So what's one-- 64 00:02:57,790 --> 00:03:00,100 I'll give you one example of something 65 00:03:00,100 --> 00:03:02,560 in which the picture being less clear 66 00:03:02,560 --> 00:03:04,630 could actually mislead you. 67 00:03:04,630 --> 00:03:06,670 So if you calculate some renormalized quantities 68 00:03:06,670 --> 00:03:09,640 in MS bar, you could have some a-priori knowledge 69 00:03:09,640 --> 00:03:11,890 that these renormalized quantities should be positive. 70 00:03:11,890 --> 00:03:13,682 Maybe they're supposed to be kinetic energy 71 00:03:13,682 --> 00:03:16,060 or some operator, higher-dimension operator, that 72 00:03:16,060 --> 00:03:17,890 gives a kind of higher-order kinetic energy 73 00:03:17,890 --> 00:03:20,140 term, some kind of physical intuition 74 00:03:20,140 --> 00:03:21,950 that it should be positive. 75 00:03:21,950 --> 00:03:23,920 Well, that might not be true in MS bar 76 00:03:23,920 --> 00:03:27,530 because MS bar loses this physical picture. 77 00:03:27,530 --> 00:03:29,350 You're removing just the 1 over epsilons. 78 00:03:29,350 --> 00:03:32,790 You're not really having any control over the constants, 79 00:03:32,790 --> 00:03:35,380 or the constants are kind of predefined. 80 00:03:35,380 --> 00:03:37,610 If those constants are kind of too negative, 81 00:03:37,610 --> 00:03:40,140 then you can end up with renormalized matrix elements 82 00:03:40,140 --> 00:03:41,560 in MS bar that could be negative, 83 00:03:41,560 --> 00:03:44,470 your physical intuition tells you they should be positive. 84 00:03:52,102 --> 00:03:53,560 So that's something to be aware of. 85 00:04:00,790 --> 00:04:03,310 Where a physical picture could guide you and tell you 86 00:04:03,310 --> 00:04:06,310 what to expect, but you might lose that by using MS bar. 87 00:04:14,320 --> 00:04:16,959 There's another thing that's kind of technical 88 00:04:16,959 --> 00:04:20,950 but is worth knowing. 89 00:04:20,950 --> 00:04:23,230 And we will talk about it a little bit more later. 90 00:04:25,960 --> 00:04:28,330 You could ask the question, is the Wilsonian picture 91 00:04:28,330 --> 00:04:29,800 and the Wilsonian scale separation 92 00:04:29,800 --> 00:04:31,750 really equivalent to the scale separation 93 00:04:31,750 --> 00:04:34,300 that you do in dimensional regularization? 94 00:04:34,300 --> 00:04:37,930 And the answer is, almost but not quite. 95 00:04:41,140 --> 00:04:43,000 It certainly is for all the log terms. 96 00:04:43,000 --> 00:04:44,932 And I mentioned last time that the log terms 97 00:04:44,932 --> 00:04:47,140 that we saw in the two ways of doing the calculation, 98 00:04:47,140 --> 00:04:50,170 you could just see a direct correspondence between them. 99 00:04:50,170 --> 00:04:53,410 But there's a leftover residual effect 100 00:04:53,410 --> 00:04:58,450 called renormalons that does show up in the MS bar scheme 101 00:04:58,450 --> 00:05:01,900 and is related to power divergences. 102 00:05:01,900 --> 00:05:06,370 And so this effect is carefully hidden. 103 00:05:06,370 --> 00:05:08,260 It's actually hidden in the asymptotics 104 00:05:08,260 --> 00:05:11,232 of the expansion in alpha s. 105 00:05:11,232 --> 00:05:12,940 And it goes under the rubric of something 106 00:05:12,940 --> 00:05:14,470 that people call renormalons. 107 00:05:24,070 --> 00:05:25,900 So there is a leftover physical effect 108 00:05:25,900 --> 00:05:29,710 from kind of having the freedom to drop all the power 109 00:05:29,710 --> 00:05:31,960 divergent terms. 110 00:05:31,960 --> 00:05:35,110 But it carefully hides itself in a strange place 111 00:05:35,110 --> 00:05:37,108 in the gauge theory. 112 00:05:37,108 --> 00:05:38,650 And we'll talk more about this later. 113 00:05:42,335 --> 00:05:44,710 It's not so important if you're working to order alpha s. 114 00:05:44,710 --> 00:05:48,340 But if you start working toward alpha squared or higher, then-- 115 00:05:48,340 --> 00:05:50,950 or if you start going to higher orders in perturbation theory, 116 00:05:50,950 --> 00:05:53,490 then this could come in and be important. 117 00:05:53,490 --> 00:05:55,660 And it does come in in QCD at fairly low orders. 118 00:05:55,660 --> 00:05:58,720 At order alpha squared, it can cause numerical effects 119 00:05:58,720 --> 00:06:00,810 if you ignore this. 120 00:06:00,810 --> 00:06:03,340 And there are ways of taking it into account without losing 121 00:06:03,340 --> 00:06:04,243 the nice things here. 122 00:06:04,243 --> 00:06:05,660 So we'll talk about that later on. 123 00:06:08,870 --> 00:06:10,270 And then there's a final thing-- 124 00:06:10,270 --> 00:06:13,840 we had three good points, we should have three bad points-- 125 00:06:13,840 --> 00:06:15,590 the final thing we'll deal with right now. 126 00:06:15,590 --> 00:06:17,470 And that is the fact that MS bar does not 127 00:06:17,470 --> 00:06:23,050 satisfy something that's a theorem called 128 00:06:23,050 --> 00:06:24,130 the decoupling theorem. 129 00:06:34,623 --> 00:06:36,040 So what is the decoupling theorem? 130 00:06:44,300 --> 00:06:47,820 So this goes back to Appelquist and Carrazone. 131 00:06:47,820 --> 00:06:49,258 And it says the following. 132 00:06:52,330 --> 00:06:54,520 So you're thinking about an effective field theory, 133 00:06:54,520 --> 00:06:56,978 you're thinking about deriving a low-energy effect of field 134 00:06:56,978 --> 00:07:00,295 theory by integrating out some mass of particle. 135 00:07:02,910 --> 00:07:06,700 And the decoupling theorem says that if the remaining 136 00:07:06,700 --> 00:07:15,490 low-energy theory is renormalizable, 137 00:07:15,490 --> 00:07:36,280 and we use a physical renormalization scheme, 138 00:07:36,280 --> 00:07:38,890 then you kind of get what expect, 139 00:07:38,890 --> 00:07:45,550 that all the effects of the heavy particles 140 00:07:45,550 --> 00:08:07,374 turn up in couplings or effects, they're suppressed. 141 00:08:11,090 --> 00:08:13,118 So this seems very physical from what 142 00:08:13,118 --> 00:08:15,410 we've described about what effective field theories do. 143 00:08:19,330 --> 00:08:20,580 So in that sense, it's-- 144 00:08:20,580 --> 00:08:23,480 I'm not going to try to prove it to you or anything, 145 00:08:23,480 --> 00:08:25,373 but there is this caveat that you 146 00:08:25,373 --> 00:08:27,290 need to use a physical renormalization scheme, 147 00:08:27,290 --> 00:08:28,490 of which MS bar is not one. 148 00:08:32,140 --> 00:08:34,120 So we decided that we liked MS bar, 149 00:08:34,120 --> 00:08:36,190 but then we found some problems. 150 00:08:36,190 --> 00:08:38,960 And this problem here, this last problem, we're 151 00:08:38,960 --> 00:08:40,210 going to deal with right away. 152 00:08:43,720 --> 00:08:45,110 So MS bar is not physical. 153 00:08:49,800 --> 00:08:55,130 And hence, it doesn't satisfy this theorem. 154 00:08:55,130 --> 00:08:57,110 It's mass-independent. 155 00:08:57,110 --> 00:08:58,760 And that is something that we like, 156 00:08:58,760 --> 00:09:01,010 but it also is causing exactly this problem. 157 00:09:11,280 --> 00:09:13,850 That's how it-- it's why it violates the decoupling 158 00:09:13,850 --> 00:09:16,370 theorem. 159 00:09:16,370 --> 00:09:17,770 Because it's mass-independent, it 160 00:09:17,770 --> 00:09:19,340 doesn't know enough about the mass 161 00:09:19,340 --> 00:09:21,802 to know that the effect of massive particles 162 00:09:21,802 --> 00:09:22,760 should always decouple. 163 00:09:28,630 --> 00:09:30,220 So in MS bar, what you have to do 164 00:09:30,220 --> 00:09:32,290 is you have to implement that decoupling by hand. 165 00:10:00,580 --> 00:10:03,250 And we've gotten actually so used to this logic 166 00:10:03,250 --> 00:10:06,220 that it's often even not mentioned that this 167 00:10:06,220 --> 00:10:08,250 is something that we're doing. 168 00:10:08,250 --> 00:10:10,990 So go over that a bit. 169 00:10:14,693 --> 00:10:16,360 So I'll just go over this in the context 170 00:10:16,360 --> 00:10:17,770 of a particular example, which is 171 00:10:17,770 --> 00:10:19,170 the most common example, which is 172 00:10:19,170 --> 00:10:24,520 QCD with massive particles in MS bar. 173 00:10:28,700 --> 00:10:33,164 And I'll show you how it works. 174 00:10:33,164 --> 00:10:40,840 So QCD has a beta function of the renormalized coupling 175 00:10:40,840 --> 00:10:43,880 mu d by d mu of g of mu. 176 00:10:43,880 --> 00:10:45,587 And just to establish some notation, 177 00:10:45,587 --> 00:10:48,170 I'm going to remind you of some facts that you've seen before. 178 00:11:01,490 --> 00:11:05,610 So this is the beta function at lowest order. 179 00:11:05,610 --> 00:11:09,600 I'll call this factor that depends on the group 180 00:11:09,600 --> 00:11:18,550 that you're dealing with and how many light fermions you have, 181 00:11:18,550 --> 00:11:19,740 I'll call it b0. 182 00:11:23,590 --> 00:11:24,660 And this is less than 0. 183 00:11:24,660 --> 00:11:28,200 And this is only the first-order term in a series, 184 00:11:28,200 --> 00:11:31,630 but it'll be enough for our discussion right now. 185 00:11:31,630 --> 00:11:35,280 So this is g to the fifth terms, et cetera. 186 00:11:35,280 --> 00:11:36,930 That's the beta function for g. 187 00:11:36,930 --> 00:11:39,270 We can also think about using-- 188 00:11:39,270 --> 00:11:45,720 since g likes to come squared, we can switch to alpha. 189 00:11:45,720 --> 00:11:48,090 QCD is asymptotically free. 190 00:11:51,970 --> 00:12:00,845 When we solve this equation for alpha, it looks as follows. 191 00:12:12,120 --> 00:12:14,715 And the coupling constant decreases 192 00:12:14,715 --> 00:12:15,840 as you go to higher energy. 193 00:12:21,640 --> 00:12:28,300 So this is alpha at md, and this is alpha at mW. 194 00:12:28,300 --> 00:12:30,330 Alpha at mW is less than alpha at md. 195 00:12:32,858 --> 00:12:34,275 This is the lowest-order solution. 196 00:12:59,020 --> 00:13:02,520 So just a little bit more in the way of setting the stage, 197 00:13:02,520 --> 00:13:05,790 you can also associate to this solution here an intrinsic mass 198 00:13:05,790 --> 00:13:09,360 scale dimensional transmutation. 199 00:13:15,090 --> 00:13:20,010 If you form the combination mu exponential 200 00:13:20,010 --> 00:13:24,420 of minus 2 pi over beta 0-- 201 00:13:24,420 --> 00:13:34,740 b0 alpha of mu, and you use this equation over here, 202 00:13:34,740 --> 00:13:36,990 then what you'll find is that this combination is also 203 00:13:36,990 --> 00:13:40,860 equal to mu 0, same thing alpha of mu 0. 204 00:13:40,860 --> 00:13:43,700 So it's independent of what choice of scale you use, mu 205 00:13:43,700 --> 00:13:44,770 or mu 0. 206 00:13:44,770 --> 00:13:46,520 Therefore, you define it to be a constant, 207 00:13:46,520 --> 00:13:48,145 and that constant is called lambda QCD. 208 00:13:51,300 --> 00:13:55,500 Once you do that, you can also just take this equation 209 00:13:55,500 --> 00:14:03,260 and write it like this. 210 00:14:03,260 --> 00:14:06,147 Write that, rewrite that solution like this. 211 00:14:06,147 --> 00:14:07,980 So it's another way of specifying a boundary 212 00:14:07,980 --> 00:14:10,230 condition, if you like, for the differential equation. 213 00:14:10,230 --> 00:14:13,590 One way is to pick a value of the coupling somewhere, 214 00:14:13,590 --> 00:14:15,690 another way is to fix this constant. 215 00:14:23,500 --> 00:14:26,850 So this guy is independent of mu, as I said. 216 00:14:26,850 --> 00:14:29,503 And it's the scale where QCD becomes nonperturbative. 217 00:14:39,948 --> 00:14:41,490 And the reason I have to mention this 218 00:14:41,490 --> 00:14:42,990 is because we're going to be talking 219 00:14:42,990 --> 00:14:44,610 about anomalous dimensions, and I 220 00:14:44,610 --> 00:14:47,977 have to tell you when the things that I write down are valid. 221 00:14:47,977 --> 00:14:49,560 And it's all going to be valid as long 222 00:14:49,560 --> 00:14:50,770 as we're not near this scale. 223 00:14:50,770 --> 00:14:52,145 Because if we're near this scale, 224 00:14:52,145 --> 00:14:53,700 the coupling gets too large for us 225 00:14:53,700 --> 00:14:56,220 to do the perturbation theory that we're doing when we 226 00:14:56,220 --> 00:14:59,100 calculate anomalous dimensions. 227 00:14:59,100 --> 00:15:02,980 So you can ask the question, what does this thing depend on? 228 00:15:02,980 --> 00:15:05,170 And if we look at the right-hand side here, 229 00:15:05,170 --> 00:15:07,140 we can already see some things it depends on, 230 00:15:07,140 --> 00:15:09,560 because it depends on b0. 231 00:15:09,560 --> 00:15:12,960 And if we looked back at what b0 is, b0 depended on the fact 232 00:15:12,960 --> 00:15:15,175 that we were at su3 if it's QCD. 233 00:15:15,175 --> 00:15:17,175 It also depended on the number of light flavors. 234 00:15:24,670 --> 00:15:30,180 So this scale that you fix here, well, it 235 00:15:30,180 --> 00:15:33,710 depends on the order of the loop expansion. 236 00:15:33,710 --> 00:15:37,830 We wrote down a formula that was valid at first order, 237 00:15:37,830 --> 00:15:40,838 but we could extend that formula at higher orders as well. 238 00:15:40,838 --> 00:15:42,630 And the formula would change, and the value 239 00:15:42,630 --> 00:15:44,338 would change if we went to higher orders. 240 00:15:46,830 --> 00:15:50,060 Depends on the number of light flavors. 241 00:15:50,060 --> 00:15:52,290 And it actually also starts to depend on the scheme 242 00:15:52,290 --> 00:15:53,460 but only at two loops. 243 00:15:57,180 --> 00:15:59,580 So it'll depend on MS bar versus MS, 244 00:15:59,580 --> 00:16:03,960 for example, but only beyond two loops. 245 00:16:03,960 --> 00:16:09,740 Now, the issue with this is that a priori, nothing tells us 246 00:16:09,740 --> 00:16:11,720 whether if you have a top quark or an up 247 00:16:11,720 --> 00:16:14,450 quark or a bottom quark. 248 00:16:14,450 --> 00:16:16,610 In MS bar, nothing a prior tells us 249 00:16:16,610 --> 00:16:20,030 what to do with these formulas, what to include in the b0. 250 00:16:20,030 --> 00:16:22,490 Just seems like we should include everything that exists, 251 00:16:22,490 --> 00:16:27,701 all the known light fermions that couple to the gauge field. 252 00:16:27,701 --> 00:16:30,050 But there might be some we don't know about. 253 00:16:30,050 --> 00:16:31,830 Should we include those, too? 254 00:16:31,830 --> 00:16:35,270 MS bar, with the logic we've presented so far, 255 00:16:35,270 --> 00:16:37,770 doesn't tell us what to include there in that nf. 256 00:16:45,540 --> 00:16:47,030 So let me phrase it this way. 257 00:16:51,292 --> 00:16:52,750 But the top quark and the up quark, 258 00:16:52,750 --> 00:16:56,020 which have very different masses, both contribute to b0. 259 00:17:00,870 --> 00:17:03,270 And it seems like they do that for any mu. 260 00:17:07,180 --> 00:17:09,680 Even though-- even if I'm at a very low-energy scale 261 00:17:09,680 --> 00:17:11,110 and the top is very heavy, MS bar 262 00:17:11,110 --> 00:17:13,631 is not smart enough to decouple the top quark. 263 00:17:13,631 --> 00:17:15,339 If you were to work in a physical scheme, 264 00:17:15,339 --> 00:17:17,579 then this decoupling theorem guarantees, actually, 265 00:17:17,579 --> 00:17:19,720 that the top quark would decouple, 266 00:17:19,720 --> 00:17:22,990 and it would drop out of the beta function in that scheme. 267 00:17:22,990 --> 00:17:25,099 But in this MS bar scheme, it doesn't happen. 268 00:17:28,530 --> 00:17:31,070 So the solution to this is to build back 269 00:17:31,070 --> 00:17:33,290 in the thing that happened in the physical schemes 270 00:17:33,290 --> 00:17:35,740 into our MS bar scheme. 271 00:17:35,740 --> 00:17:38,480 And we do that by implementing the decoupling by hand. 272 00:17:50,450 --> 00:17:53,330 And we just say that when we get to a mass threshold, 273 00:17:53,330 --> 00:17:56,480 we're going to integrate out the heavy fermion in this case. 274 00:18:06,833 --> 00:18:09,590 So at some scale of order of the mass of that fermion, 275 00:18:09,590 --> 00:18:13,040 we're going to integrate out the heavy fermion. 276 00:18:13,040 --> 00:18:15,350 And that's an example of during matching, actually. 277 00:18:20,030 --> 00:18:21,950 You're moving from one theory to another, 278 00:18:21,950 --> 00:18:23,870 because you're changing the field content. 279 00:18:23,870 --> 00:18:27,432 You're removing some fields, and that's 280 00:18:27,432 --> 00:18:28,640 an example of doing matching. 281 00:18:31,400 --> 00:18:36,050 So what this means is that we'll define different b0's 282 00:18:36,050 --> 00:18:37,430 depending on what scale we're at. 283 00:18:43,410 --> 00:18:47,380 So nf would be 6 if we're at scales above the top quark. 284 00:18:55,820 --> 00:18:58,490 But then we drop that nf down to 5 285 00:18:58,490 --> 00:19:05,360 once we're below the top quark mass, et cetera. 286 00:19:10,670 --> 00:19:12,920 So it's a discrete jump in what the b0 is. 287 00:19:17,525 --> 00:19:18,900 So in some sense, what that means 288 00:19:18,900 --> 00:19:22,680 is that this kind of matching is forced upon you in order 289 00:19:22,680 --> 00:19:25,230 to preserve physics in MS bar. 290 00:19:25,230 --> 00:19:27,930 If you were to do some other scheme like a physical scheme, 291 00:19:27,930 --> 00:19:30,180 it's not to say that you couldn't take the same logic. 292 00:19:33,240 --> 00:19:35,790 You could do matching to integrate our heavy particles 293 00:19:35,790 --> 00:19:37,240 in those schemes as well. 294 00:19:37,240 --> 00:19:40,470 And in fact, some sometimes people have. 295 00:19:40,470 --> 00:19:42,220 But in MS bar It's really forced upon you. 296 00:19:42,220 --> 00:19:44,260 There's no way out, because you otherwise 297 00:19:44,260 --> 00:19:46,300 wouldn't get the physics right. 298 00:19:48,725 --> 00:19:49,850 So we've got to be careful. 299 00:19:49,850 --> 00:19:52,017 If you gain something, sometimes you lose something. 300 00:19:52,017 --> 00:19:53,862 And if it's physics that you're losing, 301 00:19:53,862 --> 00:19:55,070 you have to build it back in. 302 00:19:59,050 --> 00:19:59,550 OK. 303 00:19:59,550 --> 00:20:02,165 So how do we actually do this matching? 304 00:20:04,670 --> 00:20:07,790 Well, we use what are called matching conditions. 305 00:20:20,790 --> 00:20:22,553 So what is a matching condition? 306 00:20:25,940 --> 00:20:28,370 At some scale which I'll call mu m-- 307 00:20:28,370 --> 00:20:30,290 that could be equal to m, but it could also 308 00:20:30,290 --> 00:20:35,180 be equal to twice m or 1/2 m, some scale that's of order m-- 309 00:20:35,180 --> 00:20:37,370 we're going to demand something about the theory 310 00:20:37,370 --> 00:20:38,210 above and below. 311 00:20:43,040 --> 00:20:47,660 And that is that S matrix elements in the two theories 312 00:20:47,660 --> 00:20:48,470 are going to agree. 313 00:20:53,150 --> 00:20:55,670 Of course, we should be careful to make sure 314 00:20:55,670 --> 00:20:58,842 that we work within the realm of things 315 00:20:58,842 --> 00:21:00,800 that we can calculate in the low-energy theory. 316 00:21:00,800 --> 00:21:03,860 So we should consider only S matrix elements 317 00:21:03,860 --> 00:21:05,660 with light external particles. 318 00:21:07,875 --> 00:21:09,500 If we're getting rid of the heavy ones, 319 00:21:09,500 --> 00:21:11,208 we don't want them on the external lines. 320 00:21:17,900 --> 00:21:22,700 So the basic idea is that we set up matching conditions which 321 00:21:22,700 --> 00:21:25,190 are conditions that say that S matrix elements should agree 322 00:21:25,190 --> 00:21:30,500 between theories 1 and 2, what we called theories 1 323 00:21:30,500 --> 00:21:31,270 and 2 last time. 324 00:21:35,790 --> 00:21:39,200 So if we do that for QCD, the example 325 00:21:39,200 --> 00:21:42,238 we've been talking about, and we do it at lowest order, 326 00:21:42,238 --> 00:21:44,030 then the conditions are actually very easy. 327 00:21:48,870 --> 00:21:53,210 So here's the picture. 328 00:21:53,210 --> 00:22:02,070 Let's imagine that the top quark is here, bottom quark is here. 329 00:22:02,070 --> 00:22:03,310 My picture is not to scale. 330 00:22:06,190 --> 00:22:08,830 Charm quark, et cetera. 331 00:22:17,720 --> 00:22:20,630 Then what we do when we want to say that the theory 332 00:22:20,630 --> 00:22:24,320 above and below, that that threshold is the same, if we 333 00:22:24,320 --> 00:22:27,650 just match QCD above and below, it just gives a continuity 334 00:22:27,650 --> 00:22:30,150 condition on alpha. 335 00:22:30,150 --> 00:22:33,840 So let me write it down at this scale. 336 00:22:33,840 --> 00:22:36,560 So here we would have alpha 6. 337 00:22:36,560 --> 00:22:38,570 Alpha depends on the number of flavors. 338 00:22:38,570 --> 00:22:40,887 Here we have alpha 5. 339 00:22:40,887 --> 00:22:42,470 A different definition of the coupling 340 00:22:42,470 --> 00:22:44,012 a different number of active flavors, 341 00:22:44,012 --> 00:22:45,620 a different value of b0. 342 00:22:51,060 --> 00:22:51,560 3. 343 00:22:54,417 --> 00:22:56,000 And what these arrows are representing 344 00:22:56,000 --> 00:22:59,600 are just kind of renormalization group evolution, 345 00:22:59,600 --> 00:23:02,133 which I'll just call running. 346 00:23:02,133 --> 00:23:03,800 And then what the lines are representing 347 00:23:03,800 --> 00:23:06,120 is doing some matching. 348 00:23:06,120 --> 00:23:10,130 So every time we reach a threshold, we do a matching, 349 00:23:10,130 --> 00:23:13,460 we switch the field contact, and we 350 00:23:13,460 --> 00:23:17,160 get a new effective field theory with a new coupling constant. 351 00:23:17,160 --> 00:23:21,920 So the coupling just depends on what theory content we have. 352 00:23:21,920 --> 00:23:27,440 And the matching condition is, say, at this b quark scale 353 00:23:27,440 --> 00:23:33,650 that alpha s at 5 at some scale that's of order mb 354 00:23:33,650 --> 00:23:40,190 is equal to alpha s at 4. 355 00:23:40,190 --> 00:23:44,340 That's the leading order matching condition here. 356 00:23:44,340 --> 00:23:48,560 And similarly, the same condition at all scales. 357 00:23:48,560 --> 00:23:52,620 And this is mu b, just to be clear, 358 00:23:52,620 --> 00:23:57,860 is something that's of order mb, could be equal to mb, 359 00:23:57,860 --> 00:24:01,830 could be equal to mb/2, could be twice mb. 360 00:24:01,830 --> 00:24:03,830 And sometimes people use these different choices 361 00:24:03,830 --> 00:24:06,060 to get uncertainties. 362 00:24:06,060 --> 00:24:06,560 OK. 363 00:24:06,560 --> 00:24:08,018 So it seems fairly straightforward, 364 00:24:08,018 --> 00:24:10,850 just continuity of the coupling. 365 00:24:10,850 --> 00:24:13,003 But that has some caveats. 366 00:24:13,003 --> 00:24:14,420 But go ahead and ask the question. 367 00:24:14,420 --> 00:24:15,045 AUDIENCE: Yeah. 368 00:24:15,045 --> 00:24:18,270 So how do you know that that's the physical observable 369 00:24:18,270 --> 00:24:19,740 that you care about? 370 00:24:19,740 --> 00:24:22,740 What if I was trying to measure the [INAUDIBLE] alpha s? 371 00:24:25,740 --> 00:24:27,370 IAIN STEWART: So this is the-- 372 00:24:27,370 --> 00:24:27,870 yeah. 373 00:24:27,870 --> 00:24:30,310 So if you're going-- 374 00:24:30,310 --> 00:24:34,218 so you have to think about it as not the derivative of alpha s, 375 00:24:34,218 --> 00:24:36,510 but you have to think about it as an S matrix element-- 376 00:24:36,510 --> 00:24:38,880 something two-to-two scattering, right? 377 00:24:38,880 --> 00:24:40,770 If you do two-to-two scattering, then it's 378 00:24:40,770 --> 00:24:43,500 just going to be proportional to alpha of mu. 379 00:24:43,500 --> 00:24:46,870 But the mu dependence in your leading order prediction 380 00:24:46,870 --> 00:24:48,180 is not really fixed. 381 00:24:48,180 --> 00:24:49,930 You need to go to higher order to do that. 382 00:24:49,930 --> 00:24:51,472 So if you're just ensuring continuity 383 00:24:51,472 --> 00:24:54,210 at leading order of S matrix elements, 384 00:24:54,210 --> 00:24:56,925 then this is all you need. 385 00:24:56,925 --> 00:24:58,800 And if you want to construct something that's 386 00:24:58,800 --> 00:25:00,570 like a derivative of alpha, you have 387 00:25:00,570 --> 00:25:02,940 to think of the derivatives of the S matrix. 388 00:25:02,940 --> 00:25:05,610 But you can't take derivatives of mu, 389 00:25:05,610 --> 00:25:08,470 because that's a higher-order question. 390 00:25:08,470 --> 00:25:10,576 So this is all you need at leading order. 391 00:25:10,576 --> 00:25:11,118 AUDIENCE: OK. 392 00:25:11,118 --> 00:25:15,997 So maybe my derivative example is [INAUDIBLE] necessarily know 393 00:25:15,997 --> 00:25:18,270 the alpha s, you'd get only-- 394 00:25:18,270 --> 00:25:19,570 IAIN STEWART: So in general-- 395 00:25:19,570 --> 00:25:20,070 right. 396 00:25:20,070 --> 00:25:21,700 AUDIENCE: --but I don't necessarily know that-- 397 00:25:21,700 --> 00:25:21,930 IAIN STEWART: Yeah. 398 00:25:21,930 --> 00:25:23,250 No, in general, you have to-- 399 00:25:23,250 --> 00:25:25,110 this is like one particular example. 400 00:25:25,110 --> 00:25:28,080 In general, you have to ensure continuity of all S matrix 401 00:25:28,080 --> 00:25:29,190 elements. 402 00:25:29,190 --> 00:25:31,275 And that will-- in this case, we'll 403 00:25:31,275 --> 00:25:33,150 give you conditions also and what's happening 404 00:25:33,150 --> 00:25:35,240 with masses and stuff. 405 00:25:35,240 --> 00:25:37,650 So it's all the parameters of the theory. 406 00:25:37,650 --> 00:25:41,350 There should be conditions for all of them. 407 00:25:41,350 --> 00:25:43,650 AUDIENCE: I guess these words seem more complicated 408 00:25:43,650 --> 00:25:47,700 than just asking for [INAUDIBLE] some different equation. 409 00:25:47,700 --> 00:25:49,200 Does it ever boil down to anything-- 410 00:25:49,200 --> 00:25:50,760 do you ever have to do anything more complicated than just 411 00:25:50,760 --> 00:25:51,210 like-- 412 00:25:51,210 --> 00:25:51,600 IAIN STEWART: I'm just-- 413 00:25:51,600 --> 00:25:51,990 yeah, so-- 414 00:25:51,990 --> 00:25:53,960 AUDIENCE: --change the beta function [INAUDIBLE] continuity 415 00:25:53,960 --> 00:25:54,853 and-- 416 00:25:54,853 --> 00:25:56,895 IAIN STEWART: It's really-- it's not complicated. 417 00:25:56,895 --> 00:25:58,790 It's just continuity of S matrix elements. 418 00:25:58,790 --> 00:26:01,380 But you have to just know that it's S matrix elements and not 419 00:26:01,380 --> 00:26:02,760 the Lagrangian. 420 00:26:02,760 --> 00:26:05,890 Because those are two different things. 421 00:26:05,890 --> 00:26:10,150 So let me give you an example why it's not the Lagrangian. 422 00:26:10,150 --> 00:26:12,090 So this kind of thing seems simple, right? 423 00:26:12,090 --> 00:26:14,750 Just say, well, the Lagrangian is continuous. 424 00:26:14,750 --> 00:26:17,040 The alpha-- the g that appears in the Lagrangian 425 00:26:17,040 --> 00:26:17,700 is continuous. 426 00:26:17,700 --> 00:26:21,000 But that's not true once you go to higher orders. 427 00:26:21,000 --> 00:26:23,190 That's only a leading order statement. 428 00:26:23,190 --> 00:26:25,830 If I write down the analog of this condition at higher 429 00:26:25,830 --> 00:26:27,370 orders, it looks as follows. 430 00:26:33,390 --> 00:26:37,170 So the coupling is actually not continuous at MC bar 431 00:26:37,170 --> 00:26:38,460 at higher orders. 432 00:26:41,420 --> 00:26:43,940 So demanding continuity of the S matrix elements 433 00:26:43,940 --> 00:26:46,190 does not lead to continuity of the coupling. 434 00:26:46,190 --> 00:26:53,450 And the matching condition at some scale looks like this. 435 00:27:07,250 --> 00:27:09,525 So the words are carefully crafted to be correct. 436 00:27:12,043 --> 00:27:13,710 And they sound a little more complicated 437 00:27:13,710 --> 00:27:16,500 than they need to be, because I want them to be correct, 438 00:27:16,500 --> 00:27:17,217 even if I want-- 439 00:27:17,217 --> 00:27:19,800 even if I were to do matching at higher orders in perturbation 440 00:27:19,800 --> 00:27:20,300 theory. 441 00:27:27,030 --> 00:27:28,920 And it's really at this alpha squared level 442 00:27:28,920 --> 00:27:33,500 that you start to see things a little more interesting. 443 00:27:50,260 --> 00:27:50,760 OK. 444 00:27:50,760 --> 00:27:52,718 So now we've gone two others beyond what I just 445 00:27:52,718 --> 00:27:54,980 told you before. 446 00:27:54,980 --> 00:27:57,590 So continuity at lowest order-- 447 00:27:57,590 --> 00:27:59,460 at the next order, you can retain continuity 448 00:27:59,460 --> 00:28:01,377 as long as you pick the particular point where 449 00:28:01,377 --> 00:28:02,655 mu b is equal to mb. 450 00:28:02,655 --> 00:28:04,530 And you'd have-- then this log would go away, 451 00:28:04,530 --> 00:28:06,547 and you'd still have continuity. 452 00:28:06,547 --> 00:28:08,880 But even that doesn't work out once you go to one higher 453 00:28:08,880 --> 00:28:10,297 order, there's this constant. 454 00:28:10,297 --> 00:28:12,630 You could get rid of the logs by picking a scale choice, 455 00:28:12,630 --> 00:28:14,005 but there's no scale choice which 456 00:28:14,005 --> 00:28:19,660 will make it continuous once you get to alpha squared 457 00:28:19,660 --> 00:28:22,600 in this matching condition. 458 00:28:22,600 --> 00:28:25,530 So this is the condition that's necessary to ensure continuity 459 00:28:25,530 --> 00:28:27,060 of S matrix elements once you get 460 00:28:27,060 --> 00:28:30,210 to that level of perturbation theory. 461 00:28:34,458 --> 00:28:38,050 OK, so that is why I said S matrix and not 462 00:28:38,050 --> 00:28:40,185 just continuity of couplings. 463 00:28:40,185 --> 00:28:43,102 AUDIENCE: So this condition is for 2-by-2 scattering, 464 00:28:43,102 --> 00:28:44,560 And then you get other conditions-- 465 00:28:44,560 --> 00:28:46,227 IAIN STEWART: This is the only condition 466 00:28:46,227 --> 00:28:49,103 you need for all the S matrix-- 467 00:28:49,103 --> 00:28:50,770 you can use different S matrix elements, 468 00:28:50,770 --> 00:28:52,940 and they'll lead to the same condition. 469 00:28:52,940 --> 00:28:55,220 AUDIENCE: How did you-- where does it come from? 470 00:28:55,220 --> 00:28:57,095 IAIN STEWART: So this, it comes from ensuring 471 00:28:57,095 --> 00:28:58,840 that I calculate S matrix elements, say, 472 00:28:58,840 --> 00:29:02,650 2-to-2 scattering, in the theory with five flavors and four 473 00:29:02,650 --> 00:29:04,096 flavors. 474 00:29:04,096 --> 00:29:05,320 AUDIENCE: At two loops? 475 00:29:05,320 --> 00:29:08,010 IAIN STEWART: And I demand that-- yeah, up to two loops. 476 00:29:08,010 --> 00:29:09,730 So where would this guy here come from? 477 00:29:09,730 --> 00:29:12,910 This guy here would come from the graph 478 00:29:12,910 --> 00:29:15,850 with an explicit b quark, which is 479 00:29:15,850 --> 00:29:19,823 in the theory with the b quark but not in the theory without. 480 00:29:19,823 --> 00:29:21,490 Once you get to this level, then there's 481 00:29:21,490 --> 00:29:23,200 more complicated diagrams. 482 00:29:23,200 --> 00:29:26,120 Just think of generalizations of this. 483 00:29:26,120 --> 00:29:28,630 And they involve constants as well as logs. 484 00:29:28,630 --> 00:29:32,170 This guy just involves a log. 485 00:29:32,170 --> 00:29:33,670 And really, what you're ensuring is 486 00:29:33,670 --> 00:29:36,100 that in the theory with the b quark, which 487 00:29:36,100 --> 00:29:39,700 is this five-flavor theory, you get the same S matrix 488 00:29:39,700 --> 00:29:42,250 elements as in the theory with four flavors. 489 00:29:42,250 --> 00:29:46,253 Since the diagrams differ in those two theories, 490 00:29:46,253 --> 00:29:47,920 they're kind of the same at lowest order 491 00:29:47,920 --> 00:29:49,180 because you're just doing two graphs. 492 00:29:49,180 --> 00:29:51,263 But once you have the b quark and it can go around 493 00:29:51,263 --> 00:29:55,840 in the loop, then they differ, and the conditions 494 00:29:55,840 --> 00:29:57,285 become more complicated. 495 00:30:00,523 --> 00:30:01,940 So any other questions about that? 496 00:30:09,580 --> 00:30:10,230 OK. 497 00:30:10,230 --> 00:30:12,930 So one other thing that we see from this 498 00:30:12,930 --> 00:30:14,550 is related to these logarithms. 499 00:30:20,180 --> 00:30:21,980 We also see from these conditions 500 00:30:21,980 --> 00:30:26,060 here that there is going to be no large logarithms as 501 00:30:26,060 --> 00:30:33,440 long as we pick the scale where we do this matching to be 502 00:30:33,440 --> 00:30:34,100 of order mb. 503 00:30:43,690 --> 00:30:47,440 So you don't want to pick mu to be the gut scale 504 00:30:47,440 --> 00:30:52,670 or something you want to pick it to be some scale so mu 505 00:30:52,670 --> 00:30:56,510 is equal to mu b, which is of order mb. 506 00:30:56,510 --> 00:30:58,293 Because you don't want, for example, 507 00:30:58,293 --> 00:30:59,960 to make this log so large that it starts 508 00:30:59,960 --> 00:31:02,608 to overcome the coupling. 509 00:31:02,608 --> 00:31:04,400 You want it to be-- you want this to really 510 00:31:04,400 --> 00:31:06,050 be a perturbative thing so it makes sense 511 00:31:06,050 --> 00:31:08,175 to think about this, and then this is a correction, 512 00:31:08,175 --> 00:31:11,950 and that's another correction. 513 00:31:11,950 --> 00:31:12,680 OK. 514 00:31:12,680 --> 00:31:18,200 So the general procedure is the same idea. 515 00:31:22,770 --> 00:31:25,390 So if I just kind of adopt a more general procedure 516 00:31:25,390 --> 00:31:27,150 for massive particles [INAUDIBLE] 517 00:31:27,150 --> 00:31:28,150 a more general notation. 518 00:31:35,592 --> 00:31:37,425 So this is with any operators and couplings. 519 00:31:37,425 --> 00:31:39,000 It doesn't have to be gauge theory. 520 00:31:45,800 --> 00:31:47,650 And if we have a hierarchy of particles-- 521 00:31:54,210 --> 00:31:58,310 let's say we have n of them-- 522 00:31:58,310 --> 00:32:06,220 when we want to pass from an L1 to an L2, an L3, to an Ln-- 523 00:32:08,780 --> 00:32:13,890 and we're doing the same type of thing we just did over there. 524 00:32:13,890 --> 00:32:16,620 So the steps are the following. 525 00:32:16,620 --> 00:32:20,970 Consider S matrix elements in theory 1. 526 00:32:20,970 --> 00:32:25,290 Do so at a scale which I'll call mu 1 that's of order m1, 527 00:32:25,290 --> 00:32:29,550 and match that onto in the same way of assuring the continuity 528 00:32:29,550 --> 00:32:33,600 and do-- using these matching conditions, match that onto L2. 529 00:32:37,440 --> 00:32:39,480 So this is the technical step by which 530 00:32:39,480 --> 00:32:40,830 I said that you could do-- 531 00:32:40,830 --> 00:32:43,980 you could in a top-down approach take theory 1 532 00:32:43,980 --> 00:32:46,550 and determine the parameters of theory 2. 533 00:32:46,550 --> 00:32:49,260 The matching conditions are determining the parameters. 534 00:32:49,260 --> 00:32:53,460 Alpha s 4 is of parameter of theory 2 to alpha s 5. 535 00:32:53,460 --> 00:32:54,860 This is theory 1. 536 00:32:54,860 --> 00:32:57,270 And we're just determining what this alpha s 4 should be, 537 00:32:57,270 --> 00:32:59,103 and this is the condition that relates them. 538 00:33:02,530 --> 00:33:04,620 So after you do that step, if you 539 00:33:04,620 --> 00:33:07,150 want to go through this picture here, 540 00:33:07,150 --> 00:33:11,090 or the analog of that picture for this case over here, 541 00:33:11,090 --> 00:33:14,095 then you need to compute the beta functions 542 00:33:14,095 --> 00:33:15,720 in anomalous dimensions in this theory. 543 00:33:29,630 --> 00:33:32,390 So it's the theory that doesn't have particle 1. 544 00:33:32,390 --> 00:33:37,080 And then you evolve/run the couplings down, 545 00:33:37,080 --> 00:33:39,420 which just means using the evolution 546 00:33:39,420 --> 00:33:43,302 equation for the couplings, whatever they may be. 547 00:33:43,302 --> 00:33:45,510 So we had the evolution equation for alpha S a minute 548 00:33:45,510 --> 00:33:46,740 ago on the board. 549 00:33:46,740 --> 00:33:51,053 And I would just use that to go from a scale of order mb 550 00:33:51,053 --> 00:33:52,220 down to a scale of order mc. 551 00:33:57,470 --> 00:33:58,550 And then I repeat. 552 00:34:04,670 --> 00:34:06,550 And this is the general kind of paradigm 553 00:34:06,550 --> 00:34:09,190 of matching and running that you hear about in effective theory 554 00:34:09,190 --> 00:34:09,790 all the time. 555 00:34:15,610 --> 00:34:17,554 So we just keep going. 556 00:34:17,554 --> 00:34:19,929 And at the end of the day, we're going to stop somewhere. 557 00:34:19,929 --> 00:34:21,304 And the place we're going to stop 558 00:34:21,304 --> 00:34:23,895 is the place we want to do lower-energy physics. 559 00:34:23,895 --> 00:34:25,520 So let's say we stop at the n-th level, 560 00:34:25,520 --> 00:34:27,060 since that's the last level I wrote. 561 00:34:36,037 --> 00:34:38,120 So if you're interested in dynamics at that scale, 562 00:34:38,120 --> 00:34:39,037 that's where you stop. 563 00:34:48,949 --> 00:34:52,170 And that's where you compute your final matrix elements. 564 00:34:52,170 --> 00:34:54,179 So everything up until that stage 565 00:34:54,179 --> 00:34:57,440 is just to determine the theory Ln 566 00:34:57,440 --> 00:35:00,720 and what are the values of decoupling in that theory-- 567 00:35:00,720 --> 00:35:05,140 determined from knowing information at the high scale. 568 00:35:05,140 --> 00:35:06,730 Knowing information in theory 1, how 569 00:35:06,730 --> 00:35:09,010 do I propagate that knowledge all the way down 570 00:35:09,010 --> 00:35:13,630 to low energies consistently without losing 571 00:35:13,630 --> 00:35:14,987 information I need? 572 00:35:14,987 --> 00:35:16,570 And then once I'm at the lowest scale, 573 00:35:16,570 --> 00:35:20,453 I just do my computations of observables. 574 00:35:42,138 --> 00:35:42,638 Yeah. 575 00:35:42,638 --> 00:35:43,131 AUDIENCE: Professor. 576 00:35:43,131 --> 00:35:43,923 IAIN STEWART: Sure. 577 00:35:43,923 --> 00:35:47,450 AUDIENCE: [INAUDIBLE] mention at the scale where the particle is 578 00:35:47,450 --> 00:35:48,851 there-- 579 00:35:48,851 --> 00:35:51,020 actually, I was a bit confused because if we 580 00:35:51,020 --> 00:35:53,470 write the full theory, then, say, 581 00:35:53,470 --> 00:35:55,990 2-to-2 scattering in that mass scale, 582 00:35:55,990 --> 00:35:59,090 isn't there any like [INAUDIBLE] say resonance effect, 583 00:35:59,090 --> 00:36:02,570 and then you want to match that to a theory 584 00:36:02,570 --> 00:36:05,240 without the particle? 585 00:36:05,240 --> 00:36:06,680 I thought we should match the s-- 586 00:36:06,680 --> 00:36:08,730 IAIN STEWART: So there is no resonance effect. 587 00:36:08,730 --> 00:36:12,560 And the reason is because the momentum on the external lines 588 00:36:12,560 --> 00:36:13,850 that you're taking-- 589 00:36:13,850 --> 00:36:16,780 you're thinking of it as small, right? 590 00:36:16,780 --> 00:36:18,680 The scale is chosen to be the mass 591 00:36:18,680 --> 00:36:22,860 of the particle, the cutoff, mu, the soft cutoff. 592 00:36:22,860 --> 00:36:25,520 But not the momentum of the particles. 593 00:36:25,520 --> 00:36:26,926 Good question. 594 00:36:26,926 --> 00:36:28,315 Any other questions? 595 00:36:32,020 --> 00:36:33,310 OK. 596 00:36:33,310 --> 00:36:38,080 So we're not really done talking about subtleties here. 597 00:36:41,100 --> 00:36:43,120 And we're not really done talking about some 598 00:36:43,120 --> 00:36:46,280 of the key ideas that come into these calculations. 599 00:36:46,280 --> 00:36:49,420 I've given you a very simple example just the coupling. 600 00:36:49,420 --> 00:36:50,740 That's a little bit too simple. 601 00:36:50,740 --> 00:36:52,365 So I want to do something a little more 602 00:36:52,365 --> 00:36:57,750 sophisticated but still fairly simple and widely used. 603 00:36:57,750 --> 00:36:59,548 And that is in the standard model just 604 00:36:59,548 --> 00:37:01,090 to take the heaviest particles, which 605 00:37:01,090 --> 00:37:04,750 are the top, the Higgs, the W, and the Z, and remove them. 606 00:37:12,330 --> 00:37:15,100 So we'll spend a bit of time talking about what 607 00:37:15,100 --> 00:37:17,870 happens when you do that. 608 00:37:17,870 --> 00:37:19,650 That's not the only thing you could do. 609 00:37:19,650 --> 00:37:22,355 And in particular, before people do how heavy the Higgs was, 610 00:37:22,355 --> 00:37:23,980 you could think of other possibilities. 611 00:37:23,980 --> 00:37:25,870 And before people knew how heavy the top was, 612 00:37:25,870 --> 00:37:27,988 people did think of other possibilities. 613 00:37:32,470 --> 00:37:35,830 This is actually a reasonable thing to do, though. 614 00:37:35,830 --> 00:37:38,930 let me give you one example of another possibility. 615 00:37:38,930 --> 00:37:42,010 Or you could say the top is heavier than the W and the Z, 616 00:37:42,010 --> 00:37:45,700 so why don't I do the top first, and then 617 00:37:45,700 --> 00:37:48,235 I'll just follow your diagram over there. 618 00:37:48,235 --> 00:37:50,740 I erased it, but it'd follow the picture. 619 00:37:50,740 --> 00:37:52,810 And then I'll do some running in the theory 620 00:37:52,810 --> 00:37:54,310 without the top quark, and then I'll 621 00:37:54,310 --> 00:37:56,530 get down to the W and the Z scales, 622 00:37:56,530 --> 00:37:59,840 then I'll remove the W and the Z. 623 00:37:59,840 --> 00:38:00,340 OK. 624 00:38:00,340 --> 00:38:02,530 That would be a valid thing to do. 625 00:38:02,530 --> 00:38:05,530 But it introduces complications that are actually not 626 00:38:05,530 --> 00:38:06,250 worth the effort. 627 00:38:09,850 --> 00:38:11,900 And benefit is actually not that great. 628 00:38:11,900 --> 00:38:15,155 So I want to emphasize that. 629 00:38:15,155 --> 00:38:17,530 When should you think of things as being comparably heavy 630 00:38:17,530 --> 00:38:19,840 versus when should you think of things being hierarchically 631 00:38:19,840 --> 00:38:20,340 heavy? 632 00:38:23,620 --> 00:38:27,190 Well, one complication of removing the top quark 633 00:38:27,190 --> 00:38:32,200 is that it breaks su2 cross u1 gauge invariance, 634 00:38:32,200 --> 00:38:35,480 because the top quark was in a doublet with the b, 635 00:38:35,480 --> 00:38:44,140 and you're trying to keep the b and remove the top, which 636 00:38:44,140 --> 00:38:46,280 doesn't sound good. 637 00:38:46,280 --> 00:38:49,200 You're trying to keep the gauge particles, the W and the Z, 638 00:38:49,200 --> 00:38:50,570 in your theory. 639 00:38:50,570 --> 00:38:53,260 So you should still have that gauge symmetry, even 640 00:38:53,260 --> 00:38:55,870 if it's spontaneously broken. 641 00:38:55,870 --> 00:38:59,410 But you're trying to remove one member of a doublet. 642 00:38:59,410 --> 00:39:01,542 And that leads to [INAUDIBLE] terms 643 00:39:01,542 --> 00:39:03,500 that you'd have to clue to the effective theory 644 00:39:03,500 --> 00:39:04,930 so you can deal with it. 645 00:39:04,930 --> 00:39:06,430 And it's just a little bit annoying. 646 00:39:10,460 --> 00:39:14,110 But, you know, it's something you can deal with. 647 00:39:14,110 --> 00:39:16,090 The real crux of the matter is that if you 648 00:39:16,090 --> 00:39:21,790 compare mZ over m top, or mW over m top, 649 00:39:21,790 --> 00:39:23,685 that's about a half. 650 00:39:23,685 --> 00:39:25,060 So if you think about what you're 651 00:39:25,060 --> 00:39:27,670 expanding in when you take external particles that 652 00:39:27,670 --> 00:39:31,690 have momentum of order mZ, integrate out particles 653 00:39:31,690 --> 00:39:34,840 of order mt, you're expanding in a half, which is not such 654 00:39:34,840 --> 00:39:36,933 a great expansion parameter. 655 00:39:36,933 --> 00:39:38,350 Usually in effective field theory, 656 00:39:38,350 --> 00:39:41,020 you want to expand in something that's at least a third, 657 00:39:41,020 --> 00:39:44,110 hopefully a quarter. 658 00:39:44,110 --> 00:39:47,960 And a tenth if you really want to have a good expansion. 659 00:39:47,960 --> 00:39:50,530 So half is not really that good. 660 00:39:50,530 --> 00:39:57,590 So that's the real reason not to do it. 661 00:40:05,510 --> 00:40:08,250 And you could ask, well, why do we lose by not doing that? 662 00:40:10,265 --> 00:40:12,015 And that, of course, is the real question. 663 00:40:19,332 --> 00:40:20,790 Well, you miss some running, right? 664 00:40:20,790 --> 00:40:23,990 Because you don't have a theory that has no top quark 665 00:40:23,990 --> 00:40:25,740 but still has a W and a Z, and that theory 666 00:40:25,740 --> 00:40:27,490 could have anomalous dimensions, and you'd 667 00:40:27,490 --> 00:40:29,970 miss any running in that theory. 668 00:40:29,970 --> 00:40:32,463 You're kind of collapsing two of the lines in my picture 669 00:40:32,463 --> 00:40:34,380 with the arrow between, you're collapsing them 670 00:40:34,380 --> 00:40:37,810 down to just a single line. 671 00:40:37,810 --> 00:40:41,998 So you're missing the running that would go between mt to mW. 672 00:40:41,998 --> 00:40:44,040 Well, of course, that's not a very big hierarchy, 673 00:40:44,040 --> 00:40:45,730 and it's logs of two. 674 00:40:45,730 --> 00:40:50,840 And that's why we don't care that much. 675 00:40:50,840 --> 00:40:55,945 What it boils down to is that you're treating alpha s at mW. 676 00:40:55,945 --> 00:40:57,320 And if we want to be generous, we 677 00:40:57,320 --> 00:40:59,820 can say it's mW squared over m top squared. 678 00:40:59,820 --> 00:41:01,620 So they're logs of four. 679 00:41:01,620 --> 00:41:03,525 But you're treating this perturbatively. 680 00:41:08,280 --> 00:41:11,153 If you remove both the top and the W at the same time, 681 00:41:11,153 --> 00:41:13,570 then this is going to show up in your matching conditions, 682 00:41:13,570 --> 00:41:15,925 and you're not thinking of that is something as large. 683 00:41:15,925 --> 00:41:17,400 Thinking of it just as order 1. 684 00:41:23,550 --> 00:41:25,710 So that's the cost, which is not a big cost. 685 00:41:28,607 --> 00:41:30,440 Especially when you give, say, that the cost 686 00:41:30,440 --> 00:41:33,140 of going the other way would be expanding in the half. 687 00:41:33,140 --> 00:41:35,300 Better to take the cost in the logarithms 688 00:41:35,300 --> 00:41:39,105 than expansion in the half. 689 00:41:39,105 --> 00:41:40,480 So that's a general kind of rule, 690 00:41:40,480 --> 00:41:42,860 that if you're thinking about removing massive particles, 691 00:41:42,860 --> 00:41:46,670 you should ask, how close are they to each other? 692 00:41:46,670 --> 00:41:50,170 Should I integrate out a slew of them at the same time, 693 00:41:50,170 --> 00:41:51,170 or one at a time? 694 00:41:51,170 --> 00:41:54,340 That'll depend on exactly the scales in the problem. 695 00:41:57,170 --> 00:42:00,410 And there's-- and it means you're organizing the theory 696 00:42:00,410 --> 00:42:02,764 differently if you do it, the two different approaches. 697 00:42:06,360 --> 00:42:06,860 All right. 698 00:42:06,860 --> 00:42:09,830 So we'll do an example of this. 699 00:42:16,870 --> 00:42:18,890 We'll take a very simple example, 700 00:42:18,890 --> 00:42:23,050 although not completely trivial. 701 00:42:23,050 --> 00:42:26,500 So just b quarks changing to charm quarks u bar and d. 702 00:42:32,190 --> 00:42:34,450 So on the standard model, which is where we start, 703 00:42:34,450 --> 00:42:38,020 let's say we have a W boson. 704 00:42:38,020 --> 00:42:41,750 We have an up quark that's left-handed. 705 00:42:41,750 --> 00:42:46,810 We have a CKM matrix connecting flavors together. 706 00:42:46,810 --> 00:42:48,040 And we have down quarks. 707 00:42:48,040 --> 00:42:50,470 So this is-- there's a matrix space here 708 00:42:50,470 --> 00:42:53,350 in the flavor, which includes bottom charm up and down. 709 00:42:57,700 --> 00:43:00,987 And tree level matching is easy. 710 00:43:00,987 --> 00:43:03,070 And some of the complications I want to talk about 711 00:43:03,070 --> 00:43:05,210 will come into the loops. 712 00:43:05,210 --> 00:43:07,270 So let's first get through the tree level 713 00:43:07,270 --> 00:43:08,600 and set up some notation. 714 00:43:08,600 --> 00:43:12,940 And then we'll talk about what happens with the loop. 715 00:43:12,940 --> 00:43:17,270 So tree level, this is the diagram in the theory 1. 716 00:43:17,270 --> 00:43:19,510 Just calculate it at tree level. 717 00:43:29,710 --> 00:43:33,670 W propagator, unitary gauge. 718 00:43:46,578 --> 00:43:47,870 And then there's some spinners. 719 00:44:04,945 --> 00:44:06,150 So we have an antiquark. 720 00:44:06,150 --> 00:44:09,630 So we have a v spinner for the antiquark. 721 00:44:09,630 --> 00:44:11,520 That's an up. 722 00:44:11,520 --> 00:44:12,860 Everything else is a u spinner. 723 00:44:17,510 --> 00:44:20,580 I've put in explicitly the P lefts to denote 724 00:44:20,580 --> 00:44:22,165 the fact that it's left-handed. 725 00:44:25,352 --> 00:44:27,310 So there's some momentum in this diagram, which 726 00:44:27,310 --> 00:44:31,750 is momentum transfer, which is the be momentum minus the c 727 00:44:31,750 --> 00:44:34,970 momentum and the kind of obvious notation. 728 00:44:34,970 --> 00:44:36,850 And of course, by momentum conservation, 729 00:44:36,850 --> 00:44:42,270 that's also the d momentum plus the u momentum. 730 00:44:42,270 --> 00:44:46,260 And that's the momentum that's going through the propagator. 731 00:44:46,260 --> 00:44:49,135 We're going to count momenta as being of order masses. 732 00:44:51,730 --> 00:44:54,372 And the heaviest mass in this case is the b quark mass. 733 00:44:54,372 --> 00:44:56,080 And we'll take it to be the b quark mass. 734 00:44:59,100 --> 00:45:01,616 We can use the equations of motion. 735 00:45:01,616 --> 00:45:06,330 So Pb slash on ub, spinner for the b quark 736 00:45:06,330 --> 00:45:10,350 is mb ub, et cetera. 737 00:45:10,350 --> 00:45:13,740 And we can simplify the diagram by doing that. 738 00:45:13,740 --> 00:45:15,630 And of course, the key thing is that we 739 00:45:15,630 --> 00:45:18,000 can expand the propagator since the momentum is smaller 740 00:45:18,000 --> 00:45:22,050 than the mass of the W for integrating out 741 00:45:22,050 --> 00:45:32,680 the W. So the leading term when we do that is this term. 742 00:45:32,680 --> 00:45:41,380 And any other terms that we drop are down by that much. 743 00:45:41,380 --> 00:45:43,020 So for example, this term here, we just 744 00:45:43,020 --> 00:45:45,850 said that the k's are of order mb so it's down. 745 00:45:45,850 --> 00:45:47,600 And then we could drop the k squared here, 746 00:45:47,600 --> 00:45:49,530 I'll get the mW squared, so we just have that. 747 00:45:53,430 --> 00:45:55,932 And then we put that together, we can calculate the Feynman 748 00:45:55,932 --> 00:45:57,140 rule in the effective theory. 749 00:45:57,140 --> 00:45:59,160 The Feynman rule in the effective theory, 750 00:45:59,160 --> 00:46:01,422 we can determine the coefficient of it. 751 00:46:01,422 --> 00:46:02,880 So let's start out with just saying 752 00:46:02,880 --> 00:46:08,250 it's some four-quark operator that couples together 753 00:46:08,250 --> 00:46:11,490 those flavors, conveniently chosen so that they're all 754 00:46:11,490 --> 00:46:13,130 different to avoid [INAUDIBLE] factors. 755 00:46:16,390 --> 00:46:18,240 And in some conventional normalization 756 00:46:18,240 --> 00:46:23,370 for this Wilson coefficient, we call it GF, which is for Fermi. 757 00:46:25,940 --> 00:46:28,090 And the Feynman rule would again give spinners, 758 00:46:28,090 --> 00:46:33,360 the same spinners as before, with the same Lagrangians 759 00:46:33,360 --> 00:46:35,550 for these light quarks. 760 00:46:35,550 --> 00:46:37,780 Haven't changed anything about that. 761 00:46:37,780 --> 00:46:39,405 We're just removing the heavy particle. 762 00:46:43,627 --> 00:46:45,210 And it's conventional also to pull out 763 00:46:45,210 --> 00:46:48,850 the CKM factor [INAUDIBLE]. 764 00:46:48,850 --> 00:46:49,350 OK. 765 00:46:49,350 --> 00:46:52,237 So if you like, you can say that there's a coefficient here, 766 00:46:52,237 --> 00:46:54,070 and that coefficient has been fixed to be 1. 767 00:46:57,110 --> 00:46:58,860 And what this G Fermi is, then, would just 768 00:46:58,860 --> 00:47:03,180 be the leftover factors of the gauge coupling 769 00:47:03,180 --> 00:47:07,943 and the mass of the W. And that's the usual convention. 770 00:47:17,810 --> 00:47:23,160 So G Fermi is just fixed in this case, not as the thing that 771 00:47:23,160 --> 00:47:27,927 is going to get corrected at higher orders in QCD 772 00:47:27,927 --> 00:47:30,260 but would get corrected at higher orders in electroweak, 773 00:47:30,260 --> 00:47:31,910 if you like. 774 00:47:31,910 --> 00:47:35,510 And then you sort of put in with the QCD corrections 775 00:47:35,510 --> 00:47:38,750 into some other coefficient that we can call C. 776 00:47:38,750 --> 00:47:41,990 And we'll just say that it's 1 at the level of this. 777 00:47:41,990 --> 00:47:44,640 And this line agrees with the expansion of that line, 778 00:47:44,640 --> 00:47:46,760 and that's the matching of something 779 00:47:46,760 --> 00:47:50,727 that's measurable, which is this four-point function. 780 00:47:50,727 --> 00:47:52,310 So we can call it an S matrix element. 781 00:48:04,790 --> 00:48:07,010 So this theory is very popular. 782 00:48:07,010 --> 00:48:10,220 It's called the electroweak Hamiltonian. 783 00:48:10,220 --> 00:48:11,440 It's used all over the place. 784 00:48:15,250 --> 00:48:17,410 And it involves more than just doing what I did, 785 00:48:17,410 --> 00:48:20,145 because I chose a particular set of flavors 786 00:48:20,145 --> 00:48:21,520 for a particular channel, and you 787 00:48:21,520 --> 00:48:23,183 have to do it for the whole-- 788 00:48:23,183 --> 00:48:24,850 for all sorts of other channels as well. 789 00:48:28,630 --> 00:48:31,410 So we'll study some aspects of this theory. 790 00:48:31,410 --> 00:48:33,150 We won't study every possible aspect. 791 00:48:33,150 --> 00:48:36,000 But I've given you a handout that studies a lot more. 792 00:48:36,000 --> 00:48:38,490 It's 250 pages long. 793 00:48:38,490 --> 00:48:40,510 I'm not asking you to even read all of that. 794 00:48:40,510 --> 00:48:44,003 I've pointed you at some pages of that 795 00:48:44,003 --> 00:48:46,170 in the reading, which are relevant to the discussion 796 00:48:46,170 --> 00:48:47,330 we're having here. 797 00:48:47,330 --> 00:48:49,080 If you really want to dig deeper, you can. 798 00:48:51,720 --> 00:48:55,020 So people often call this an electroweak Hamiltonian, which 799 00:48:55,020 --> 00:48:59,120 is just minus the Lagrangian. 800 00:48:59,120 --> 00:49:02,860 So then that's 4 GF over 2. 801 00:49:02,860 --> 00:49:07,940 And as a Hamiltonian, we write fields instead of spinners. 802 00:49:07,940 --> 00:49:11,320 So we have this four-quark operator. 803 00:49:11,320 --> 00:49:13,890 And we would have determined the Hamiltonian from tree level 804 00:49:13,890 --> 00:49:15,430 matching, from what we've done. 805 00:49:15,430 --> 00:49:29,460 And this would be the result. Standard stuff. 806 00:49:34,530 --> 00:49:37,080 How do we want to go further than that? 807 00:49:37,080 --> 00:49:39,975 Well, if we want to think about theory 2, which is this theory, 808 00:49:39,975 --> 00:49:41,850 and we want to think about it in more detail, 809 00:49:41,850 --> 00:49:43,597 we should worry about whether we're-- 810 00:49:43,597 --> 00:49:45,180 with just this operator, whether we've 811 00:49:45,180 --> 00:49:50,033 got a complete set of structures that could possibly occur. 812 00:49:50,033 --> 00:49:51,700 So we should think about the symmetries. 813 00:49:54,475 --> 00:49:56,850 And it's useful to, therefore, construct the most general 814 00:49:56,850 --> 00:50:04,018 basis of operators in theory 2. 815 00:50:06,580 --> 00:50:08,400 And we'll just do that. 816 00:50:08,400 --> 00:50:10,460 So if you read this 250-page review, 817 00:50:10,460 --> 00:50:14,178 then it's done for the full electroweak Hamiltonian, 818 00:50:14,178 --> 00:50:15,720 and we'll just stick with the flavors 819 00:50:15,720 --> 00:50:18,660 we're talking about here. 820 00:50:18,660 --> 00:50:23,220 So what are the most general bases of operators? 821 00:50:23,220 --> 00:50:25,670 How should we think about that? 822 00:50:25,670 --> 00:50:27,170 Well, we're constructing some theory 823 00:50:27,170 --> 00:50:31,630 that's going to-- that we're matching onto it mu equals mW. 824 00:50:31,630 --> 00:50:33,800 And at that scale, when we're determining the theory 825 00:50:33,800 --> 00:50:35,758 and determining the coefficients of the theory, 826 00:50:35,758 --> 00:50:38,900 we can treat the bottom the charm the down 827 00:50:38,900 --> 00:50:42,740 and the up respectively as if they're massless. 828 00:50:45,710 --> 00:50:50,960 And really, what I mean by that is that the masses of these 829 00:50:50,960 --> 00:50:53,990 particles are only going to show up in the operators and not 830 00:50:53,990 --> 00:51:00,630 in the coefficients, which I've-- 831 00:51:00,630 --> 00:51:03,598 you see that I talked about over here. 832 00:51:03,598 --> 00:51:05,390 The thing that shows up in the coefficients 833 00:51:05,390 --> 00:51:06,848 are the mass scales we're removing. 834 00:51:06,848 --> 00:51:11,150 And we're not removing the mass scales of these things. 835 00:51:11,150 --> 00:51:13,420 So that means, if we can think of them as massless, 836 00:51:13,420 --> 00:51:16,420 that we can think about the matching in terms of using 837 00:51:16,420 --> 00:51:18,680 something like chirality. 838 00:51:18,680 --> 00:51:22,600 And we can use the fact that QCD for massless quarks 839 00:51:22,600 --> 00:51:26,038 does not change chirality. 840 00:51:26,038 --> 00:51:27,955 And we can use that in constructing our basis. 841 00:51:32,940 --> 00:51:38,580 So that means even though the b quark and the charm quark 842 00:51:38,580 --> 00:51:40,710 really have masses, for the purpose of constructing 843 00:51:40,710 --> 00:51:43,710 the operator basis, we can think of them as massless. 844 00:51:43,710 --> 00:51:45,970 And we only have left-handed guys to worry about here. 845 00:51:48,650 --> 00:51:50,070 Chirality means more than that. 846 00:51:50,070 --> 00:51:52,850 It also-- well, it means effectively that. 847 00:51:52,850 --> 00:51:57,240 But it means also that we will only get one gamma matrix here. 848 00:51:57,240 --> 00:51:59,940 And we can think about why that is. 849 00:51:59,940 --> 00:52:04,190 Chirality means that you get an odd number. 850 00:52:04,190 --> 00:52:08,550 You need an odd number to have left on both sides. 851 00:52:08,550 --> 00:52:10,010 But you can reduce 3 to 1. 852 00:52:14,440 --> 00:52:19,660 So there's an identity which I won't write out for you, 853 00:52:19,660 --> 00:52:28,830 but you can reduce 3 back down to 1. 854 00:52:28,830 --> 00:52:31,780 So any higher odd number can be reduced back down to 1. 855 00:52:31,780 --> 00:52:37,920 So once we use chirality and that fact, 856 00:52:37,920 --> 00:52:44,660 then we have a fairly restrictive basis 857 00:52:44,660 --> 00:52:46,435 in terms of Dirac structures. 858 00:52:54,200 --> 00:52:57,610 You could ask, what are the most general possible ways 859 00:52:57,610 --> 00:52:59,738 of contracting spinner indices? 860 00:52:59,738 --> 00:53:02,280 Who said I had to put this charm quirk with the bottom quark? 861 00:53:02,280 --> 00:53:04,190 I could have put the up quark over here. 862 00:53:04,190 --> 00:53:06,955 And maybe in higher orders that happens. 863 00:53:06,955 --> 00:53:08,330 Well, at higher orders, you could 864 00:53:08,330 --> 00:53:09,455 think about that happening. 865 00:53:09,455 --> 00:53:11,450 But you can always get back to the form 866 00:53:11,450 --> 00:53:14,700 that I wrote over there using what's called the spin Fierz. 867 00:53:20,385 --> 00:53:25,430 So if I write it for fields, it means 868 00:53:25,430 --> 00:53:28,760 that there's an identity that I can rearrange 869 00:53:28,760 --> 00:53:29,910 this guy the other way. 870 00:53:29,910 --> 00:53:32,420 And so these are equivalent operators. 871 00:53:39,223 --> 00:53:41,140 And sometimes, you have to use these relations 872 00:53:41,140 --> 00:53:42,820 when you're doing matching calculations, 873 00:53:42,820 --> 00:53:44,908 because you construct a complete basis, which 874 00:53:44,908 --> 00:53:45,700 is the minimal one. 875 00:53:45,700 --> 00:53:47,530 And then maybe when you do your calculation, 876 00:53:47,530 --> 00:53:48,947 you get this operator, so you have 877 00:53:48,947 --> 00:53:52,000 to turn it back into this one. 878 00:53:52,000 --> 00:53:55,060 There's two minus signs in this relation. 879 00:53:55,060 --> 00:53:58,163 One is from the Fierz identity, and one 880 00:53:58,163 --> 00:53:59,830 is because when you do the manipulations 881 00:53:59,830 --> 00:54:01,288 you end up anticommuting to fields. 882 00:54:08,000 --> 00:54:10,102 So from the statistics. 883 00:54:10,102 --> 00:54:12,310 So if you were to use the same relation for spinners, 884 00:54:12,310 --> 00:54:14,800 then there would be one less minus sign. 885 00:54:14,800 --> 00:54:17,050 Something to be careful, though. 886 00:54:17,050 --> 00:54:21,100 So what this tells you, once you put that information together, 887 00:54:21,100 --> 00:54:27,580 is that you know that you can write the operators this way, 888 00:54:27,580 --> 00:54:30,430 and you also know that this gamma, capital gamma, 889 00:54:30,430 --> 00:54:36,860 has that gamma u p left form in terms of the Dirac structure. 890 00:54:36,860 --> 00:54:41,380 So the only really thing that can happen is color. 891 00:54:41,380 --> 00:54:43,937 And you can contract the color of these 3's-- 892 00:54:43,937 --> 00:54:46,270 so you have four 3's, you can contract the color of them 893 00:54:46,270 --> 00:54:47,710 in different ways. 894 00:54:47,710 --> 00:54:49,780 Well, I have two 3 bars and two 3's. 895 00:54:49,780 --> 00:54:52,450 Think about having different color contractions. 896 00:54:52,450 --> 00:54:54,310 And that will expand our operator basis 897 00:54:54,310 --> 00:54:57,160 by one more operator. 898 00:54:57,160 --> 00:55:00,750 There is also a color Fierz. 899 00:55:00,750 --> 00:55:04,870 And I can use that color Fierz to get rid of having explicit 900 00:55:04,870 --> 00:55:06,870 TA's. 901 00:55:06,870 --> 00:55:09,100 So color is a lot like spin in the sense 902 00:55:09,100 --> 00:55:11,650 that if I had more TA's, then I can always 903 00:55:11,650 --> 00:55:15,440 reduce a higher number of TA's back down to a lower number. 904 00:55:15,440 --> 00:55:18,770 So I've had two, I can reduce it down to one or zero. 905 00:55:18,770 --> 00:55:19,270 OK. 906 00:55:19,270 --> 00:55:21,478 So I only really have to think about having a TA here 907 00:55:21,478 --> 00:55:23,470 and a TA there. 908 00:55:23,470 --> 00:55:26,290 If I have a TA there and a TA there, there's a spinner-- 909 00:55:26,290 --> 00:55:30,270 or there's a color Fierz relation for this guy 910 00:55:30,270 --> 00:55:33,280 that you could use. 911 00:55:33,280 --> 00:55:35,062 And there's a handout that I posted 912 00:55:35,062 --> 00:55:37,270 on the web that has a summary of all these relations. 913 00:55:37,270 --> 00:55:42,020 I'm not going to bother writing them down in lecture here. 914 00:55:42,020 --> 00:55:47,030 So once I take that into account, 915 00:55:47,030 --> 00:55:49,090 then it ends up in two operators. 916 00:55:52,010 --> 00:55:55,920 And if I were to write them in a kind of renormalized notation 917 00:55:55,920 --> 00:56:00,460 where I introduce the scale mu, then I 918 00:56:00,460 --> 00:56:02,767 would write them as follows. 919 00:56:02,767 --> 00:56:03,850 Let's call them O1 and O2. 920 00:56:07,540 --> 00:56:11,650 And the difference between O1 and O2 is just simply color. 921 00:56:11,650 --> 00:56:14,860 So in O1, the color index alpha is 922 00:56:14,860 --> 00:56:18,150 contracted between charm bottom and up down and up. 923 00:56:24,820 --> 00:56:27,370 And then in O2 it's the the colors 924 00:56:27,370 --> 00:56:28,880 contracted the other way. 925 00:56:28,880 --> 00:56:30,490 So I don't write explicit TA's, but I 926 00:56:30,490 --> 00:56:33,620 do have to consider the other possible way of contracting 927 00:56:33,620 --> 00:56:34,120 the indices. 928 00:56:47,690 --> 00:56:50,400 Up, down, doesn't matter. 929 00:56:50,400 --> 00:56:50,900 OK. 930 00:56:50,900 --> 00:56:54,085 So these operators, these are the two operators they have 931 00:56:54,085 --> 00:56:55,460 once I satisfy all the symmetries 932 00:56:55,460 --> 00:56:57,130 and I have some coefficients. 933 00:57:00,630 --> 00:57:03,080 And if you ask really what the coefficients can depend 934 00:57:03,080 --> 00:57:07,690 on here, well, they can depend on mass scales like top 935 00:57:07,690 --> 00:57:10,920 and the other particles that I'm integrating out, 936 00:57:10,920 --> 00:57:14,427 Z. I'm not going to make that explicit in my notation. 937 00:57:14,427 --> 00:57:16,010 If I tried to make every possible mass 938 00:57:16,010 --> 00:57:18,900 scale like the Higgs or the top, it would just be too much. 939 00:57:18,900 --> 00:57:22,820 So let me just denote two things that it can depend on. 940 00:57:22,820 --> 00:57:25,940 Mu over mW can show up, and alpha of mu 941 00:57:25,940 --> 00:57:29,502 can show up, as well as some ratios of other particles 942 00:57:29,502 --> 00:57:30,460 which we will suppress. 943 00:57:45,000 --> 00:57:47,060 So then what tree level matching is saying 944 00:57:47,060 --> 00:57:48,810 is that you've got one of these operators, 945 00:57:48,810 --> 00:57:50,790 and you didn't get the other one. 946 00:57:50,790 --> 00:57:55,235 So it says that at tree level, what we did before-- 947 00:57:57,845 --> 00:57:59,840 and so say we do matching at mu equals 948 00:57:59,840 --> 00:58:11,860 mW and C1 at 1, which is mu over mW equals 1 and alpha mW is 1. 949 00:58:11,860 --> 00:58:14,630 And then corrections to that would be suppressed, 950 00:58:14,630 --> 00:58:17,070 no large logs, just by alpha. 951 00:58:17,070 --> 00:58:27,770 And C2 of 1 alpha mW is 0 plus something of this order. 952 00:58:27,770 --> 00:58:32,300 So that's what we determined by doing our tree level matching. 953 00:58:32,300 --> 00:58:35,600 Now, when we did our tree level matching we did something. 954 00:58:35,600 --> 00:58:40,472 We used external states which were quarks. 955 00:58:40,472 --> 00:58:44,190 We matched up the S matrix elements in that way. 956 00:58:44,190 --> 00:58:46,065 If you were really interested in this process 957 00:58:46,065 --> 00:58:48,433 b goes to c u bar d, you'd not really 958 00:58:48,433 --> 00:58:49,850 be interested in measuring quarks. 959 00:58:49,850 --> 00:58:51,050 We don't see them. 960 00:58:51,050 --> 00:58:53,890 You'd be interested in thinking about the process for mesons-- 961 00:58:53,890 --> 00:58:58,365 B meson changes to a D meson, and a pion, for example. 962 00:58:58,365 --> 00:59:00,740 And who's to say that the matching that we did for quarks 963 00:59:00,740 --> 00:59:04,760 is also the matching that's valid for hadrons? 964 00:59:04,760 --> 00:59:05,420 Well, it is. 965 00:59:08,040 --> 00:59:11,760 So there's a key fact about matching which is important. 966 00:59:11,760 --> 00:59:15,470 One of the things I wanted to emphasize to you 967 00:59:15,470 --> 00:59:18,050 is that it doesn't depend on what states you pick. 968 00:59:23,340 --> 00:59:26,790 It's independent of the choice of states. 969 00:59:26,790 --> 00:59:29,340 And it's independent of the IR regulators that you pick. 970 00:59:31,890 --> 00:59:34,560 So when you do matching, you often 971 00:59:34,560 --> 00:59:38,165 pick some IR regulators to regulate IR divergences. 972 00:59:38,165 --> 00:59:39,540 And the only thing you have to do 973 00:59:39,540 --> 00:59:41,482 is pick the same states and same IR regulators 974 00:59:41,482 --> 00:59:42,315 in the two theories. 975 00:59:45,180 --> 00:59:48,638 And once you do that, your results for your coefficients 976 00:59:48,638 --> 00:59:50,430 will be independent of the choice you made. 977 01:00:00,540 --> 01:00:03,680 So this is intuitively something I think that's, once you 978 01:00:03,680 --> 01:00:08,110 think about it, fairly obvious. 979 01:00:08,110 --> 01:00:11,030 What this sentence is saying is that the matching, which 980 01:00:11,030 --> 01:00:13,280 is supposed to be a high-energy property, 981 01:00:13,280 --> 01:00:15,465 is independent of the low-energy physics. 982 01:00:15,465 --> 01:00:17,840 The choice of the states and the choice of IR regulators, 983 01:00:17,840 --> 01:00:21,530 that's parameterizing something about low-energy physics-- 984 01:00:21,530 --> 01:00:23,780 lower energy than the scale we're trying to determine. 985 01:00:23,780 --> 01:00:26,700 These things should only depend on higher-energy physics. 986 01:00:26,700 --> 01:00:30,365 So the outcome for them will be independent of this choice. 987 01:00:34,350 --> 01:00:49,670 So just to emphasize, even when we use hadronic states, 988 01:00:49,670 --> 01:00:57,290 the result that we obtain from quark states is valid. 989 01:01:01,860 --> 01:01:06,092 That's just different choice of state. 990 01:01:06,092 --> 01:01:08,300 And what we're doing when we do the matching is we're 991 01:01:08,300 --> 01:01:13,560 picking a convenient choice of state, something that 992 01:01:13,560 --> 01:01:15,378 makes it easy to calculate. 993 01:01:15,378 --> 01:01:17,670 If we tried to calculate the matrix elements of B and D 994 01:01:17,670 --> 01:01:20,100 and pi mesons, then I'm saying if you're a strong enough, 995 01:01:20,100 --> 01:01:22,188 you'd get the same matching coefficient. 996 01:01:22,188 --> 01:01:23,730 Of course, you should pick the result 997 01:01:23,730 --> 01:01:25,320 that makes the calculation easy, and that 998 01:01:25,320 --> 01:01:26,280 would be to use quarks. 999 01:01:28,808 --> 01:01:29,850 Any questions about that? 1000 01:01:33,740 --> 01:01:34,240 OK. 1001 01:01:34,240 --> 01:01:36,400 So this is something that you should keep in mind. 1002 01:01:36,400 --> 01:01:38,280 And you should keep in mind that it is-- 1003 01:01:38,280 --> 01:01:39,900 you do have to make sure that you're 1004 01:01:39,900 --> 01:01:43,110 using the same states and IR regulator in the two theories. 1005 01:01:43,110 --> 01:01:44,920 In this case, that's pretty easy. 1006 01:01:44,920 --> 01:01:48,330 The states are really the same states, because you're 1007 01:01:48,330 --> 01:01:50,330 defining the states with the Lagrangian machine 1008 01:01:50,330 --> 01:01:52,610 for the b quarks, say, and the b quark Lagrangian 1009 01:01:52,610 --> 01:01:53,950 hasn't really changed. 1010 01:01:53,950 --> 01:01:56,460 So there's no change in the state. 1011 01:01:56,460 --> 01:01:58,710 And you can set things up so that you have the same IR 1012 01:01:58,710 --> 01:01:59,730 regulator. 1013 01:01:59,730 --> 01:02:01,650 And that'll become important in the example 1014 01:02:01,650 --> 01:02:04,270 that we're just about to do. 1015 01:02:04,270 --> 01:02:08,280 So let's do an example of carrying out 1016 01:02:08,280 --> 01:02:13,970 this matching for C1 and C2 in a little more detail. 1017 01:02:13,970 --> 01:02:15,810 And before we carry out matching, 1018 01:02:15,810 --> 01:02:19,462 we actually have to renormalize the two theories. 1019 01:02:19,462 --> 01:02:21,420 Well, we've been talking as if the theory above 1020 01:02:21,420 --> 01:02:22,820 is the standard model. 1021 01:02:22,820 --> 01:02:25,823 So imagine that we've already renormalized that. 1022 01:02:25,823 --> 01:02:27,240 And so we only have to renormalize 1023 01:02:27,240 --> 01:02:29,640 the effective theory in MS bar. 1024 01:02:32,930 --> 01:02:37,080 So we'll talk about doing that, remind you of doing that-- 1025 01:02:37,080 --> 01:02:40,660 maybe something you've seen before. 1026 01:02:40,660 --> 01:02:42,945 So there's going to be wave function renormalization. 1027 01:02:52,450 --> 01:02:53,780 So you have this 4/3. 1028 01:02:56,860 --> 01:03:01,390 I'm going to leave out the prefactor, this prefactor. 1029 01:03:01,390 --> 01:03:02,770 It's always going to be there. 1030 01:03:02,770 --> 01:03:05,020 I'm going to stop writing it and start just 1031 01:03:05,020 --> 01:03:08,890 focusing on the thing in square brackets here. 1032 01:03:08,890 --> 01:03:14,267 We'll do calculations with Feynman gauge. 1033 01:03:14,267 --> 01:03:16,100 I'm not going to really do the calculations. 1034 01:03:16,100 --> 01:03:19,640 I'm just going to quote results to you. 1035 01:03:19,640 --> 01:03:31,150 And in order to simplify what we have to write down, 1036 01:03:31,150 --> 01:03:33,880 let me define the set of spinners 1037 01:03:33,880 --> 01:03:38,860 that you get from taking the tree level matrix element of O1 1038 01:03:38,860 --> 01:03:41,850 to be S1 and of O2 to be S2. 1039 01:03:47,090 --> 01:03:48,970 So this is just the spinners that we 1040 01:03:48,970 --> 01:03:51,370 wrote down before for the case of S1, 1041 01:03:51,370 --> 01:03:53,465 exactly what we had before. 1042 01:03:53,465 --> 01:03:56,090 And then for S2, just a slightly different contraction of color 1043 01:03:56,090 --> 01:03:56,590 indices. 1044 01:04:08,700 --> 01:04:09,200 OK. 1045 01:04:09,200 --> 01:04:11,633 So that's just trying to make the lowest-error result 1046 01:04:11,633 --> 01:04:13,550 simple to write down so that when I write down 1047 01:04:13,550 --> 01:04:15,440 the higher-error result, we can focus on the things that 1048 01:04:15,440 --> 01:04:18,290 are changing and mattering, and not on the complications that 1049 01:04:18,290 --> 01:04:21,210 come in from what we're talking about. 1050 01:04:21,210 --> 01:04:23,120 So if we think about diagrams here, 1051 01:04:23,120 --> 01:04:24,530 we have four-quark operator. 1052 01:04:24,530 --> 01:04:39,160 And we just have to draw all the loop graphs Right. 1053 01:04:39,160 --> 01:04:40,780 We have to regulate them in some way, 1054 01:04:40,780 --> 01:04:44,380 because they are IR divergent. 1055 01:04:44,380 --> 01:04:45,795 So let's regulate them with-- 1056 01:04:45,795 --> 01:04:47,920 if we really want to talk about the matching, which 1057 01:04:47,920 --> 01:04:49,545 is what I really want to do eventually, 1058 01:04:49,545 --> 01:04:51,280 we should regulate them in some way. 1059 01:04:51,280 --> 01:04:54,130 And so let's regulate them with off-shell momenta 1060 01:04:54,130 --> 01:04:55,480 on the external lines. 1061 01:04:55,480 --> 01:05:00,117 And I'll take it to be a common off-shell momenta p. 1062 01:05:00,117 --> 01:05:02,200 And I'll just set the masses of the external lines 1063 01:05:02,200 --> 01:05:06,040 to 0, since they're not going to matter. 1064 01:05:08,960 --> 01:05:09,460 OK. 1065 01:05:09,460 --> 01:05:12,590 So then what does it look like? 1066 01:05:12,590 --> 01:05:14,660 So the matrix element of O1-- 1067 01:05:14,660 --> 01:05:15,940 the 0 means Bayer-- 1068 01:05:18,520 --> 01:05:20,080 is going to have divergences. 1069 01:05:20,080 --> 01:05:38,098 And it has the following structure, times spinner 1. 1070 01:05:38,098 --> 01:05:40,390 And then there's another piece that involves spinner 1. 1071 01:05:57,770 --> 01:06:00,230 And then there's a piece that involves spinner 2 that shows 1072 01:06:00,230 --> 01:06:03,170 up in from these loop graphs. 1073 01:06:03,170 --> 01:06:05,840 So even though we're calculating the Bayer matrix [INAUDIBLE] 1074 01:06:05,840 --> 01:06:08,690 of operator 1, the spinner combination 2 1075 01:06:08,690 --> 01:06:13,010 shows up, because this is supposedly a gluon, 1076 01:06:13,010 --> 01:06:15,050 and the gluon moves the color around. 1077 01:06:20,210 --> 01:06:21,968 And I'm only writing the divergent terms, 1078 01:06:21,968 --> 01:06:23,510 although later on we'll be interested 1079 01:06:23,510 --> 01:06:24,760 in the constant terms as well. 1080 01:06:27,660 --> 01:06:31,320 So the dots are the constant terms underneath logarithms, 1081 01:06:31,320 --> 01:06:32,830 and all these little round brackets 1082 01:06:32,830 --> 01:06:34,202 have constant terms in them. 1083 01:06:37,803 --> 01:06:39,220 Part of the reason for introducing 1084 01:06:39,220 --> 01:06:41,320 the basis in the way I did is that when 1085 01:06:41,320 --> 01:06:43,680 I want to quote you the result for O2, 1086 01:06:43,680 --> 01:06:46,360 it's the same, just switching 1 and 2. 1087 01:06:54,100 --> 01:06:56,380 And so the statement that you would have from this-- 1088 01:06:56,380 --> 01:06:57,820 these are ultraviolet divergences, 1089 01:06:57,820 --> 01:06:59,598 the infrared divergences are regulated. 1090 01:06:59,598 --> 01:07:01,390 The statement that you would have from this 1091 01:07:01,390 --> 01:07:04,320 is that you can look at these various terms, and you can ask, 1092 01:07:04,320 --> 01:07:07,210 what's going on with those divergences? 1093 01:07:07,210 --> 01:07:10,300 Well, this one here is actually cancelled by wave function 1094 01:07:10,300 --> 01:07:16,030 renormalization, which I haven't put in yet. 1095 01:07:21,060 --> 01:07:23,670 And this one here is O1 mixing into O1. 1096 01:07:28,080 --> 01:07:31,800 So this one here is a counterterm, if you like, 1097 01:07:31,800 --> 01:07:32,920 for O1. 1098 01:07:32,920 --> 01:07:41,070 And this one here is a counterterm for O2. 1099 01:07:41,070 --> 01:07:44,160 And the language you use as you say that O1, which we're 1100 01:07:44,160 --> 01:07:45,600 calculating, has mixed into O2. 1101 01:08:00,900 --> 01:08:02,910 So there's two different methods we 1102 01:08:02,910 --> 01:08:09,810 could use to carry out the renormalization here 1103 01:08:09,810 --> 01:08:11,858 that are actually equivalent. 1104 01:08:21,130 --> 01:08:24,550 So method one is called composite operator 1105 01:08:24,550 --> 01:08:25,860 renormalization. 1106 01:08:33,038 --> 01:08:35,080 It's useful to know what things are equivalent so 1107 01:08:35,080 --> 01:08:39,140 that you know what you can get away with ignoring. 1108 01:08:39,140 --> 01:08:42,790 So what is composite operator renormalization? 1109 01:08:42,790 --> 01:08:46,189 Well, you think about having a Bayer operator. 1110 01:08:46,189 --> 01:08:49,431 And you need to introduce some constants to renormalize. 1111 01:08:52,408 --> 01:08:54,700 These are not related to wave function renormalization. 1112 01:08:54,700 --> 01:08:56,859 It's actually operator renormalization. 1113 01:08:56,859 --> 01:08:59,263 You have multiple fields at the same spacetime point, 1114 01:08:59,263 --> 01:09:01,180 you need additional renormalization constants, 1115 01:09:01,180 --> 01:09:02,263 that's what these Z's are. 1116 01:09:05,255 --> 01:09:06,880 When you go to take the matrix element, 1117 01:09:06,880 --> 01:09:08,830 you have to include the wave function renormalization 1118 01:09:08,830 --> 01:09:09,350 as well. 1119 01:09:09,350 --> 01:09:17,760 So the Bayer matrix element has a wave function renormalization 1120 01:09:17,760 --> 01:09:21,029 as well as this Z. And this is the relation 1121 01:09:21,029 --> 01:09:26,370 between the renormalized matrix element and the Bayer one. 1122 01:09:26,370 --> 01:09:33,270 So this guy here is renormalized and amputated. 1123 01:09:40,510 --> 01:09:41,010 OK. 1124 01:09:41,010 --> 01:09:43,759 So that's the relation you can calculate O0. 1125 01:09:43,759 --> 01:09:45,330 That's what we were doing. 1126 01:09:45,330 --> 01:09:47,109 We just calculated O0. 1127 01:09:47,109 --> 01:09:48,223 We could remove Z psi. 1128 01:09:48,223 --> 01:09:49,890 And I just told you if you remove Z psi, 1129 01:09:49,890 --> 01:09:51,399 it's going to get rid of this. 1130 01:09:51,399 --> 01:09:53,732 And then you use the remainder of this stuff to get Zij. 1131 01:10:11,820 --> 01:10:13,570 I'm going to need it the other way around. 1132 01:10:13,570 --> 01:10:14,445 So let me write the-- 1133 01:10:18,680 --> 01:10:21,502 so I can write this relation the other way around. 1134 01:10:21,502 --> 01:10:22,880 And then it looks like this. 1135 01:10:26,512 --> 01:10:27,920 OK, so that's one method. 1136 01:10:27,920 --> 01:10:32,360 The other one is related more to how we usually 1137 01:10:32,360 --> 01:10:35,640 think about things in terms of gauge theory. 1138 01:10:35,640 --> 01:10:37,700 And that is to have counterterm coefficients 1139 01:10:37,700 --> 01:10:41,480 for the various operators. 1140 01:10:44,870 --> 01:10:48,290 So start with the Hamiltonian written in terms of Bayer 1141 01:10:48,290 --> 01:10:54,560 coefficients and operators that have Bayer fields-- 1142 01:10:54,560 --> 01:10:58,080 Bayer operators, if you like. 1143 01:10:58,080 --> 01:11:00,540 And now switch over to renormalized quantities, 1144 01:11:00,540 --> 01:11:03,230 so you get Z psi squared. 1145 01:11:03,230 --> 01:11:07,533 We have to switch over from the Bayer coefficient 1146 01:11:07,533 --> 01:11:08,450 to a renormalized one. 1147 01:11:08,450 --> 01:11:10,370 So let me do that first. 1148 01:11:10,370 --> 01:11:13,510 So we get some Z for the coefficient, 1149 01:11:13,510 --> 01:11:16,540 some renormalized coefficients. 1150 01:11:16,540 --> 01:11:19,370 And then we get a Z psi squared from the four fermions that 1151 01:11:19,370 --> 01:11:21,170 are in this operator. 1152 01:11:21,170 --> 01:11:26,440 Then we get an O. 1153 01:11:26,440 --> 01:11:35,200 And then we can write this in terms of Ci Oi plus Z psi 1154 01:11:35,200 --> 01:11:44,863 squared Zij C minus delta ij Cj Oi. 1155 01:11:44,863 --> 01:11:46,530 And then this would be our counterterms. 1156 01:11:49,500 --> 01:11:53,670 So we would stick in this guy, that operator 1157 01:11:53,670 --> 01:11:56,110 we haven't renormalized-- we haven't done our operator 1158 01:11:56,110 --> 01:11:57,000 renormalization. 1159 01:11:57,000 --> 01:11:58,583 We're not doing method 1. 1160 01:11:58,583 --> 01:12:00,750 So this, the matrix element here will still diverge, 1161 01:12:00,750 --> 01:12:02,981 but we cancel them off with the counterterms. 1162 01:12:05,750 --> 01:12:09,310 So that's the logic of method 2. 1163 01:12:09,310 --> 01:12:11,380 And once you do that, the divergences 1164 01:12:11,380 --> 01:12:17,410 cancel between these two terms, and you're left, if you like-- 1165 01:12:17,410 --> 01:12:19,090 or even if you don't-- 1166 01:12:19,090 --> 01:12:21,090 with the coefficients times the operators, which 1167 01:12:21,090 --> 01:12:25,125 are renormalized, both renormalized, 1168 01:12:25,125 --> 01:12:28,455 even renormalized separately. 1169 01:12:34,500 --> 01:12:35,765 And these two are equivalent. 1170 01:12:35,765 --> 01:12:38,627 These are equivalent ways of thinking about the same thing. 1171 01:12:38,627 --> 01:12:41,210 And we can even go further and derive the equivalence of them. 1172 01:12:44,547 --> 01:12:46,880 So if we take this theory and we take the matrix element 1173 01:12:46,880 --> 01:12:51,380 of the Hamiltonian in this second way of doing things, 1174 01:12:51,380 --> 01:12:52,070 what do we get? 1175 01:12:57,240 --> 01:12:58,860 We get this. 1176 01:12:58,860 --> 01:13:00,620 So even though as Oi, I said, we haven't 1177 01:13:00,620 --> 01:13:02,540 done operator renormalization, so we still 1178 01:13:02,540 --> 01:13:12,780 have the divergences there, that's Cj Oj. 1179 01:13:12,780 --> 01:13:15,260 And if we write the OJ using the relation 1180 01:13:15,260 --> 01:13:22,010 at the top of the board, then we get Cj Zji inverse Z 1181 01:13:22,010 --> 01:13:27,388 psi squared Oi 0. 1182 01:13:27,388 --> 01:13:29,180 And then we can just look at the two sides. 1183 01:13:29,180 --> 01:13:31,070 They both have the Z psi squared. 1184 01:13:31,070 --> 01:13:33,590 They both have a Cj, they both have Oi 0. 1185 01:13:33,590 --> 01:13:38,400 The only thing that's different is this. 1186 01:13:38,400 --> 01:13:43,440 So we find the Zij for the coefficient 1187 01:13:43,440 --> 01:13:48,410 is the transpose of the inverse of the Z's for the operators. 1188 01:13:48,410 --> 01:13:51,420 So that's a more definite way of saying the two would 1189 01:13:51,420 --> 01:13:52,462 lead to the same results. 1190 01:13:52,462 --> 01:13:53,878 And this is how you would actually 1191 01:13:53,878 --> 01:13:56,050 get the relation between the two ways of doing it. 1192 01:13:59,630 --> 01:14:02,420 So it's not completely trivial. 1193 01:14:02,420 --> 01:14:05,430 You have to know you have to take the inverse and transpose. 1194 01:14:05,430 --> 01:14:09,290 But it's the same information. 1195 01:14:09,290 --> 01:14:09,790 OK. 1196 01:14:09,790 --> 01:14:19,950 So in our example, the Z is a matrix. 1197 01:14:19,950 --> 01:14:22,950 It starts out as the unit matrix. 1198 01:14:22,950 --> 01:14:28,270 And in MS bar, we collect the divergences that are not 1199 01:14:28,270 --> 01:14:32,437 cancelled by wave function renormalization. 1200 01:14:32,437 --> 01:14:34,270 And that gives us this little 2-by-2 matrix. 1201 01:14:56,720 --> 01:14:58,660 And from that little 2-by-2 matrix, 1202 01:14:58,660 --> 01:15:02,051 we can construct anomalous dimension for the operators. 1203 01:15:02,051 --> 01:15:02,830 So let's do that. 1204 01:15:12,440 --> 01:15:13,360 So how do we do that? 1205 01:15:16,040 --> 01:15:18,250 Well, if we take the renormalization of the Bayer 1206 01:15:18,250 --> 01:15:21,130 operators, remember those dependent on the regulator, 1207 01:15:21,130 --> 01:15:23,260 but they didn't depend on the scale mu. 1208 01:15:23,260 --> 01:15:25,450 So if we take mu d by d mu of Oi 0, 1209 01:15:25,450 --> 01:15:27,770 which is the Bayer operator at 0. 1210 01:15:27,770 --> 01:15:30,560 That's what I've written. 1211 01:15:30,560 --> 01:15:33,550 And then we can write O in terms of the Z, 1212 01:15:33,550 --> 01:15:36,690 in the equation I raised. 1213 01:15:36,690 --> 01:15:40,040 And both Z and the renormalized operator 1214 01:15:40,040 --> 01:15:42,157 depend on the scale mu. 1215 01:15:42,157 --> 01:15:43,615 And so that gives us this equation. 1216 01:15:52,170 --> 01:15:54,490 So I'm doing method 1 here, if you like, 1217 01:15:54,490 --> 01:15:56,650 using the notations from method 1. 1218 01:16:01,062 --> 01:16:02,770 If I take that equation and rearrange it, 1219 01:16:02,770 --> 01:16:04,330 I can write it as anomalous dimension 1220 01:16:04,330 --> 01:16:05,720 equation for the operator. 1221 01:16:05,720 --> 01:16:08,830 So it's mu d by d mu is one of the terms. 1222 01:16:08,830 --> 01:16:11,230 I isolate this, and I move that over by taking 1223 01:16:11,230 --> 01:16:12,400 an inverse of it. 1224 01:16:16,620 --> 01:16:19,540 And I just call everything that I 1225 01:16:19,540 --> 01:16:23,288 get there the anomalous dimension of the operator. 1226 01:16:27,590 --> 01:16:33,215 So gamma ji is all the other stuff, Zjk inverse. 1227 01:16:43,310 --> 01:16:45,050 And I put the explicit minus sign here 1228 01:16:45,050 --> 01:16:46,800 so there's no minus sign in that equation. 1229 01:16:50,095 --> 01:16:52,220 Anomalous dimension is determined by the Z factors. 1230 01:16:54,952 --> 01:16:56,660 And we determine the Z factors, and which 1231 01:16:56,660 --> 01:16:59,660 we can stick this equation into that equation 1232 01:16:59,660 --> 01:17:01,319 and get the anomalous dimension. 1233 01:17:11,600 --> 01:17:14,900 When we do that, we have to be careful about the fact 1234 01:17:14,900 --> 01:17:18,355 that alpha s in d dimensions has a little extra piece that's 1235 01:17:18,355 --> 01:17:20,230 actually important for this discussion, which 1236 01:17:20,230 --> 01:17:22,870 is this piece. 1237 01:17:22,870 --> 01:17:26,645 So when we take mu d by d mu, you ask, what depends on mu? 1238 01:17:26,645 --> 01:17:29,190 And it's the alpha here that depends on mu. 1239 01:17:29,190 --> 01:17:30,960 And it's still in-- 1240 01:17:30,960 --> 01:17:34,180 depends on epsilon as well in that equation. 1241 01:17:34,180 --> 01:17:37,320 And so this term is the term that matters at one loop. 1242 01:17:40,410 --> 01:17:43,390 And this matters at two loops, and so does that. 1243 01:17:43,390 --> 01:17:47,610 But at one loop, we could just replace this by the identity, 1244 01:17:47,610 --> 01:17:48,630 and we can drop this. 1245 01:17:57,820 --> 01:17:58,920 OK. 1246 01:17:58,920 --> 01:18:00,280 Put those things together. 1247 01:18:00,280 --> 01:18:02,730 Anomalous dimension, of course, is something 1248 01:18:02,730 --> 01:18:05,610 that doesn't depend on epsilon. 1249 01:18:05,610 --> 01:18:07,545 And it's, in this case, a 2-by-2 matrix. 1250 01:18:19,060 --> 01:18:21,657 And we can solve this problem. 1251 01:18:21,657 --> 01:18:23,865 If we want to solve this matrix, then what do you do? 1252 01:18:23,865 --> 01:18:25,660 Well, you could diagonalize it. 1253 01:18:25,660 --> 01:18:28,920 That's how you would solve it. 1254 01:18:28,920 --> 01:18:31,420 So if you want to run the operators in this theory just 1255 01:18:31,420 --> 01:18:44,220 for completeness, you would diagonalize 1256 01:18:44,220 --> 01:18:49,120 by forming O1 plus or minus O2, call the coefficients of those 1257 01:18:49,120 --> 01:18:53,715 new operators, C plus or minus, and an obvious notation. 1258 01:18:56,450 --> 01:18:58,830 And when you write down an anomalous dimension 1259 01:18:58,830 --> 01:19:01,020 equations for these guys, there's 1260 01:19:01,020 --> 01:19:02,720 no mixing anymore at one loop. 1261 01:19:05,855 --> 01:19:07,230 In general, this is the procedure 1262 01:19:07,230 --> 01:19:08,855 if you do this diagonalization that you 1263 01:19:08,855 --> 01:19:12,210 have to carry out again when you get to two loops. 1264 01:19:12,210 --> 01:19:14,550 It's not like the basis that you pick at one loop 1265 01:19:14,550 --> 01:19:17,698 will be fine for two loops. 1266 01:19:17,698 --> 01:19:18,490 We're not so lucky. 1267 01:19:21,380 --> 01:19:27,435 But at one loop, this is a perfectly valid basis 1268 01:19:27,435 --> 01:19:28,430 for the problem. 1269 01:19:31,790 --> 01:19:34,870 And in an hopefully self-evident notation, 1270 01:19:34,870 --> 01:19:36,580 I either take all pluses or all minuses. 1271 01:19:57,430 --> 01:19:58,880 OK. 1272 01:19:58,880 --> 01:20:03,470 And if I write my Hamiltonian, I could write it 1273 01:20:03,470 --> 01:20:05,500 in the original basis. 1274 01:20:05,500 --> 01:20:08,810 Then if I switch to the other basis, 1275 01:20:08,810 --> 01:20:12,110 I'll set up my convention so that it's simple. 1276 01:20:12,110 --> 01:20:15,190 And that means I picked C plus or minus 1277 01:20:15,190 --> 01:20:18,334 to be C1 plus or minus C2/2. 1278 01:20:18,334 --> 01:20:21,707 AUDIENCE: [INAUDIBLE] minus? 1279 01:20:21,707 --> 01:20:23,040 IAIN STEWART: Whoops, thank you. 1280 01:20:29,140 --> 01:20:31,430 Yeah, they have the opposite sign. 1281 01:20:31,430 --> 01:20:36,980 First one's positive, second one's negative. 1282 01:20:36,980 --> 01:20:42,180 And in this new basis tree level matching, 1283 01:20:42,180 --> 01:20:48,970 which is your boundary condition for your revolution, 1284 01:20:48,970 --> 01:20:50,830 is that both coefficients are 1/2. 1285 01:20:54,970 --> 01:20:55,470 OK. 1286 01:20:55,470 --> 01:20:59,370 So then we could solve the differential equation 1287 01:20:59,370 --> 01:21:00,990 the same way we were doing for QCD-- 1288 01:21:00,990 --> 01:21:02,810 write down a result that would sum 1289 01:21:02,810 --> 01:21:05,540 logs below the scale of the W mass. 1290 01:21:10,020 --> 01:21:10,590 OK. 1291 01:21:10,590 --> 01:21:14,860 So we'll continue along these lines next time, 1292 01:21:14,860 --> 01:21:17,092 say a few more words about renormalization, 1293 01:21:17,092 --> 01:21:19,300 which is really just a lead-up of doing the matching, 1294 01:21:19,300 --> 01:21:20,810 which is what I want to-- 1295 01:21:20,810 --> 01:21:23,360 kind of the thing I want to spend a little more time on. 1296 01:21:23,360 --> 01:21:25,995 So we'll finish that up also next time 1297 01:21:25,995 --> 01:21:29,400 and move on to some other things. 1298 01:21:29,400 --> 01:21:30,900 So there's a few things to emphasize 1299 01:21:30,900 --> 01:21:31,710 when we do the matching. 1300 01:21:31,710 --> 01:21:33,543 And I want to write down the matrix elements 1301 01:21:33,543 --> 01:21:36,020 and the full theory, which are box diagrams, 1302 01:21:36,020 --> 01:21:38,143 I'll write down the results for those. 1303 01:21:38,143 --> 01:21:39,560 We'll compare those to the results 1304 01:21:39,560 --> 01:21:40,977 of the effective theory, and we'll 1305 01:21:40,977 --> 01:21:42,195 draw some lessons from that. 1306 01:21:42,195 --> 01:21:43,570 And we'll carry out the matching, 1307 01:21:43,570 --> 01:21:48,690 figure out what the next order coefficients would be here. 1308 01:21:48,690 --> 01:21:51,535 What would those alpha s terms be? 1309 01:21:51,535 --> 01:21:53,430 What's the procedure I would go through 1310 01:21:53,430 --> 01:21:55,440 to get those coefficients? 1311 01:21:55,440 --> 01:21:55,940 OK. 1312 01:21:55,940 --> 01:21:58,340 So we'll stop there for today.