1 00:00:00,000 --> 00:00:02,520 The following content is provided under a Creative 2 00:00:02,520 --> 00:00:03,970 Commons license. 3 00:00:03,970 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,660 continue to offer high quality educational resources for free. 5 00:00:10,660 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,160 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,160 --> 00:00:18,370 at ocw.mit.edu. 8 00:00:20,980 --> 00:00:22,540 IAIN STEWART: --play with each other. 9 00:00:22,540 --> 00:00:24,350 We did the standard model as an effective field 10 00:00:24,350 --> 00:00:25,780 theory, higher dimension operators 11 00:00:25,780 --> 00:00:27,220 in the standard model. 12 00:00:27,220 --> 00:00:29,950 And then we started talking about taking the standard model 13 00:00:29,950 --> 00:00:33,340 as a theory one and removing things from it, 14 00:00:33,340 --> 00:00:35,380 in particular constructing what's 15 00:00:35,380 --> 00:00:38,590 called the weak Hamiltonian by removing the top W, Z, 16 00:00:38,590 --> 00:00:41,260 and Higgs from the standard model. 17 00:00:41,260 --> 00:00:45,670 And last time, we were focusing on the anomalous dimensions 18 00:00:45,670 --> 00:00:48,640 and things about renormalization. 19 00:00:48,640 --> 00:00:51,790 So we had this equation for the weak Hamiltonian 20 00:00:51,790 --> 00:00:57,220 for a particular case of b goes to c u bar d. 21 00:00:57,220 --> 00:00:58,900 That was the case that we decided 22 00:00:58,900 --> 00:01:01,660 to study rather than the full Hamiltonian. 23 00:01:01,660 --> 00:01:03,130 So there was some pre-factor. 24 00:01:03,130 --> 00:01:06,680 We had two operators with Wilson coefficients and operators. 25 00:01:06,680 --> 00:01:10,790 These are four-fermion operators. 26 00:01:10,790 --> 00:01:12,040 And there was different bases. 27 00:01:12,040 --> 00:01:14,577 We could write them in the bare form, 28 00:01:14,577 --> 00:01:16,660 or we could write them in renormalized coefficient 29 00:01:16,660 --> 00:01:18,650 and renormalized operator. 30 00:01:18,650 --> 00:01:20,800 And then we could do it either in the 1, 2 basis 31 00:01:20,800 --> 00:01:22,970 or the plus minus basis. 32 00:01:22,970 --> 00:01:25,330 So the plus minus basis is just linear combinations 33 00:01:25,330 --> 00:01:26,050 of the 1, 2. 34 00:01:29,410 --> 00:01:31,000 If you're in the 1, 2 basis, then you 35 00:01:31,000 --> 00:01:32,300 have this mixing matrix. 36 00:01:32,300 --> 00:01:33,325 So it's a 2 by 2 matrix. 37 00:01:33,325 --> 00:01:36,760 So if you're in the plus minus basis, 38 00:01:36,760 --> 00:01:42,100 then at least, at the lowest order, 39 00:01:42,100 --> 00:01:44,540 it's a simple product equation. 40 00:01:44,540 --> 00:01:46,690 So plus doesn't interfere with minus. 41 00:01:50,610 --> 00:01:51,110 OK. 42 00:01:51,110 --> 00:01:56,730 So that's where we got to, and we'll just continue today. 43 00:01:56,730 --> 00:01:59,390 So we have these anomalous dimension equations 44 00:01:59,390 --> 00:02:00,493 for the operators. 45 00:02:00,493 --> 00:02:02,660 We can also write down anomalous dimension equations 46 00:02:02,660 --> 00:02:04,770 for the Wilson coefficients. 47 00:02:04,770 --> 00:02:06,396 How do we do that? 48 00:02:06,396 --> 00:02:10,542 Well, the way that we do that is we make use of the fact 49 00:02:10,542 --> 00:02:14,060 that, if we look at the first line here, there's no mu. 50 00:02:14,060 --> 00:02:17,173 So the Hamiltonian is mu independent. 51 00:02:17,173 --> 00:02:19,340 That means that the mu dependence of the coefficient 52 00:02:19,340 --> 00:02:23,420 cancels the mu dependence of the operator. 53 00:02:23,420 --> 00:02:26,458 And you can use that to take an anomalous dimension 54 00:02:26,458 --> 00:02:28,000 equation for the operator and turn it 55 00:02:28,000 --> 00:02:29,729 into one for the coefficient. 56 00:02:40,210 --> 00:02:42,365 So last time, we talked about the fact 57 00:02:42,365 --> 00:02:43,990 that the normalization of the operators 58 00:02:43,990 --> 00:02:45,310 was equivalent to-- you can think 59 00:02:45,310 --> 00:02:46,330 about it two different ways. 60 00:02:46,330 --> 00:02:48,538 There's renormalization of operators, renormalization 61 00:02:48,538 --> 00:02:49,240 of coefficients. 62 00:02:49,240 --> 00:02:51,633 Likewise, you can think of anomalous dimensions 63 00:02:51,633 --> 00:02:53,050 is either running the coefficients 64 00:02:53,050 --> 00:02:54,160 or running the operators. 65 00:02:54,160 --> 00:02:56,320 So those are equivalent things. 66 00:02:56,320 --> 00:02:59,260 And the thing that makes them equivalent 67 00:02:59,260 --> 00:03:04,720 is just imposing that the derivative with respect 68 00:03:04,720 --> 00:03:06,310 to mu of the Hamiltonian is 0. 69 00:03:15,860 --> 00:03:18,890 So if I use my equation up here for anomalous dimension 70 00:03:18,890 --> 00:03:27,080 of the operator, I get, with my sign convention, a minus sign 71 00:03:27,080 --> 00:03:28,715 here. 72 00:03:28,715 --> 00:03:30,560 Or I had a minus sign here. 73 00:03:30,560 --> 00:03:31,640 I had no minus sign here. 74 00:03:31,640 --> 00:03:34,010 These things are conventions. 75 00:03:34,010 --> 00:03:37,650 I picked some convention, and I'll stick with it. 76 00:03:37,650 --> 00:03:41,235 So from this equation, we basically 77 00:03:41,235 --> 00:03:43,610 have an equation that has to be true for the coefficients 78 00:03:43,610 --> 00:03:45,830 if we think of just stripping off the operator. 79 00:04:05,540 --> 00:04:08,737 So we could write it this way, just reading it off right 80 00:04:08,737 --> 00:04:10,820 from here, just putting this guy on the other side 81 00:04:10,820 --> 00:04:13,640 and then reading it off, dropping the operator. 82 00:04:13,640 --> 00:04:15,800 Or we could write it this way if we 83 00:04:15,800 --> 00:04:17,630 wanted to write it in a way that's 84 00:04:17,630 --> 00:04:19,610 more similar to this equation up here, where 85 00:04:19,610 --> 00:04:22,780 you have the anomalous dimension matrix times the coefficient. 86 00:04:22,780 --> 00:04:24,620 It's either from the right or from the left, 87 00:04:24,620 --> 00:04:27,593 and it's just a matter of transposing. 88 00:04:27,593 --> 00:04:29,510 So the anomalous dimension for the coefficient 89 00:04:29,510 --> 00:04:32,660 is determined from the one, this one here. 90 00:04:32,660 --> 00:04:34,160 If you know it, then you immediately 91 00:04:34,160 --> 00:04:37,490 know the one for the coefficient, which 92 00:04:37,490 --> 00:04:40,040 shouldn't be surprising given that we could think of the Z 93 00:04:40,040 --> 00:04:43,820 factors as being related to renormalization factors. 94 00:04:43,820 --> 00:04:47,970 Anomalous dimensions, therefore, should also be related. 95 00:04:47,970 --> 00:04:51,030 OK, so we'll solve this equation. 96 00:04:51,030 --> 00:04:53,080 It's a little bit simpler to think about. 97 00:04:53,080 --> 00:04:54,538 Although we could have equivalently 98 00:04:54,538 --> 00:04:56,687 solve the operator equation. 99 00:04:56,687 --> 00:04:58,020 So how do we solve the equation? 100 00:05:03,490 --> 00:05:07,300 Well, we go over to our plus minus basis. 101 00:05:07,300 --> 00:05:11,850 So take the coefficient of either C+ or C-. 102 00:05:11,850 --> 00:05:13,740 And the equation that we need to solve 103 00:05:13,740 --> 00:05:17,115 can be written as follows. 104 00:05:30,510 --> 00:05:31,750 It's a simple equation. 105 00:05:31,750 --> 00:05:34,790 And I can even take the C, which is on the right-hand side, 106 00:05:34,790 --> 00:05:36,950 move it over to the left, if I write log C here. 107 00:05:36,950 --> 00:05:39,560 Because the derivative of the log is giving me a 1 over C. 108 00:05:39,560 --> 00:05:42,560 If I put it back over here, it would just multiply. 109 00:05:42,560 --> 00:05:44,330 OK, so that's the analog. 110 00:05:44,330 --> 00:05:46,550 This equation here is the analog of this equation 111 00:05:46,550 --> 00:05:50,260 here for the operators, but now for the coefficients. 112 00:05:50,260 --> 00:05:52,760 Obviously, if these are numbers, there's no transpose to do. 113 00:05:56,050 --> 00:05:58,810 So you have to solve this equation simultaneously 114 00:05:58,810 --> 00:06:00,350 with another differential equation. 115 00:06:00,350 --> 00:06:02,698 This is a coupled equation because alpha also 116 00:06:02,698 --> 00:06:03,865 has a differential equation. 117 00:06:20,150 --> 00:06:23,350 So at lowest order, this is the beta function equation. 118 00:06:23,350 --> 00:06:27,340 And so if we want to solve, take into account this equation 119 00:06:27,340 --> 00:06:30,610 and integrate this equation, the simple trick for doing that 120 00:06:30,610 --> 00:06:34,250 is to make a change of variable. 121 00:06:34,250 --> 00:06:39,040 So we use this equation here to make a change of variable. 122 00:06:39,040 --> 00:06:41,350 This is a very useful trick because it actually 123 00:06:41,350 --> 00:06:43,360 works to whatever order in the expansion 124 00:06:43,360 --> 00:06:46,896 that you might want to work. 125 00:06:46,896 --> 00:06:49,780 So let me explain what the trick is, and then I'll 126 00:06:49,780 --> 00:06:51,480 explain why that is. 127 00:06:51,480 --> 00:06:53,890 So we're going to change variables in this equation 128 00:06:53,890 --> 00:06:54,880 from mu to alpha. 129 00:07:01,658 --> 00:07:03,950 You see we could think about solving this equation here 130 00:07:03,950 --> 00:07:07,820 by integrating, just move this thing to the other side 131 00:07:07,820 --> 00:07:09,230 and integrate. 132 00:07:09,230 --> 00:07:12,080 But we'd be integrating, in mu, a function that's 133 00:07:12,080 --> 00:07:14,910 a function of alpha of mu. 134 00:07:14,910 --> 00:07:16,985 So if we can switch from mu to alpha, 135 00:07:16,985 --> 00:07:19,110 then we'll be just integrating a function of alpha. 136 00:07:19,110 --> 00:07:21,770 And that's what we're going to do using this equation. 137 00:07:38,240 --> 00:07:41,050 So if I write it in general, that's this equality here. 138 00:07:41,050 --> 00:07:45,610 For any function of alpha, I say that d mu over mu-- 139 00:07:45,610 --> 00:07:49,730 just rearranging this equation-- is d alpha over beta of alpha. 140 00:07:49,730 --> 00:07:51,940 So I can switch the integration d mu over mu 141 00:07:51,940 --> 00:07:55,040 to d alpha over beta of alpha. 142 00:07:55,040 --> 00:07:56,800 And that's exactly what I want to do 143 00:07:56,800 --> 00:07:59,650 if I want to move this operator to the other side, 144 00:07:59,650 --> 00:08:01,082 this operation. 145 00:08:01,082 --> 00:08:03,040 If I work at lowest order in the beta function, 146 00:08:03,040 --> 00:08:06,710 then that's just I just plug in this result. 147 00:08:06,710 --> 00:08:10,180 So we can switch variables from mu to alpha. 148 00:08:10,180 --> 00:08:13,360 And if I had high order terms in this equation and high order 149 00:08:13,360 --> 00:08:16,810 terms in this equation, I could use the same trick. 150 00:08:16,810 --> 00:08:18,575 I just have other integrals to do. 151 00:08:18,575 --> 00:08:20,450 And the integrals are pretty straightforward, 152 00:08:20,450 --> 00:08:24,260 so this is a useful way of proceeding. 153 00:08:24,260 --> 00:08:24,760 OK. 154 00:08:24,760 --> 00:08:25,750 So what does that do? 155 00:08:25,750 --> 00:08:26,770 So now, let's do that. 156 00:08:26,770 --> 00:08:28,764 Let's move this over and integrate. 157 00:08:36,669 --> 00:08:41,400 Well, we'll do a definite integral from mu w up to mu. 158 00:08:46,846 --> 00:08:49,300 So here, we just have d log this. 159 00:08:49,300 --> 00:08:54,610 Integrate that-- just gives log between the limits. 160 00:09:06,200 --> 00:09:09,010 And if I didn't change variable, I would have that. 161 00:09:09,010 --> 00:09:11,950 But if I make the change variable, 162 00:09:11,950 --> 00:09:13,600 then it becomes a very simple integral. 163 00:09:36,400 --> 00:09:38,890 And remember that this guy here is also 164 00:09:38,890 --> 00:09:46,150 just a number times alpha that we worked out last time. 165 00:09:48,830 --> 00:09:51,590 So this is just d alpha over alpha. 166 00:09:51,590 --> 00:09:53,410 And that's a simple logarithmic integral. 167 00:09:53,410 --> 00:09:55,030 Yeah. 168 00:09:55,030 --> 00:09:56,530 AUDIENCE: I don't know if it matter, 169 00:09:56,530 --> 00:10:00,340 but mu w should be greater than mu, right? 170 00:10:00,340 --> 00:10:02,380 IAIN STEWART: Mu w should be greater than mu. 171 00:10:02,380 --> 00:10:03,040 That's right. 172 00:10:03,040 --> 00:10:04,957 AUDIENCE: OK, so you're just writing integrals 173 00:10:04,957 --> 00:10:06,324 like that to avoid signs? 174 00:10:06,324 --> 00:10:07,596 OK. 175 00:10:07,596 --> 00:10:08,708 I have another question. 176 00:10:08,708 --> 00:10:09,500 IAIN STEWART: Yeah. 177 00:10:09,500 --> 00:10:12,830 AUDIENCE: How do you know that the anomalous dimensions, 178 00:10:12,830 --> 00:10:15,474 including the beta function, are only functions of alpha S 179 00:10:15,474 --> 00:10:16,630 rather than [INAUDIBLE]. 180 00:10:16,630 --> 00:10:18,530 IAIN STEWART: Ah, yeah. 181 00:10:18,530 --> 00:10:21,750 So I'm sneaking that in here. 182 00:10:21,750 --> 00:10:25,580 So it follows from the renormalization structure 183 00:10:25,580 --> 00:10:28,070 of this effective field theory that there's only 184 00:10:28,070 --> 00:10:31,250 single logarithmic divergences. 185 00:10:31,250 --> 00:10:33,890 So in the standard model, if you're at one loop, 186 00:10:33,890 --> 00:10:36,500 you only have 1 over epsilon poles for the renormalization. 187 00:10:36,500 --> 00:10:38,470 And you're renormalizing the coupling. 188 00:10:38,470 --> 00:10:40,220 The same is true of this effective theory. 189 00:10:40,220 --> 00:10:43,640 At one loop, you only have 1 over epsilon divergences. 190 00:10:43,640 --> 00:10:47,450 And that implies that your anomalous dimensions 191 00:10:47,450 --> 00:10:50,340 won't depend on anything more complicated. 192 00:10:50,340 --> 00:10:53,180 We will discuss more complicated cases in the future, 193 00:10:53,180 --> 00:10:54,740 as you know. 194 00:10:54,740 --> 00:11:00,080 But the structure of this effective theory and its UV 195 00:11:00,080 --> 00:11:04,460 structure, which I didn't go into on a lot of detail about, 196 00:11:04,460 --> 00:11:06,510 implies that fact. 197 00:11:06,510 --> 00:11:07,010 Yeah. 198 00:11:07,010 --> 00:11:08,300 AUDIENCE: [INAUDIBLE] 199 00:11:08,300 --> 00:11:09,092 IAIN STEWART: Yeah. 200 00:11:11,540 --> 00:11:12,040 OK. 201 00:11:12,040 --> 00:11:13,860 So do the integral. 202 00:11:13,860 --> 00:11:19,120 There's some pre-factor, which I'll call a+-. 203 00:11:19,120 --> 00:11:22,100 And then I get a log, as I mentioned. 204 00:11:27,160 --> 00:11:29,620 And just for the record, this a+-, 205 00:11:29,620 --> 00:11:32,020 if I put all the factors together and put in what this 206 00:11:32,020 --> 00:11:43,990 number is, these would be some factors like this. 207 00:11:43,990 --> 00:11:47,350 And I've put in that Nc is 3. 208 00:11:51,540 --> 00:11:55,340 So this alpha at mu w and this C+ of mu w, 209 00:11:55,340 --> 00:11:57,960 you should think of mu w as the boundary condition scale. 210 00:12:02,202 --> 00:12:03,660 So this is a differential equation. 211 00:12:03,660 --> 00:12:06,330 We needed a boundary condition to solve it. 212 00:12:06,330 --> 00:12:08,220 And the boundary condition is the value 213 00:12:08,220 --> 00:12:13,195 of the coefficients at the scale uw, which 214 00:12:13,195 --> 00:12:14,445 is supposed to be of order Mw. 215 00:12:23,760 --> 00:12:26,100 Typically, what that means is you 216 00:12:26,100 --> 00:12:28,110 could take a common choice, which 217 00:12:28,110 --> 00:12:31,020 would be just to take it equal. 218 00:12:31,020 --> 00:12:34,740 Or you could pick twice or half. 219 00:12:34,740 --> 00:12:38,220 And these are the most common choices that people pick. 220 00:12:41,400 --> 00:12:44,490 So the way that you should think of that, 221 00:12:44,490 --> 00:12:48,060 this guy in the denominator, is you 222 00:12:48,060 --> 00:12:52,860 should think that he's really a fixed order series in alpha 223 00:12:52,860 --> 00:13:02,970 of mu w, something that you would calculate order by order 224 00:13:02,970 --> 00:13:06,180 and perturbation theory. 225 00:13:06,180 --> 00:13:09,060 And you'd be determining the boundary condition. 226 00:13:09,060 --> 00:13:13,420 We'll talk about how you would do that a little later today. 227 00:13:13,420 --> 00:13:16,770 But for now, just think of it as a series in alpha of mu w. 228 00:13:16,770 --> 00:13:19,500 And it doesn't have any large logarithms. 229 00:13:19,500 --> 00:13:21,480 And as Elia said, you want to think 230 00:13:21,480 --> 00:13:26,040 of this mu as some small scale, some scale that's less than uw 231 00:13:26,040 --> 00:13:27,690 because you're thinking of evolving 232 00:13:27,690 --> 00:13:30,145 the operators to a scale less than the scale 233 00:13:30,145 --> 00:13:31,770 where you integrated out the particles. 234 00:13:34,322 --> 00:13:35,780 I'll draw that picture in a second. 235 00:13:53,030 --> 00:13:56,470 So we can take the exponential of that equation, 236 00:13:56,470 --> 00:14:00,220 and then we can write C of mu is equal to something 237 00:14:00,220 --> 00:14:01,638 that we've determined. 238 00:14:10,610 --> 00:14:12,090 Take the exponential. 239 00:14:12,090 --> 00:14:13,520 Move this guy to the other side. 240 00:14:41,280 --> 00:14:44,910 Remember the a+- are just numbers. 241 00:14:44,910 --> 00:14:47,730 We can write the solution in this way, where we determine 242 00:14:47,730 --> 00:14:50,250 this guy by thinking about doing a matching calculation 243 00:14:50,250 --> 00:14:51,090 at the high scale. 244 00:14:51,090 --> 00:14:54,810 We determine it to be 1/2 already at the lowest order. 245 00:14:54,810 --> 00:14:57,100 So think about sticking in 1/2 here. 246 00:14:57,100 --> 00:15:00,030 And then this factor here is what you get 247 00:15:00,030 --> 00:15:02,192 from the renormalization group. 248 00:15:02,192 --> 00:15:03,900 And you can see from this form right here 249 00:15:03,900 --> 00:15:06,192 that you've summed up an infinite number of logarithms. 250 00:15:06,192 --> 00:15:07,860 It's exponential of a number times log. 251 00:15:07,860 --> 00:15:14,280 And if I were to expand that out in alpha of some fixed scale, 252 00:15:14,280 --> 00:15:16,280 there would be an infinite series in logarithms. 253 00:15:20,310 --> 00:15:20,810 OK. 254 00:15:20,810 --> 00:15:22,610 So what do we want to pick this mu to be? 255 00:15:22,610 --> 00:15:29,180 Well, we're thinking about the process mu goes to c u bar d. 256 00:15:29,180 --> 00:15:31,700 If you think about this process in nature, 257 00:15:31,700 --> 00:15:36,350 the scale in the initial state here is the b has a mass. 258 00:15:36,350 --> 00:15:39,668 So you'd like to take this scale here not at Mw, but down 259 00:15:39,668 --> 00:15:40,460 at the b core mass. 260 00:15:48,290 --> 00:15:50,350 So we want mu to be of order Mb. 261 00:15:55,776 --> 00:15:58,430 And then you have a large hierarchy because this 262 00:15:58,430 --> 00:16:01,280 is much less than Mw-- 263 00:16:01,280 --> 00:16:04,160 5 g of e-ish, 80 g of e. 264 00:16:12,150 --> 00:16:23,620 So our result here sums what are called the leading logarithms, 265 00:16:23,620 --> 00:16:27,360 which is denoted by LL. 266 00:16:27,360 --> 00:16:30,960 And schematically, the lowest order term was 1/2. 267 00:16:30,960 --> 00:16:37,130 And if we were to expand higher order terms 268 00:16:37,130 --> 00:16:39,380 and think about what logarithms we're talking about, 269 00:16:39,380 --> 00:16:43,130 we're talking about logarithms of Mw over Mb. 270 00:16:45,930 --> 00:16:47,810 And the series, if we were to expand it, 271 00:16:47,810 --> 00:16:53,195 would look like this schematically 272 00:16:53,195 --> 00:16:58,100 without worrying about the coefficients, 273 00:16:58,100 --> 00:17:02,330 an infinite series where each term has one alpha and one log. 274 00:17:05,900 --> 00:17:07,480 So the counting that you're doing 275 00:17:07,480 --> 00:17:11,500 in this type of setup, where we would think 276 00:17:11,500 --> 00:17:25,020 of using this equation to go down to the scale Mb, 277 00:17:25,020 --> 00:17:30,845 is that you're counting this parameter as order 1. 278 00:17:30,845 --> 00:17:32,220 And you're saying, any time I see 279 00:17:32,220 --> 00:17:34,388 a log of Mw over Mb times an alpha, 280 00:17:34,388 --> 00:17:36,180 I'm not going to count that as order alpha. 281 00:17:36,180 --> 00:17:38,010 I'm going to count that as order 1. 282 00:17:38,010 --> 00:17:40,170 That's why I have to sum up this infinite series. 283 00:17:46,820 --> 00:17:49,390 So the physical picture of what we've said here 284 00:17:49,390 --> 00:17:50,617 is the following. 285 00:17:54,395 --> 00:17:55,853 So the basic physical picture would 286 00:17:55,853 --> 00:17:58,990 be that there's two scales, Mw an Mb, which 287 00:17:58,990 --> 00:18:00,220 are physical scales. 288 00:18:00,220 --> 00:18:02,590 You want to get rid of the scale Mw. 289 00:18:02,590 --> 00:18:05,290 You do that by going over to this electroweak Hamiltonian. 290 00:18:05,290 --> 00:18:06,850 But then you have to renormalization 291 00:18:06,850 --> 00:18:10,300 group evolve Hamiltonian down to the scale where you want to do 292 00:18:10,300 --> 00:18:13,300 physics, which is the scale Mb. 293 00:18:13,300 --> 00:18:15,890 And when you do that, there's some choice in the matter. 294 00:18:15,890 --> 00:18:19,400 And we've been careful to parameterize that choice. 295 00:18:19,400 --> 00:18:22,420 We said that you pick a scale that's 296 00:18:22,420 --> 00:18:25,943 of order Mw, which we called mu w. 297 00:18:25,943 --> 00:18:27,610 And then we said, you pick another scale 298 00:18:27,610 --> 00:18:30,355 that's of order Mb, which we called mu. 299 00:18:30,355 --> 00:18:32,230 And you actually do the renormalization group 300 00:18:32,230 --> 00:18:34,570 between these two. 301 00:18:34,570 --> 00:18:39,400 You could pick Mw and mu equal exactly Mb. 302 00:18:39,400 --> 00:18:42,580 That would be another simpler story. 303 00:18:42,580 --> 00:18:45,100 But it is actually important that, 304 00:18:45,100 --> 00:18:46,960 once you go beyond the lowest order, 305 00:18:46,960 --> 00:18:49,900 to keep track of the fact that you have this freedom. 306 00:18:49,900 --> 00:18:53,240 And that's why I've kept track over here. 307 00:18:53,240 --> 00:18:55,660 So what that means is that, in terms of counting, 308 00:18:55,660 --> 00:19:05,280 you've counted this, but logs of mu or Mw 309 00:19:05,280 --> 00:19:07,380 were counted as order 1. 310 00:19:07,380 --> 00:19:11,760 And then down here, logs of mu over B 311 00:19:11,760 --> 00:19:15,900 are counted as order 1 numbers. 312 00:19:15,900 --> 00:19:21,810 It could be 0, but 0 is order 1, not enhanced 313 00:19:21,810 --> 00:19:25,230 such that they would compensate for a factor of alpha. 314 00:19:25,230 --> 00:19:30,810 And this is the renormalization group evolution or the running 315 00:19:30,810 --> 00:19:34,560 that sums up these logarithms here, which are the large logs. 316 00:19:38,760 --> 00:19:39,940 And that's pretty simple. 317 00:19:39,940 --> 00:19:42,580 It just gave this factor. 318 00:19:42,580 --> 00:19:45,253 And that's pretty common in QCD to get factors 319 00:19:45,253 --> 00:19:46,920 like that, alpha at one scale over alpha 320 00:19:46,920 --> 00:19:49,260 at another scale raised to a power. 321 00:19:49,260 --> 00:19:51,780 That's a very common thing to get from renormalization group 322 00:19:51,780 --> 00:19:52,817 evolution. 323 00:19:55,800 --> 00:19:57,860 OK, any questions so far? 324 00:20:14,600 --> 00:20:17,720 How many people have done the calculation 325 00:20:17,720 --> 00:20:20,480 of the anomalous dimension for four-fermion operators 326 00:20:20,480 --> 00:20:21,680 in some other course? 327 00:20:21,680 --> 00:20:23,870 It's a common problem. 328 00:20:23,870 --> 00:20:24,560 Nobody? 329 00:20:24,560 --> 00:20:25,905 All right. 330 00:20:25,905 --> 00:20:27,530 That means you'll see it on a homework. 331 00:20:41,210 --> 00:20:46,510 So let's come back here and think 332 00:20:46,510 --> 00:20:49,960 about what the general structure of what we've done is. 333 00:20:55,937 --> 00:20:57,020 And I'll put back indices. 334 00:21:09,738 --> 00:21:12,030 So you can think about taking the solution that we have 335 00:21:12,030 --> 00:21:16,320 at the top of the board here and generalizing it 336 00:21:16,320 --> 00:21:18,937 to a form that would be valid at higher orders. 337 00:21:18,937 --> 00:21:21,270 And basically, it says that the coefficient at one scale 338 00:21:21,270 --> 00:21:23,310 is connected to the coefficient at another scale 339 00:21:23,310 --> 00:21:25,350 times some evolution factor. 340 00:21:25,350 --> 00:21:26,850 In this case, the evolution factor 341 00:21:26,850 --> 00:21:29,010 is just the ratio of these alphas to a power. 342 00:21:29,010 --> 00:21:31,140 It could be some more complicated function 343 00:21:31,140 --> 00:21:32,100 at higher orders. 344 00:21:32,100 --> 00:21:33,520 And it could even be a matrix. 345 00:21:33,520 --> 00:21:34,895 That's why I've given it indices. 346 00:21:57,950 --> 00:22:01,420 So we can put our results back together 347 00:22:01,420 --> 00:22:03,520 using this higher order form, so that they're 348 00:22:03,520 --> 00:22:06,430 generally true, into our Hamiltonian 349 00:22:06,430 --> 00:22:08,035 and see what we've achieved. 350 00:22:21,740 --> 00:22:24,840 And let me call this scale that I was calling mu a minute 351 00:22:24,840 --> 00:22:28,440 ago mu b just to remind you that it's a scale of order Mb. 352 00:22:34,140 --> 00:22:36,390 So previously, we had the coefficient and the operator 353 00:22:36,390 --> 00:22:38,247 at the same scale. 354 00:22:38,247 --> 00:22:40,580 But now, using this equation, I can move the coefficient 355 00:22:40,580 --> 00:22:42,510 to a different scale. 356 00:22:42,510 --> 00:22:45,140 And so let me think of sticking this equation in. 357 00:22:45,140 --> 00:22:46,700 And then I have mu w. 358 00:22:46,700 --> 00:22:48,810 I've called mu equals mu b. 359 00:22:48,810 --> 00:22:50,860 So now, this is mu w mu b. 360 00:22:50,860 --> 00:22:57,890 So this here is the coefficient Ci at mu b, 361 00:22:57,890 --> 00:22:59,540 but I find it useful to write it out. 362 00:23:05,968 --> 00:23:07,760 So the thing in square brackets is Ci mu b, 363 00:23:07,760 --> 00:23:11,120 but I write it out using the renormalization group 364 00:23:11,120 --> 00:23:12,800 equation that way. 365 00:23:12,800 --> 00:23:15,560 And this tells you how you're doing the calculation. 366 00:23:15,560 --> 00:23:17,250 This is a fixed order calculation. 367 00:23:22,130 --> 00:23:26,330 This comes from anomalous dimensions 368 00:23:26,330 --> 00:23:28,940 and gives you the evolution. 369 00:23:28,940 --> 00:23:31,730 And then you have operators involving the B quark 370 00:23:31,730 --> 00:23:38,960 that you would calculate matrix elements of at mu 371 00:23:38,960 --> 00:23:41,330 b of order Mb. 372 00:23:41,330 --> 00:23:45,500 And there's no dependence at all in those operators on the scale 373 00:23:45,500 --> 00:23:47,270 Mw. 374 00:23:47,270 --> 00:23:49,338 All the Mw's are in the pre-factors here. 375 00:23:49,338 --> 00:23:50,630 You've taken that into account. 376 00:23:50,630 --> 00:23:53,720 You've calculated it. 377 00:23:53,720 --> 00:23:58,180 OK, so that's how this organizes the physics 378 00:23:58,180 --> 00:24:01,030 of the different scales. 379 00:24:01,030 --> 00:24:03,117 So you could ask, if I had this story, 380 00:24:03,117 --> 00:24:04,450 how would I go to higher orders? 381 00:24:04,450 --> 00:24:06,730 And we will have some discussion of what 382 00:24:06,730 --> 00:24:09,400 goes on at higher orders because there are some things that 383 00:24:09,400 --> 00:24:11,912 happen at higher orders that you don't see at leading order. 384 00:24:11,912 --> 00:24:13,870 And they're actually important physical things, 385 00:24:13,870 --> 00:24:16,147 so important things to know about and keep 386 00:24:16,147 --> 00:24:18,230 track of if you ever want to use things like this. 387 00:24:18,230 --> 00:24:21,040 So let's talk a little bit about what it 388 00:24:21,040 --> 00:24:22,270 takes to go to higher orders. 389 00:24:33,970 --> 00:24:35,550 So let's just first think about what 390 00:24:35,550 --> 00:24:39,990 it would look like if we went to higher orders. 391 00:24:39,990 --> 00:24:43,680 Well, leading order was a series that I could schematically say 392 00:24:43,680 --> 00:24:46,500 is alpha times large logs. 393 00:24:46,500 --> 00:24:49,920 And I summed them all up, and I called that leading log. 394 00:24:49,920 --> 00:24:51,750 When I go to higher orders, I am going 395 00:24:51,750 --> 00:24:57,315 to continue to get series, but I got extra factors of alpha. 396 00:25:05,880 --> 00:25:10,490 So something that you call Next to Leading Log, or NLL, 397 00:25:10,490 --> 00:25:13,400 is the same type of thing, a different series than that 1 398 00:25:13,400 --> 00:25:14,970 times an extra factor of alpha. 399 00:25:14,970 --> 00:25:18,110 So it's down compared to this by alpha S. 400 00:25:18,110 --> 00:25:19,160 And then you keep going. 401 00:25:23,680 --> 00:25:28,370 This is the general structure of the renormalization group 402 00:25:28,370 --> 00:25:29,690 improved perturbation theory. 403 00:25:32,690 --> 00:25:36,380 Just keep adding Ns and keep adding alphas, always 404 00:25:36,380 --> 00:25:40,250 summing up some series which changes from order to order. 405 00:25:40,250 --> 00:25:41,960 And that summation of that series 406 00:25:41,960 --> 00:25:43,700 is determined by determining higher order 407 00:25:43,700 --> 00:25:45,710 anomalous dimensions. 408 00:25:45,710 --> 00:25:56,690 So this kind of thing is called renormalization group improved 409 00:25:56,690 --> 00:25:57,620 perturbation theory. 410 00:26:00,680 --> 00:26:04,490 Every time you take alpha S and you take it at some scale, 411 00:26:04,490 --> 00:26:06,680 you're already doing renormalization group 412 00:26:06,680 --> 00:26:08,120 improved perturbation theory. 413 00:26:08,120 --> 00:26:10,247 It's just that, once you have theories 414 00:26:10,247 --> 00:26:12,830 that have other things that run and have anomalous dimensions, 415 00:26:12,830 --> 00:26:16,340 then it can be more complicated than just simply picking alpha 416 00:26:16,340 --> 00:26:18,380 S at the appropriate scale. 417 00:26:18,380 --> 00:26:20,870 Here, in this theory, we have these coefficients. 418 00:26:20,870 --> 00:26:21,835 We have to run them. 419 00:26:21,835 --> 00:26:23,960 Then we have to pick them at the appropriate scale. 420 00:26:23,960 --> 00:26:25,670 And that's what we're doing by solving 421 00:26:25,670 --> 00:26:29,750 these renormalization group equations. 422 00:26:29,750 --> 00:26:30,250 OK. 423 00:26:30,250 --> 00:26:31,520 So what do we need to do? 424 00:26:31,520 --> 00:26:32,410 We determined this. 425 00:26:32,410 --> 00:26:34,660 I showed you what you needed to do to get that. 426 00:26:34,660 --> 00:26:37,210 What would we need to do to get the next term in the series? 427 00:26:37,210 --> 00:26:39,340 How much would we have to compute? 428 00:26:43,980 --> 00:26:45,930 Well, we just have to go to one higher order 429 00:26:45,930 --> 00:26:48,340 in the perturbation theory. 430 00:26:48,340 --> 00:27:00,290 So let's make a little table of what 431 00:27:00,290 --> 00:27:02,900 it takes to get leading log, next leading log. 432 00:27:05,470 --> 00:27:07,350 Maybe we'll even add one more term. 433 00:27:12,813 --> 00:27:14,480 So there's two parts to the calculation. 434 00:27:14,480 --> 00:27:16,910 There's the boundary condition, and then there's 435 00:27:16,910 --> 00:27:23,150 the differential equation, which is the anomalous dimension. 436 00:27:23,150 --> 00:27:26,480 At leading log, we had tree level matching. 437 00:27:26,480 --> 00:27:29,480 We determined the C plus and minus where 1/2. 438 00:27:29,480 --> 00:27:32,990 C1 and C2 were 1 and 0. 439 00:27:32,990 --> 00:27:36,980 And we just needed the one loop anomalous dimension. 440 00:27:36,980 --> 00:27:39,090 And then we just keep going in this pattern. 441 00:27:39,090 --> 00:27:44,330 So next leading log, we need to match it one loop. 442 00:27:44,330 --> 00:27:49,220 And we would need the two-loop anomalous dimension 443 00:27:49,220 --> 00:27:50,050 and et cetera. 444 00:27:53,790 --> 00:27:55,647 So the order in which you need the running 445 00:27:55,647 --> 00:27:57,980 is one higher order than what you need for the matching. 446 00:28:01,760 --> 00:28:02,820 That's the rule. 447 00:28:02,820 --> 00:28:04,400 And given those ingredients, we would 448 00:28:04,400 --> 00:28:09,860 be able to determine exactly these series here, OK? 449 00:28:15,080 --> 00:28:18,012 So there's some things that happen at this order 450 00:28:18,012 --> 00:28:19,970 that aren't really apparent yet at leading log, 451 00:28:19,970 --> 00:28:21,980 and so I want to talk a little bit about that. 452 00:28:34,980 --> 00:28:37,590 Before we get there, let me add one other little note. 453 00:28:41,080 --> 00:28:44,290 This operator O2, we didn't see it when we thought originally 454 00:28:44,290 --> 00:28:44,980 about matching. 455 00:28:44,980 --> 00:28:48,100 It had Wilson coefficient that was 0 at tree level. 456 00:28:53,040 --> 00:28:55,410 So at leading order, you could say that this Wilson 457 00:28:55,410 --> 00:28:58,020 coefficient is 0. 458 00:28:58,020 --> 00:28:59,760 But at leading log, it's not 0. 459 00:29:05,160 --> 00:29:08,070 So I have these two different types of perturbation theory. 460 00:29:08,070 --> 00:29:10,200 Just order by order and alpha or doing 461 00:29:10,200 --> 00:29:11,847 renormalization group improvement, 462 00:29:11,847 --> 00:29:12,930 you get different results. 463 00:29:17,970 --> 00:29:21,120 And that's because you've included some higher order 464 00:29:21,120 --> 00:29:25,110 terms by using the renormalization group improved 465 00:29:25,110 --> 00:29:27,720 version. 466 00:29:27,720 --> 00:29:30,960 But you can argue, if alpha times the large log is order 1, 467 00:29:30,960 --> 00:29:35,410 then this is the right type of perturbation theory to do. 468 00:29:35,410 --> 00:29:45,030 So if you think about it as a picture where this is mu, 469 00:29:45,030 --> 00:29:51,720 this is Mw, this is Mb, then you have two coefficients, C1 470 00:29:51,720 --> 00:29:52,470 and C2. 471 00:29:52,470 --> 00:29:55,890 We call them C+ and C-, but they're just related. 472 00:29:55,890 --> 00:29:58,200 And the results that we derived at leading order 473 00:29:58,200 --> 00:30:04,590 were that, for C1, it started at 1 at the high scale, 474 00:30:04,590 --> 00:30:07,740 basically, if we mu w equal to Mw. 475 00:30:07,740 --> 00:30:09,930 And it would evolve, actually, this direction 476 00:30:09,930 --> 00:30:14,820 if we put in all the signs that came out of our calculations. 477 00:30:14,820 --> 00:30:18,360 And for C2, it starts at 0 here, and then it evolves this way 478 00:30:18,360 --> 00:30:21,248 to a negative value. 479 00:30:21,248 --> 00:30:22,212 [INAUDIBLE] 480 00:30:32,340 --> 00:30:42,330 So roughly putting in some numbers, 481 00:30:42,330 --> 00:30:46,470 the kind of thing that we would get is this. 482 00:30:46,470 --> 00:30:49,860 So a coefficient which was 0 all of a sudden becomes minus 0.3 483 00:30:49,860 --> 00:30:52,320 and becomes something that you have to keep track of. 484 00:30:52,320 --> 00:30:53,280 That's at leading log. 485 00:30:56,793 --> 00:30:58,460 Obviously, when you go to higher orders, 486 00:30:58,460 --> 00:31:00,755 those numbers will be perturbatively improved. 487 00:31:04,370 --> 00:31:05,600 OK. 488 00:31:05,600 --> 00:31:08,510 So is the physical picture here clear of what's happening 489 00:31:08,510 --> 00:31:12,180 with these operators? 490 00:31:12,180 --> 00:31:14,855 So what is the application? 491 00:31:14,855 --> 00:31:18,292 Since we spent all this time deriving these results, 492 00:31:18,292 --> 00:31:20,000 we should have some applications in mind. 493 00:31:42,630 --> 00:31:47,570 So for b to c u bar d, if you ask about what process that 494 00:31:47,570 --> 00:31:52,550 gives, well, one process that it gives 495 00:31:52,550 --> 00:31:56,285 is just a B to D pi transition. 496 00:31:56,285 --> 00:32:00,230 B bar is built over u bar b. 497 00:32:00,230 --> 00:32:04,220 And this guy is u bar c. 498 00:32:04,220 --> 00:32:06,050 And the pi is the u bar d. 499 00:32:08,783 --> 00:32:11,200 So we can think of the reason we're studying this is maybe 500 00:32:11,200 --> 00:32:13,880 we want to calculate B of D pi. 501 00:32:13,880 --> 00:32:15,850 So if we wanted to calculate B of D pi, 502 00:32:15,850 --> 00:32:20,650 we take matrix elements involving our Hamiltonian 503 00:32:20,650 --> 00:32:23,270 with a B [INAUDIBLE] in the in state and a D pi in the out 504 00:32:23,270 --> 00:32:23,770 state. 505 00:32:31,400 --> 00:32:33,670 And if we just use the original Hamiltonian 506 00:32:33,670 --> 00:32:35,740 that we wrote down with the renormalization group 507 00:32:35,740 --> 00:32:40,810 improvement, then we would have that. 508 00:32:40,810 --> 00:32:43,760 That's with that renormalization group improvement. 509 00:32:43,760 --> 00:32:46,450 So this is at mu equals Mw. 510 00:32:46,450 --> 00:32:47,890 And the problem with this formula 511 00:32:47,890 --> 00:32:49,990 is that this matrix element has large logs. 512 00:32:52,900 --> 00:32:54,190 It depends on Mw. 513 00:32:54,190 --> 00:32:55,450 It also depends on Mb. 514 00:33:00,342 --> 00:33:02,050 And if it's something we can't calculate, 515 00:33:02,050 --> 00:33:04,420 then that's kind of bad news. 516 00:33:04,420 --> 00:33:06,170 In particular, having large logs like that 517 00:33:06,170 --> 00:33:08,087 would also make it hard to calculate something 518 00:33:08,087 --> 00:33:09,140 like this on the lattice. 519 00:33:09,140 --> 00:33:12,030 So it's not just a-- 520 00:33:12,030 --> 00:33:13,290 It's really a problem. 521 00:33:13,290 --> 00:33:15,267 If you have multiple scales tied together, 522 00:33:15,267 --> 00:33:16,850 it just makes the calculations harder. 523 00:33:20,445 --> 00:33:22,320 It's also a problem for dimensional analysis. 524 00:33:22,320 --> 00:33:23,700 Because if you have large logs, that 525 00:33:23,700 --> 00:33:24,900 means you've got large numbers. 526 00:33:24,900 --> 00:33:27,090 And something that you thought was of a certain size 527 00:33:27,090 --> 00:33:29,280 might be bigger or smaller. 528 00:33:29,280 --> 00:33:31,830 So what we do is, instead, we work in the renormalization 529 00:33:31,830 --> 00:33:39,630 group improved version where we take this down at the scale Mb. 530 00:33:39,630 --> 00:33:41,580 So this guy includes the renormalization group 531 00:33:41,580 --> 00:33:44,160 evolution. 532 00:33:44,160 --> 00:33:45,690 We use our results over there. 533 00:33:49,500 --> 00:33:51,870 And then we've got the operators at the scale Mb, 534 00:33:51,870 --> 00:33:53,070 and there's no large logs. 535 00:33:58,820 --> 00:34:00,600 OK, so you'd want to calculate something 536 00:34:00,600 --> 00:34:03,090 like this on the lattice or some other way. 537 00:34:03,090 --> 00:34:05,250 There's other ways of doing it. 538 00:34:05,250 --> 00:34:07,930 And we'll talk about other ways of doing it later on. 539 00:34:07,930 --> 00:34:10,630 But so far, we've separated out the scale Mw 540 00:34:10,630 --> 00:34:13,659 into this coefficient that's evaluated 541 00:34:13,659 --> 00:34:15,070 at the scale of mu equals Mb. 542 00:34:28,730 --> 00:34:29,230 OK. 543 00:34:29,230 --> 00:34:31,000 So the one way of thinking about this 544 00:34:31,000 --> 00:34:33,670 is, if you want to do physics at the scale Mb, 545 00:34:33,670 --> 00:34:37,989 the right couplings to use in your theory are these ones. 546 00:34:37,989 --> 00:34:40,030 Forget about what's going on at the high scale. 547 00:34:40,030 --> 00:34:41,565 You have to determine the low energy 548 00:34:41,565 --> 00:34:43,690 couplings that are appropriate to the theory you're 549 00:34:43,690 --> 00:34:44,739 dealing with. 550 00:34:44,739 --> 00:34:47,800 And those are the C's at Mb, OK? 551 00:35:11,040 --> 00:35:13,880 All right. 552 00:35:13,880 --> 00:35:14,380 OK. 553 00:35:14,380 --> 00:35:16,990 So now, I want to come back to this question of thinking 554 00:35:16,990 --> 00:35:19,390 about the next leading log. 555 00:35:19,390 --> 00:35:21,850 And I'm going to do that by going back 556 00:35:21,850 --> 00:35:26,230 to our comparison of full theory to effective theory. 557 00:35:26,230 --> 00:35:29,620 We'll do a comparison of results in the full theory 558 00:35:29,620 --> 00:35:31,120 and results in the effective theory. 559 00:35:31,120 --> 00:35:35,230 And I'll show you how, by making that comparison, 560 00:35:35,230 --> 00:35:40,090 we can determine the ingredients that we need for this one loop 561 00:35:40,090 --> 00:35:41,652 matching here. 562 00:35:41,652 --> 00:35:43,210 So we'll focus on this. 563 00:35:47,990 --> 00:35:50,080 So we already renormalized the effective theory. 564 00:35:50,080 --> 00:35:52,450 So we can compare the renormalized effective theory 565 00:35:52,450 --> 00:35:53,170 and full theory. 566 00:36:00,580 --> 00:36:02,080 And that's the right way to proceed. 567 00:36:06,200 --> 00:36:09,070 So in our parlance of theory one and theory two, 568 00:36:09,070 --> 00:36:12,490 the effective theory would be theory two. 569 00:36:12,490 --> 00:36:15,310 We have to think about the full theory, which in our parlance 570 00:36:15,310 --> 00:36:19,800 would be theory one, the full theory being 571 00:36:19,800 --> 00:36:20,800 the standard model here. 572 00:36:24,370 --> 00:36:28,390 We have to think about renormalizing that theory. 573 00:36:28,390 --> 00:36:30,580 But in the standard model, our calculation 574 00:36:30,580 --> 00:36:31,795 involves conserved currents. 575 00:36:36,070 --> 00:36:38,350 These are just a weak currents. 576 00:36:38,350 --> 00:36:41,522 And so there's actually no extra UV divergences 577 00:36:41,522 --> 00:36:42,730 associated to those currents. 578 00:36:42,730 --> 00:36:44,819 We just have coupling renormalization. 579 00:36:50,920 --> 00:36:53,860 And one way of saying this is that what happens 580 00:36:53,860 --> 00:36:55,600 is that the vertex in the wave function 581 00:36:55,600 --> 00:37:03,728 graphs, the UV divergences cancel 582 00:37:03,728 --> 00:37:04,770 or the conserved current. 583 00:37:08,880 --> 00:37:13,170 So the result for the full theory 584 00:37:13,170 --> 00:37:18,170 will be some result that is independent of having-- 585 00:37:18,170 --> 00:37:20,402 it doesn't have any ultraviolet divergences. 586 00:37:25,190 --> 00:37:26,690 And like the effective theory, where 587 00:37:26,690 --> 00:37:28,815 you had to carry out a renormalization of operators 588 00:37:28,815 --> 00:37:30,342 in that theory, for the full theory, 589 00:37:30,342 --> 00:37:32,050 coupling renormalization is all there is. 590 00:37:39,040 --> 00:37:40,750 So let's draw the full theory graphs. 591 00:37:47,610 --> 00:37:51,470 Gluons should be green. 592 00:37:51,470 --> 00:37:52,880 Maybe my w should be pink. 593 00:38:29,010 --> 00:38:36,030 Six permutations-- and then there's also 594 00:38:36,030 --> 00:38:37,682 wave function normalization. 595 00:38:40,930 --> 00:38:41,430 OK. 596 00:38:41,430 --> 00:38:43,472 So if you want to do the full theory calculation, 597 00:38:43,472 --> 00:38:45,000 these are the graphs you compute. 598 00:38:45,000 --> 00:38:47,070 It's triangles as well as box integrals. 599 00:38:47,070 --> 00:38:50,074 It's actually a much harder calculation 600 00:38:50,074 --> 00:38:51,580 than in the effective theory. 601 00:38:54,750 --> 00:38:56,910 And I'm not going to do the calculation, 602 00:38:56,910 --> 00:38:58,785 but I'll tell you what the results look like. 603 00:39:06,760 --> 00:39:09,090 So let's start by thinking about the logs. 604 00:39:13,017 --> 00:39:14,850 And then we'll talk about the constants that 605 00:39:14,850 --> 00:39:15,975 are under the logs as well. 606 00:39:21,610 --> 00:39:26,100 So if we look at this calculation, 607 00:39:26,100 --> 00:39:27,390 it has the following form. 608 00:39:52,730 --> 00:39:56,810 So there's S1, which was some spinners. 609 00:39:56,810 --> 00:39:59,443 We defined it in an earlier lecture. 610 00:39:59,443 --> 00:40:00,860 There's something involving a log, 611 00:40:00,860 --> 00:40:03,260 and it has a p squared. p squared was the off-shellness 612 00:40:03,260 --> 00:40:04,790 associated to these guys. 613 00:40:04,790 --> 00:40:09,000 And we regulated the infrared divergences with p squared. 614 00:40:09,000 --> 00:40:15,710 So p squared not equal to 0 regulates IR divergences. 615 00:40:15,710 --> 00:40:19,160 And there are IR divergences in these diagrams. 616 00:40:19,160 --> 00:40:20,750 Even though I said they are UV finite, 617 00:40:20,750 --> 00:40:23,000 they're not finite in the infrared. 618 00:40:23,000 --> 00:40:25,282 And that's what leads to these logs of p squared. 619 00:40:32,370 --> 00:40:33,750 OK. 620 00:40:33,750 --> 00:40:35,250 Now, I didn't write everything. 621 00:40:35,250 --> 00:40:38,250 I only wrote the pieces proportional to S1. 622 00:40:38,250 --> 00:40:42,390 There's pieces proportional to the other spinner, S2. 623 00:40:42,390 --> 00:40:44,802 And there's mod log terms. 624 00:40:44,802 --> 00:40:46,260 And they're all hiding in the dots. 625 00:40:50,430 --> 00:40:53,070 So let's compare this result to a similar expression 626 00:40:53,070 --> 00:40:58,020 in the effective theory that we just 627 00:40:58,020 --> 00:41:00,690 set the coefficients to the values at the high scale. 628 00:41:05,530 --> 00:41:08,520 So then we have 1 for the coefficient 629 00:41:08,520 --> 00:41:12,150 times the one-loop matrix element of O1, 630 00:41:12,150 --> 00:41:14,730 which we wrote down earlier. 631 00:41:14,730 --> 00:41:18,030 And it looks kind of similar to this, but not exactly the same. 632 00:41:25,700 --> 00:41:28,400 It's very similar, but not precisely the same. 633 00:41:38,476 --> 00:41:44,480 A similar statement applies to these guys over here, OK? 634 00:41:44,480 --> 00:41:46,940 So the difference is really that, instead 635 00:41:46,940 --> 00:41:49,027 of Mw squared in this log, we have a mu squared. 636 00:41:49,027 --> 00:41:51,110 That's really the only difference for these terms. 637 00:41:56,840 --> 00:41:59,000 With constant terms, the non-logarithmic terms 638 00:41:59,000 --> 00:42:00,470 here and here won't agree. 639 00:42:00,470 --> 00:42:03,360 And we'll talk about those things in a minute. 640 00:42:03,360 --> 00:42:06,200 So what do we learn by thinking about the physics of these two 641 00:42:06,200 --> 00:42:08,130 equations? 642 00:42:08,130 --> 00:42:09,980 Well, one comment is the comment I already 643 00:42:09,980 --> 00:42:14,240 said, that the effective theory computation for this line 644 00:42:14,240 --> 00:42:17,730 here is much, much easier than this one. 645 00:42:17,730 --> 00:42:19,740 So one reason to use effective field theory 646 00:42:19,740 --> 00:42:21,590 is just that it makes computations easier. 647 00:42:25,550 --> 00:42:27,422 And the reason it makes computations easier 648 00:42:27,422 --> 00:42:28,880 is because you're basically dealing 649 00:42:28,880 --> 00:42:30,750 with one scale at a time. 650 00:42:30,750 --> 00:42:33,530 And whenever you have integrals involving only one scale, 651 00:42:33,530 --> 00:42:36,740 that's always much easier than having multiple scales. 652 00:42:41,908 --> 00:42:44,450 But if you want to encode all the physics of the full theory, 653 00:42:44,450 --> 00:42:46,700 you'll still have to do that calculation at some point 654 00:42:46,700 --> 00:42:47,480 as well. 655 00:42:47,480 --> 00:42:50,030 Although you may be able to do it in a simpler configuration 656 00:42:50,030 --> 00:42:51,738 to get off the information that you need. 657 00:42:56,750 --> 00:42:59,083 Furthermore, if you really only cared about the logs, 658 00:42:59,083 --> 00:43:01,250 then all you really need is the 1 over epsilon term. 659 00:43:01,250 --> 00:43:06,320 And that's even easier than the full triangle diagrams. 660 00:43:06,320 --> 00:43:10,250 So to compute the anomalous dimensions, 661 00:43:10,250 --> 00:43:13,200 you just have to keep the divergences. 662 00:43:13,200 --> 00:43:17,210 And you can throw away all the finite pieces. 663 00:43:17,210 --> 00:43:18,568 And that's even easier. 664 00:43:18,568 --> 00:43:21,110 So you can organize things by thinking about doing the easier 665 00:43:21,110 --> 00:43:22,443 calculations first. 666 00:43:22,443 --> 00:43:23,360 That's what people do. 667 00:43:23,360 --> 00:43:24,818 They calculate anomalous dimensions 668 00:43:24,818 --> 00:43:29,110 before they calculate matching because it's just easier. 669 00:43:29,110 --> 00:43:31,610 And it's also the thing you need for the leading log result. 670 00:43:31,610 --> 00:43:34,068 You don't need the matching at one loop for the leading log 671 00:43:34,068 --> 00:43:36,440 result. So there's a conservation of ease 672 00:43:36,440 --> 00:43:37,370 and what you need. 673 00:43:41,795 --> 00:43:43,670 These two things play nicely with each other. 674 00:43:50,270 --> 00:43:53,860 Second point-- in the effective theory, 675 00:43:53,860 --> 00:43:57,200 you're supposed to think that Mw goes to infinity. 676 00:43:57,200 --> 00:43:59,540 And that's why this thing doesn't know about Mw. 677 00:43:59,540 --> 00:44:01,810 So how could it possibly get an Mw? 678 00:44:01,810 --> 00:44:04,630 And so what happens is that Mw gets replaced by the cut 679 00:44:04,630 --> 00:44:05,680 off, which is mu here. 680 00:44:26,840 --> 00:44:29,080 Another point that we can make about this-- if you 681 00:44:29,080 --> 00:44:31,090 look at the logs of minus p squared, 682 00:44:31,090 --> 00:44:33,550 they're all the same between the two equations. 683 00:44:37,870 --> 00:44:41,118 That's actually important because what 684 00:44:41,118 --> 00:44:43,660 that means is that the infrared structure of the two theories 685 00:44:43,660 --> 00:44:44,160 agree. 686 00:44:46,990 --> 00:44:49,120 The logs of p squared are the infrared divergences. 687 00:44:49,120 --> 00:44:51,642 They agree between the higher theory and the lower theory. 688 00:44:51,642 --> 00:44:52,600 And they have to agree. 689 00:44:56,503 --> 00:44:58,420 What this tells you about the effective theory 690 00:44:58,420 --> 00:45:01,523 is that you're doing something right. 691 00:45:01,523 --> 00:45:03,940 In particular, it tells you that your effective theory has 692 00:45:03,940 --> 00:45:05,148 the right degrees of freedom. 693 00:45:08,050 --> 00:45:10,642 That's almost trivial in this example. 694 00:45:10,642 --> 00:45:12,850 What other degrees of freedom could we possibly think 695 00:45:12,850 --> 00:45:13,990 that would be missing? 696 00:45:18,460 --> 00:45:20,660 But in more complicated examples-- 697 00:45:20,660 --> 00:45:24,400 and we will deal with at least one such example later 698 00:45:24,400 --> 00:45:25,820 in the course-- 699 00:45:25,820 --> 00:45:28,510 it's not so trivial to see that these things match up. 700 00:45:28,510 --> 00:45:30,880 And people have discovered new degrees of freedom 701 00:45:30,880 --> 00:45:32,860 by doing matching computations like this. 702 00:45:32,860 --> 00:45:35,027 They said, oh, this is a relevant degree of freedom. 703 00:45:35,027 --> 00:45:35,980 And it's needed. 704 00:45:35,980 --> 00:45:39,370 Because if I do a one-loop calculation, 705 00:45:39,370 --> 00:45:41,997 I need it to get the infrared divergences right. 706 00:45:41,997 --> 00:45:44,330 So this can really teach you about the effective theory, 707 00:45:44,330 --> 00:45:46,780 doing a matching computation, teach you 708 00:45:46,780 --> 00:45:50,550 about the physics of the effective theory. 709 00:45:50,550 --> 00:45:53,455 So if you made a mistake, this would be a place 710 00:45:53,455 --> 00:45:54,580 where you'd catch yourself. 711 00:46:06,820 --> 00:46:09,670 Now, that you've analyzed fully what the differences are, 712 00:46:09,670 --> 00:46:10,870 you can subtract them. 713 00:46:15,270 --> 00:46:19,290 You take the difference of the renormalized calculations. 714 00:46:19,290 --> 00:46:23,850 And that is what gives you one-loop matching. 715 00:46:23,850 --> 00:46:25,770 Just like we compared tree level calculations 716 00:46:25,770 --> 00:46:28,500 to get tree level matching, we compare one-loop calculations 717 00:46:28,500 --> 00:46:31,770 to get one-loop matching. 718 00:46:37,620 --> 00:46:48,630 So [INAUDIBLE] we do that. 719 00:46:48,630 --> 00:46:53,580 At tree level, if we're just looking at the S1 pieces, 720 00:46:53,580 --> 00:46:56,760 we have these two terms. 721 00:46:56,760 --> 00:46:58,250 And then at one-loop-- 722 00:47:00,920 --> 00:47:02,120 let me use this notation. 723 00:47:02,120 --> 00:47:03,980 This is the full A. So it has a tree level 724 00:47:03,980 --> 00:47:05,150 piece and one-loop piece. 725 00:47:08,210 --> 00:47:10,400 Then we take the piece of the C1 that's at one loop. 726 00:47:14,000 --> 00:47:19,520 And we take the matrix element of the operator, 727 00:47:19,520 --> 00:47:21,490 evaluate it order alpha. 728 00:47:28,430 --> 00:47:29,660 And then there's C2 terms. 729 00:47:29,660 --> 00:47:30,590 Let me just put dots. 730 00:47:37,440 --> 00:47:40,692 OK, so there's two ways that I can get an alpha. 731 00:47:40,692 --> 00:47:42,150 There's an order alpha coefficient. 732 00:47:42,150 --> 00:47:45,210 That's what I want to know, what you want to determine. 733 00:47:45,210 --> 00:47:48,120 And then there's an order alpha matrix element. 734 00:47:48,120 --> 00:47:52,110 So by subtracting, putting this guy on the left-hand side, 735 00:47:52,110 --> 00:47:55,800 I get the value of C1 of 1. 736 00:47:55,800 --> 00:47:59,550 So we use this equation to determine this. 737 00:47:59,550 --> 00:48:02,250 And the story would be similar if I kept all the C2 terms. 738 00:48:05,070 --> 00:48:07,260 So the matching, i.e. the difference 739 00:48:07,260 --> 00:48:10,620 of the full and effective theory and calculations, 740 00:48:10,620 --> 00:48:12,120 determines that coefficient for you. 741 00:48:19,930 --> 00:48:21,460 So the notation here is that we'd 742 00:48:21,460 --> 00:48:27,530 write the full coefficient as 1 plus C1 plus higher order terms 743 00:48:27,530 --> 00:48:30,560 where this is order alpha. 744 00:48:30,560 --> 00:48:31,840 That's the notation I'm using. 745 00:48:36,130 --> 00:48:37,470 So let's do that for the logs. 746 00:48:37,470 --> 00:48:38,460 It's pretty simple. 747 00:48:38,460 --> 00:48:40,030 These things here are just cancel. 748 00:48:40,030 --> 00:48:43,200 And this one we can just subtract them. 749 00:48:43,200 --> 00:48:45,040 The p squareds will cancel. 750 00:48:45,040 --> 00:48:47,220 And we'll get a log of my squared over Mw squared. 751 00:49:22,180 --> 00:49:24,910 So just focusing on those terms that we have in S1-- 752 00:49:27,595 --> 00:49:29,470 and rearranging the equation in the way 753 00:49:29,470 --> 00:49:33,730 I said and then plugging in the values for the things that 754 00:49:33,730 --> 00:49:35,630 don't cancel. 755 00:49:50,140 --> 00:49:54,460 So the terms that were slightly different-- 756 00:49:54,460 --> 00:49:55,690 and dropping the S1. 757 00:50:09,090 --> 00:50:11,316 CF is 4/3. 758 00:50:11,316 --> 00:50:13,489 It's Casimir of the fundamental. 759 00:50:18,540 --> 00:50:21,420 We see that, since we've only kept the log terms, what 760 00:50:21,420 --> 00:50:24,420 we find is the one-loop correction in this guy that's 761 00:50:24,420 --> 00:50:26,130 got a logarithm. 762 00:50:26,130 --> 00:50:30,510 but it'd also be a term here that's a number times alpha. 763 00:50:33,240 --> 00:50:38,670 But we haven't kept those terms in what 764 00:50:38,670 --> 00:50:40,478 I've written on the board. 765 00:50:40,478 --> 00:50:42,270 So the way that you should think about this 766 00:50:42,270 --> 00:50:44,270 is that we've got these Wilson coefficients that 767 00:50:44,270 --> 00:50:46,560 depend on the scale Mw. 768 00:50:46,560 --> 00:50:51,240 And what the matching is doing, it's taking the full theory. 769 00:50:51,240 --> 00:50:54,813 And it's dividing it into large momentum pieces 770 00:50:54,813 --> 00:50:55,980 times small momentum pieces. 771 00:51:04,160 --> 00:51:07,660 So the large momentum pieces are in C. Small 772 00:51:07,660 --> 00:51:12,440 momentum pieces are in the matrix element of the operator. 773 00:51:12,440 --> 00:51:16,330 And this statement, we see an explicit realization of it. 774 00:51:16,330 --> 00:51:21,830 The full theory knows about high scale Mw's and the p squared. 775 00:51:21,830 --> 00:51:32,687 And we can write this as a split like this. 776 00:51:32,687 --> 00:51:35,270 The effective theory knows about p squared, doesn't know about 777 00:51:35,270 --> 00:51:35,870 Mw squared. 778 00:51:35,870 --> 00:51:37,820 The Wilson coefficient knows about Mw squared, 779 00:51:37,820 --> 00:51:39,350 doesn't know about p squared. 780 00:51:39,350 --> 00:51:41,267 The additional thing that they both know about 781 00:51:41,267 --> 00:51:42,230 is the scale mu. 782 00:51:42,230 --> 00:51:44,000 And that's providing a cutoff for where 783 00:51:44,000 --> 00:52:04,800 you split between large momentum and small momentum 784 00:52:04,800 --> 00:52:07,980 p squared was the small scale. 785 00:52:07,980 --> 00:52:10,353 Now, if you look at this equation, 786 00:52:10,353 --> 00:52:11,520 you may wonder for a minute. 787 00:52:11,520 --> 00:52:13,380 Why is it additive? 788 00:52:13,380 --> 00:52:16,170 Up here, I just said times. 789 00:52:16,170 --> 00:52:17,670 And then immediately below times, 790 00:52:17,670 --> 00:52:22,410 I wrote something that was a sum, seemed a little weird. 791 00:52:22,410 --> 00:52:26,190 That's just because, if you take something that includes the 1, 792 00:52:26,190 --> 00:52:31,890 then the product becomes a sum. 793 00:52:31,890 --> 00:52:34,890 So if I write it this way, I can write it in product form 794 00:52:34,890 --> 00:52:37,790 if I include the one tree level. 795 00:52:41,880 --> 00:52:44,360 So it really is a product. 796 00:52:44,360 --> 00:52:46,880 It's just that, if you look at the order alpha pieces, 797 00:52:46,880 --> 00:52:49,920 it breaks into the sum where we can nicely see how 798 00:52:49,920 --> 00:52:51,750 things are combining together. 799 00:52:51,750 --> 00:52:53,795 But really it has this product structure, 800 00:52:53,795 --> 00:52:55,920 and there's non-trivial relations between these two 801 00:52:55,920 --> 00:52:58,440 series that make it all work out even when 802 00:52:58,440 --> 00:53:00,520 you go to higher orders. 803 00:53:00,520 --> 00:53:02,250 So if I expand that to order alpha, 804 00:53:02,250 --> 00:53:03,750 look at the order alpha coefficient. 805 00:53:03,750 --> 00:53:05,635 I get this equation back again. 806 00:53:05,635 --> 00:53:08,010 And this is how you would think about it in product form. 807 00:53:11,070 --> 00:53:12,360 OK. 808 00:53:12,360 --> 00:53:16,560 So the other thing you see here is that order by order 809 00:53:16,560 --> 00:53:20,640 in our expansion, as we kind of already stated, 810 00:53:20,640 --> 00:53:23,070 the mu dependence between these coefficients and these 811 00:53:23,070 --> 00:53:26,010 operators is exactly cancelling because the full theory here 812 00:53:26,010 --> 00:53:27,810 didn't involve that mu. 813 00:53:31,760 --> 00:53:35,390 That's another little piece of information that we get 814 00:53:35,390 --> 00:53:37,110 or that we knew, but we see explicitly 815 00:53:37,110 --> 00:53:38,450 from looking at this. 816 00:53:48,650 --> 00:53:51,320 So I think, if I'm counting right, 817 00:53:51,320 --> 00:53:53,000 this is comment number five. 818 00:53:57,335 --> 00:53:58,210 Was there a question? 819 00:54:04,550 --> 00:54:06,940 So not surprisingly, the cut off dependence 820 00:54:06,940 --> 00:54:10,960 cancels in the product of C of mu O of mu 821 00:54:10,960 --> 00:54:13,627 because the cut off is what we introduced 822 00:54:13,627 --> 00:54:15,460 to split up the physics in these two things. 823 00:54:18,990 --> 00:54:23,810 Now, if you look at that in a little more detail, 824 00:54:23,810 --> 00:54:26,390 it's only mu independent of the order in perturbation theory 825 00:54:26,390 --> 00:54:27,223 that you're working. 826 00:54:31,343 --> 00:54:34,690 If you've worked at a fixed order in some expansion, 827 00:54:34,690 --> 00:54:38,290 then you shouldn't be surprised that everything you've derived 828 00:54:38,290 --> 00:54:39,580 is only true at that order. 829 00:54:43,820 --> 00:54:45,550 So if you stopped at one loop, then 830 00:54:45,550 --> 00:54:50,440 it's mu independent at order alpha S. 831 00:54:50,440 --> 00:54:52,390 What that technically means is that terms that 832 00:54:52,390 --> 00:54:55,480 are alpha S mu log mu cancel. 833 00:54:58,060 --> 00:55:00,610 The log mu here cancels, but there's mu dependence also 834 00:55:00,610 --> 00:55:01,540 here. 835 00:55:01,540 --> 00:55:03,760 And that mu dependence in the alpha 836 00:55:03,760 --> 00:55:06,100 is something that would be related to terms 837 00:55:06,100 --> 00:55:10,970 that are alpha squared log mu. 838 00:55:10,970 --> 00:55:12,470 And that cancels at higher order. 839 00:55:16,910 --> 00:55:18,410 So some of the mu dependents cancel. 840 00:55:18,410 --> 00:55:20,300 Some of the mu dependents doesn't cancel. 841 00:55:20,300 --> 00:55:22,640 And people actually use the fact that some of the mu dependence 842 00:55:22,640 --> 00:55:24,680 doesn't cancel as getting a handle on the higher order 843 00:55:24,680 --> 00:55:25,180 terms. 844 00:55:25,180 --> 00:55:27,770 It's doing a kind of theory uncertainty. 845 00:55:52,800 --> 00:55:54,810 If we just think about the logarithms, 846 00:55:54,810 --> 00:56:00,090 then actually the one-loop results in the full theory 847 00:56:00,090 --> 00:56:02,299 has actually less information. 848 00:56:09,235 --> 00:56:11,610 And the reason is that, if you wanted to get higher order 849 00:56:11,610 --> 00:56:14,027 terms in this leading log series that we talked about, 850 00:56:14,027 --> 00:56:16,110 if you wanted to derive those from the full theory 851 00:56:16,110 --> 00:56:19,410 point of view, you'd have to do a two-loop computation. 852 00:56:24,220 --> 00:56:29,590 So if you wanted to get alpha squared log squared of Mw 853 00:56:29,590 --> 00:56:34,060 squared over minus p squared, then you'd 854 00:56:34,060 --> 00:56:41,125 have to look at diagrams, two gluons. 855 00:56:43,995 --> 00:56:45,370 On the full theory point of view, 856 00:56:45,370 --> 00:56:50,513 that's what you'd have to do to find those terms. 857 00:56:50,513 --> 00:56:52,180 From the effective theory point of view, 858 00:56:52,180 --> 00:56:53,763 all you have to do to find those terms 859 00:56:53,763 --> 00:56:56,380 is renormalize the effective theory properly. 860 00:56:56,380 --> 00:56:58,420 And then you get those terms. 861 00:57:09,630 --> 00:57:12,750 So we just needed the one-loop anomalous dimension. 862 00:57:12,750 --> 00:57:14,660 So in that sense, the effective theory, 863 00:57:14,660 --> 00:57:16,640 because of the renormalization properties 864 00:57:16,640 --> 00:57:19,280 of the effective theory, know something 865 00:57:19,280 --> 00:57:22,550 that the full theory doesn't know so easily. 866 00:57:22,550 --> 00:57:24,290 And that kind of shows you the advantage 867 00:57:24,290 --> 00:57:27,710 of taking something that's a constant, Mw squared, 868 00:57:27,710 --> 00:57:29,132 and turning it into a scale. 869 00:57:29,132 --> 00:57:30,590 Because by turning it into a scale, 870 00:57:30,590 --> 00:57:32,798 you have the whole power of the renormalization group 871 00:57:32,798 --> 00:57:34,820 at your disposal to predict higher order 872 00:57:34,820 --> 00:57:38,030 things, like the higher order coefficients. 873 00:57:38,030 --> 00:57:39,800 And that's one way of phrasing what 874 00:57:39,800 --> 00:57:42,163 the example is of splitting scales 875 00:57:42,163 --> 00:57:43,580 and going to the effective theory. 876 00:57:51,290 --> 00:57:53,750 So the final thing that I want to talk about here 877 00:57:53,750 --> 00:57:56,360 has to do with the fact that-- 878 00:57:56,360 --> 00:57:59,600 well, actually, there's two more things I want to talk about, 879 00:57:59,600 --> 00:58:01,340 but let me make the final comment here. 880 00:58:04,020 --> 00:58:06,920 So the final comment I want to make in my list, 881 00:58:06,920 --> 00:58:10,745 which is number seven, has to do with scheme dependence. 882 00:58:14,000 --> 00:58:15,530 So scheme dependence means that we 883 00:58:15,530 --> 00:58:18,537 pick the renormalization scheme MS bar. 884 00:58:18,537 --> 00:58:20,120 And we could have done the calculation 885 00:58:20,120 --> 00:58:23,438 in a different renormalization scheme. 886 00:58:23,438 --> 00:58:25,355 And we should ask what depends on that choice. 887 00:58:28,850 --> 00:58:31,490 You may know, if you've taken a course on the beta function 888 00:58:31,490 --> 00:58:34,660 or if you've taken QFD3, that the beta function of QCD 889 00:58:34,660 --> 00:58:36,800 is scheme independent for the first two orders. 890 00:58:40,580 --> 00:58:42,680 The analog of that statement here 891 00:58:42,680 --> 00:58:45,920 is that the one-loop anomalous dimension for our operators 892 00:58:45,920 --> 00:58:47,328 is scheme independent. 893 00:58:50,674 --> 00:58:55,360 It doesn't depend on which mass independent scheme you pick. 894 00:58:55,360 --> 00:58:58,270 So in the class of mass independence schemes, 895 00:58:58,270 --> 00:59:00,535 the result is what we derived. 896 00:59:06,000 --> 00:59:08,600 We'll come back and study that in a little more detail. 897 00:59:11,430 --> 00:59:14,958 OK, so let's go back now and establish some notation 898 00:59:14,958 --> 00:59:17,000 where we actually just put the constants back in. 899 00:59:20,260 --> 00:59:22,010 And again, I'm not going to write numbers. 900 00:59:22,010 --> 00:59:24,408 I'll just give them names. 901 00:59:24,408 --> 00:59:25,950 And we'll track what happens to them. 902 00:59:28,530 --> 00:59:35,180 So let's think about the full one-loop matching 903 00:59:35,180 --> 00:59:39,780 and how we get the next leading log result. 904 00:59:39,780 --> 00:59:42,200 And really what I want to focus on, or at least one thing 905 00:59:42,200 --> 00:59:46,580 I want to focus on, is the scheme dependence. 906 00:59:46,580 --> 00:59:50,630 Because the coefficients, once you get to next leading log, 907 00:59:50,630 --> 00:59:52,940 are totally scheme dependent. 908 00:59:52,940 --> 00:59:54,920 So you can ask, what physical sense 909 00:59:54,920 --> 00:59:57,943 do they make if they're scheme dependent? 910 00:59:57,943 --> 01:00:00,110 Well, it turns out that the matrix elements are also 911 01:00:00,110 --> 01:00:02,000 scheme dependent. 912 01:00:02,000 --> 01:00:04,760 And the anomalous dimensions are scheme dependent. 913 01:00:04,760 --> 01:00:08,180 So basically, everything is scheme dependent. 914 01:00:08,180 --> 01:00:10,940 And when we put it all together, we get a scheme 915 01:00:10,940 --> 01:00:25,965 independent result. So you might think, well, 916 01:00:25,965 --> 01:00:27,590 if we can get scheme dependent results, 917 01:00:27,590 --> 01:00:30,050 we should just stop because maybe we can't 918 01:00:30,050 --> 01:00:32,030 understand what's going on. 919 01:00:32,030 --> 01:00:41,360 But C of mu times O of mu is independent of the scheme. 920 01:00:44,900 --> 01:00:46,070 It's a physical observable. 921 01:00:46,070 --> 01:00:47,840 And physical observables don't depend 922 01:00:47,840 --> 01:00:49,940 on our definitions of things. 923 01:00:54,620 --> 01:00:56,090 Nature gets to decide, not us. 924 01:01:01,020 --> 01:01:04,400 So one way of thinking about this 925 01:01:04,400 --> 01:01:07,910 is that we already saw some kind of scheme independence 926 01:01:07,910 --> 01:01:11,060 in a statement that C of mu times O of mu 927 01:01:11,060 --> 01:01:12,260 is independent of mu. 928 01:01:12,260 --> 01:01:15,080 But there's even a deeper scheme independence to it 929 01:01:15,080 --> 01:01:18,890 that it's independent of whether we chose MS bar or some others 930 01:01:18,890 --> 01:01:21,440 scheme. 931 01:01:21,440 --> 01:01:23,910 So for the context of this discussion, 932 01:01:23,910 --> 01:01:27,560 I'm going to start dropping all the matrix indices. 933 01:01:27,560 --> 01:01:29,540 And we're not going to write i and j 934 01:01:29,540 --> 01:01:33,470 just because I want to keep things a little bit simple. 935 01:01:33,470 --> 01:01:37,580 So we'll write that the effective theory is simply 936 01:01:37,580 --> 01:01:41,510 one coefficient times the matrix of one operator. 937 01:01:47,940 --> 01:01:54,050 So let's think about, in that context, trying 938 01:01:54,050 --> 01:01:56,570 to understand where all this scheme dependence is floating 939 01:01:56,570 --> 01:02:05,500 around and how the matching works. 940 01:02:10,360 --> 01:02:13,430 So we just do the same thing we did before. 941 01:02:13,430 --> 01:02:16,970 I'm leaving off some pre-factors, 942 01:02:16,970 --> 01:02:18,310 leaving off the pre-factors. 943 01:02:18,310 --> 01:02:20,102 I don't have the write the spinners anymore 944 01:02:20,102 --> 01:02:22,540 since there's only one structure. 945 01:02:22,540 --> 01:02:28,700 And let me introduce some notation for the results. 946 01:02:28,700 --> 01:02:31,090 So we had this Mw squared over p squared type term. 947 01:02:33,620 --> 01:02:36,140 And let me just focus on these terms and not the terms that 948 01:02:36,140 --> 01:02:39,290 just cancelled away. 949 01:02:39,290 --> 01:02:41,900 So let me focus on the terms that are different. 950 01:02:41,900 --> 01:02:44,000 But now, I'm also going to include the constants. 951 01:03:03,860 --> 01:03:05,920 So the constants that we get in the full theory 952 01:03:05,920 --> 01:03:07,545 and the effective theory are different. 953 01:03:07,545 --> 01:03:09,850 So I'll call one of them A and the other one B. 954 01:03:09,850 --> 01:03:12,310 So I call the A the full theory result 955 01:03:12,310 --> 01:03:14,723 and the B the effective theory result. 956 01:03:14,723 --> 01:03:16,390 So you should think of this as a number, 957 01:03:16,390 --> 01:03:21,280 like 3, just some number, same thing here. 958 01:03:21,280 --> 01:03:23,830 But just to avoid talking about numbers 959 01:03:23,830 --> 01:03:26,500 and to track also where the scheme dependence is-- 960 01:03:26,500 --> 01:03:28,240 like this 3 could be 2 in one scheme 961 01:03:28,240 --> 01:03:33,413 and this 4 could be 2 in one scheme and 5 in another scheme. 962 01:03:33,413 --> 01:03:35,830 In order to keep track of that, let me call it a variable. 963 01:03:35,830 --> 01:03:39,520 Let me call it B. 964 01:03:39,520 --> 01:03:41,310 So then the Wilson coefficient is just 965 01:03:41,310 --> 01:03:42,635 we construct the difference. 966 01:03:42,635 --> 01:03:44,260 And then we'll have an A minus B in it. 967 01:04:07,133 --> 01:04:09,300 So if you like what the Wilson coefficient is doing, 968 01:04:09,300 --> 01:04:11,400 it's compensating for the fact that the effective theory 969 01:04:11,400 --> 01:04:12,930 has the wrong value for this constant. 970 01:04:12,930 --> 01:04:13,680 It should be this. 971 01:04:13,680 --> 01:04:15,960 That's what the full theory told you it was. 972 01:04:15,960 --> 01:04:20,730 So the effective theory Wilson coefficient 973 01:04:20,730 --> 01:04:22,740 has minus the effective theory matrix element 974 01:04:22,740 --> 01:04:25,620 result plus the correct result. So this 975 01:04:25,620 --> 01:04:29,842 is the thing that's correcting the effective theory. 976 01:04:29,842 --> 01:04:31,050 So it has the right constant. 977 01:04:43,510 --> 01:04:47,280 And if we just take C at Mw, then it 978 01:04:47,280 --> 01:04:51,107 would simply be equal to that. 979 01:04:51,107 --> 01:04:52,190 And the log would go away. 980 01:05:08,130 --> 01:05:11,180 So in order to do the renormalization group 981 01:05:11,180 --> 01:05:13,520 improved perturbation theory at next leading log, 982 01:05:13,520 --> 01:05:15,290 we also need to do a two-loop computation. 983 01:05:18,530 --> 01:05:22,235 We're not going to do the two-loop consultation, 984 01:05:22,235 --> 01:05:24,110 but I'll tell you the structure of the series 985 01:05:24,110 --> 01:05:26,240 that you get if you did that computation. 986 01:05:31,780 --> 01:05:33,940 So this equation is true. 987 01:05:33,940 --> 01:05:36,420 Therefore, we can write the anomalous dimension equation 988 01:05:36,420 --> 01:05:37,922 again as log C. 989 01:05:37,922 --> 01:05:39,630 And the right-hand side will be a series. 990 01:05:43,113 --> 01:05:45,030 And the structure of the series that was there 991 01:05:45,030 --> 01:05:48,690 is the 0-th order term. 992 01:05:48,690 --> 01:05:50,640 And then there's some higher order terms. 993 01:05:55,500 --> 01:06:00,770 And we need this guy, the two-loop coefficient, 994 01:06:00,770 --> 01:06:01,610 [INAUDIBLE] gamma 1. 995 01:06:04,580 --> 01:06:06,870 Again, this is a coupled differential equation. 996 01:06:06,870 --> 01:06:09,260 And we would solve it by using the kind of thing 997 01:06:09,260 --> 01:06:10,080 that we did before. 998 01:06:10,080 --> 01:06:17,540 So d mu over mu is d alpha over beta of alpha. 999 01:06:17,540 --> 01:06:20,030 And we would write down beta to one higher 1000 01:06:20,030 --> 01:06:24,410 order, which I do in my notes. 1001 01:06:24,410 --> 01:06:29,090 But it's the same idea is I just expand it in alpha. 1002 01:06:29,090 --> 01:06:31,990 And I keep not just the coefficient beta 0, 1003 01:06:31,990 --> 01:06:33,615 but I also keep the coefficient beta 1. 1004 01:06:47,910 --> 01:06:50,370 I want to kind of not focus so much on the calculations, 1005 01:06:50,370 --> 01:06:52,710 but more the results and the implications 1006 01:06:52,710 --> 01:06:56,610 of the calculations. 1007 01:06:56,610 --> 01:07:01,850 So do some renormalization group evolution. 1008 01:07:01,850 --> 01:07:07,658 You can write the all-order solution as an integral, 1009 01:07:07,658 --> 01:07:08,450 like we did before. 1010 01:07:15,460 --> 01:07:19,410 And if I just keep it in terms of these all-order objects, 1011 01:07:19,410 --> 01:07:23,760 then it's just the ratio, which I expand that ratio in alpha. 1012 01:07:23,760 --> 01:07:26,225 And if I want to do it to second order, 1013 01:07:26,225 --> 01:07:27,600 I don't just keep the first time. 1014 01:07:27,600 --> 01:07:28,558 I keep the second term. 1015 01:07:38,220 --> 01:07:41,170 So the first term was a 1 over alpha. 1016 01:07:41,170 --> 01:07:45,930 So we're going to keep the order alpha to the 0 term. 1017 01:07:51,140 --> 01:07:53,840 And if we use our notation that we established before, 1018 01:07:53,840 --> 01:07:57,050 where we call this guy here, we call the exponential 1019 01:07:57,050 --> 01:08:09,020 of this guy u, so C of u C of mu 0, of mu w mu of mu w mu. 1020 01:08:18,140 --> 01:08:20,644 Then we can write the solution of that guy 1021 01:08:20,644 --> 01:08:24,830 as an exponential of an integral of d alpha gamma over beta. 1022 01:08:28,904 --> 01:08:31,279 So some of the steps that we were doing at one-loop, just 1023 01:08:31,279 --> 01:08:33,319 like the exponentiation, the separation, 1024 01:08:33,319 --> 01:08:34,802 they just all go through. 1025 01:08:34,802 --> 01:08:36,260 And the only thing we do have to do 1026 01:08:36,260 --> 01:08:38,177 is evaluate this integral at one higher order. 1027 01:08:43,290 --> 01:08:50,270 Let me take mu w equal to Mw and then do the integral. 1028 01:08:50,270 --> 01:08:53,420 And what you get is a result that we can 1029 01:08:53,420 --> 01:08:54,850 organize in the following way. 1030 01:09:02,248 --> 01:09:04,040 Try to get my arguments in the right order. 1031 01:09:14,640 --> 01:09:18,060 In this particular case, the next leading log solution 1032 01:09:18,060 --> 01:09:19,770 looks as follows. 1033 01:09:19,770 --> 01:09:23,460 Our leading log solution is obviously buried inside it. 1034 01:09:23,460 --> 01:09:35,660 So we have this ratio of alphas, something which is a number. 1035 01:09:35,660 --> 01:09:39,700 And then there's these extra factors 1036 01:09:39,700 --> 01:09:41,020 that depend on this then j. 1037 01:09:44,620 --> 01:09:48,189 And I can write the result this way 1038 01:09:48,189 --> 01:09:51,310 where j involves all the things that are the higher order 1039 01:09:51,310 --> 01:09:52,170 ingredients. 1040 01:09:52,170 --> 01:09:55,480 So it involves the lowest order anomalous dimension, but now 1041 01:09:55,480 --> 01:09:57,245 times beta 1. 1042 01:09:57,245 --> 01:09:59,620 That's like taking the leading order anomalous dimension, 1043 01:09:59,620 --> 01:10:02,500 but now running the coupling with the second order term 1044 01:10:02,500 --> 01:10:03,980 as well. 1045 01:10:03,980 --> 01:10:07,480 And then there's a term that involves the second order 1046 01:10:07,480 --> 01:10:11,170 anomalous dimension. 1047 01:10:11,170 --> 01:10:14,930 So it encodes that information. 1048 01:10:14,930 --> 01:10:17,380 So this is the U. We can combine that together 1049 01:10:17,380 --> 01:10:22,810 with our equation for the C over here, or this one. 1050 01:10:22,810 --> 01:10:24,300 So let me keep that one. 1051 01:10:44,610 --> 01:10:47,910 So I take this equation, multiply by that equation. 1052 01:10:47,910 --> 01:10:49,476 That gives us C of mu. 1053 01:10:52,194 --> 01:10:53,910 So I have to write this lone more time. 1054 01:11:09,830 --> 01:11:12,500 And basically, I can group that together with these other terms 1055 01:11:12,500 --> 01:11:13,870 that depend on an alpha of Mw. 1056 01:11:27,250 --> 01:11:28,240 OK. 1057 01:11:28,240 --> 01:11:31,330 So j is the anomalous dimension piece. 1058 01:11:31,330 --> 01:11:33,010 A and B are the matching. 1059 01:11:33,010 --> 01:11:35,680 A minus B is the matching piece. 1060 01:11:35,680 --> 01:11:38,710 And I can write the result this way. 1061 01:11:38,710 --> 01:11:43,590 So this is next leading order matching, 1062 01:11:43,590 --> 01:11:48,470 which is A minus B and next leading 1063 01:11:48,470 --> 01:12:00,340 log running to get the full next leading log result. 1064 01:12:00,340 --> 01:12:02,530 So this is the kind of structure that you could get. 1065 01:12:02,530 --> 01:12:03,970 That's what renormalization group 1066 01:12:03,970 --> 01:12:05,803 improved perturbation theory looks like when 1067 01:12:05,803 --> 01:12:07,180 you go to higher orders. 1068 01:12:07,180 --> 01:12:08,758 You basically have logs. 1069 01:12:08,758 --> 01:12:10,300 But then the higher order terms, when 1070 01:12:10,300 --> 01:12:12,940 you expand out this integral, are just giving you 1071 01:12:12,940 --> 01:12:14,853 polynomials in alpha. 1072 01:12:14,853 --> 01:12:16,270 So when you integrate polynomials, 1073 01:12:16,270 --> 01:12:17,390 you get back polynomials. 1074 01:12:17,390 --> 01:12:19,070 So if you integrate 1, you get alpha. 1075 01:12:19,070 --> 01:12:23,665 If you integrate alpha squared, you get alpha cubed, et cetera. 1076 01:12:23,665 --> 01:12:25,040 So you just get back polynomials. 1077 01:12:25,040 --> 01:12:28,420 And that's why you can write it this way. 1078 01:12:28,420 --> 01:12:30,520 What are the terms in this result 1079 01:12:30,520 --> 01:12:31,966 that are scheme dependent? 1080 01:12:36,430 --> 01:12:40,630 I claim that beta 1 gamma 1-- 1081 01:12:44,454 --> 01:12:48,000 oh, why did I-- not beta 1, B1. 1082 01:12:52,440 --> 01:12:59,850 B1 gamma 1 J, C, O, these are all scheme dependent. 1083 01:12:59,850 --> 01:13:01,920 They depend on what renormalization scheme 1084 01:13:01,920 --> 01:13:03,690 I use to define my effective theory. 1085 01:13:09,545 --> 01:13:10,920 And then there's a list of things 1086 01:13:10,920 --> 01:13:13,230 that are scheme independent. 1087 01:13:13,230 --> 01:13:16,890 So beta 0 and beta 1 are scheme independent. 1088 01:13:16,890 --> 01:13:20,153 I told you that gamma 0 is scheme independent. 1089 01:13:20,153 --> 01:13:21,570 You could think of that like, when 1090 01:13:21,570 --> 01:13:24,237 you do the one-loop calculation, the ultraviolet divergences are 1091 01:13:24,237 --> 01:13:25,403 always going to be the same. 1092 01:13:25,403 --> 01:13:26,370 You get 1 over epsilon. 1093 01:13:26,370 --> 01:13:28,662 And it's only the constant that depends on your scheme. 1094 01:13:31,980 --> 01:13:34,200 A1 is scheme independent. 1095 01:13:34,200 --> 01:13:36,425 That's because A1 was the full theory calculation. 1096 01:13:36,425 --> 01:13:37,800 So how could it possibly know how 1097 01:13:37,800 --> 01:13:40,930 we define the effective theory? 1098 01:13:40,930 --> 01:13:43,800 So that's scheme independent. 1099 01:13:43,800 --> 01:13:51,417 And a non-trivial one is that B1 plus J is scheme independent. 1100 01:13:51,417 --> 01:13:54,000 So there's scheme dependence in B1 and scheme dependence in J, 1101 01:13:54,000 --> 01:13:56,220 but it cancels in exactly the combination that's 1102 01:13:56,220 --> 01:14:00,220 showing up in this result. 1103 01:14:00,220 --> 01:14:07,300 And as I mentioned, C times O is scheme independent 1104 01:14:07,300 --> 01:14:09,798 because that's related to observables. 1105 01:14:16,150 --> 01:14:18,340 So I have a little proof of that in my notes, 1106 01:14:18,340 --> 01:14:21,490 which, because of time, I'm going to skip. 1107 01:14:21,490 --> 01:14:24,493 But I encourage you, when I post my notes through the website, 1108 01:14:24,493 --> 01:14:26,410 that you take a look at where that comes from. 1109 01:14:30,123 --> 01:14:31,540 So the only non-trivial one really 1110 01:14:31,540 --> 01:14:38,012 is this B1 plus J being scheme independent, OK? 1111 01:14:38,012 --> 01:14:39,970 I have a little proof of that in my notes here. 1112 01:14:44,490 --> 01:14:47,190 OK, so let's go back to the equation 1113 01:14:47,190 --> 01:14:50,520 at the top in the middle there and see 1114 01:14:50,520 --> 01:14:55,810 what conclusions we can draw once we believe this. 1115 01:14:55,810 --> 01:14:58,650 So if B1 plus J is scheme independent, then 1116 01:14:58,650 --> 01:15:03,180 this thing that's showing up in that term, A1 minus B minus J, 1117 01:15:03,180 --> 01:15:07,237 is scheme independent, as A was. 1118 01:15:07,237 --> 01:15:08,570 It was just a full theory thing. 1119 01:15:12,982 --> 01:15:14,940 And there's a cancellation of scheme dependence 1120 01:15:14,940 --> 01:15:19,170 between the one-loop anomalous dimension. 1121 01:15:19,170 --> 01:15:25,135 There's a cancellation here between 1122 01:15:25,135 --> 01:15:26,760 the two-loop anomalous dimension, which 1123 01:15:26,760 --> 01:15:30,570 we called gamma 1, and the B1. 1124 01:15:30,570 --> 01:15:32,610 That's where the scheme dependence cancels. 1125 01:15:32,610 --> 01:15:34,797 So the scheme you pick, you have to be consistent. 1126 01:15:34,797 --> 01:15:35,880 You have to keep using it. 1127 01:15:35,880 --> 01:15:38,005 If you do a matching calculation or if someone else 1128 01:15:38,005 --> 01:15:40,002 did a matching calculation, you want to use it. 1129 01:15:40,002 --> 01:15:42,210 You better figure out what scheme they're working in. 1130 01:15:42,210 --> 01:15:43,800 Because if you start working in a different scheme, 1131 01:15:43,800 --> 01:15:44,967 you're just making mistakes. 1132 01:15:48,220 --> 01:15:50,700 So this is the statement that the matching is scheme 1133 01:15:50,700 --> 01:15:54,580 dependent, the anomalous dimension scheme dependent, 1134 01:15:54,580 --> 01:15:56,490 but there's a cancellation between those two. 1135 01:16:07,540 --> 01:16:09,787 If we look at the gamma 0 over beta 0 term, 1136 01:16:09,787 --> 01:16:10,870 that's scheme independent. 1137 01:16:10,870 --> 01:16:11,590 So that's good. 1138 01:16:11,590 --> 01:16:16,215 If we look over here, J was not scheme dependent. 1139 01:16:16,215 --> 01:16:19,168 J is scheme dependent. 1140 01:16:19,168 --> 01:16:20,710 So we still have to worry about that. 1141 01:16:35,181 --> 01:16:39,170 So leading log result was scheme independent, 1142 01:16:39,170 --> 01:16:46,080 but we still have scheme dependence 1143 01:16:46,080 --> 01:16:57,310 of this factor 1 plus alpha of mu J over 4 pi in our C of mu. 1144 01:16:57,310 --> 01:17:01,920 And the thing that cancels that scheme dependence 1145 01:17:01,920 --> 01:17:04,150 is the fact that the Wilson coefficient alone 1146 01:17:04,150 --> 01:17:05,890 is not a physical observable. 1147 01:17:05,890 --> 01:17:08,985 It's really the Wilson coefficient times the operator. 1148 01:17:08,985 --> 01:17:10,360 And so there is scheme dependence 1149 01:17:10,360 --> 01:17:12,834 in the matrix element of the operator. 1150 01:17:28,650 --> 01:17:31,430 So a matrix element of the operator at the scale mu 1151 01:17:31,430 --> 01:17:34,610 is scheme dependent. 1152 01:17:34,610 --> 01:17:37,720 And this is at the lower end of our integration. 1153 01:17:42,220 --> 01:17:43,775 So this is the final matrix element, 1154 01:17:43,775 --> 01:17:45,400 like the matrix element at the B scale. 1155 01:17:45,400 --> 01:17:46,730 That's a scheme dependent thing. 1156 01:17:46,730 --> 01:17:48,480 So if you think of these things as numbers 1157 01:17:48,480 --> 01:17:51,040 that you want to determine from data, one way of thinking 1158 01:17:51,040 --> 01:17:52,630 about it, those numbers are going 1159 01:17:52,630 --> 01:17:54,490 to depend on what scheme you're using. 1160 01:17:54,490 --> 01:17:56,650 If you extract some numbers in one scheme 1161 01:17:56,650 --> 01:17:58,400 and your friend does it in another scheme, 1162 01:17:58,400 --> 01:18:00,560 you could get totally different numbers. 1163 01:18:00,560 --> 01:18:02,080 So you have to know what scheme you're working in. 1164 01:18:02,080 --> 01:18:03,538 And you have to combine it together 1165 01:18:03,538 --> 01:18:06,247 with the Wilson coefficient in the same scheme. 1166 01:18:06,247 --> 01:18:08,080 If you take some numbers from the literature 1167 01:18:08,080 --> 01:18:09,788 and you don't know what scheme they're in 1168 01:18:09,788 --> 01:18:11,450 and you're working at next leading log, 1169 01:18:11,450 --> 01:18:13,280 you have a problem. 1170 01:18:13,280 --> 01:18:14,950 You got to know what the scheme is 1171 01:18:14,950 --> 01:18:17,995 because you have to work in the same scheme consistently. 1172 01:18:17,995 --> 01:18:19,120 And that's the lesson here. 1173 01:18:25,960 --> 01:18:28,270 If you really want to do this whole program that I've 1174 01:18:28,270 --> 01:18:30,970 talked about, which is done in this 250 page review article-- 1175 01:18:30,970 --> 01:18:33,790 and I'm not asking you to read that. 1176 01:18:33,790 --> 01:18:36,160 If you really want to do this whole program, 1177 01:18:36,160 --> 01:18:37,480 there are some subtleties. 1178 01:18:37,480 --> 01:18:43,270 And I should at least mention them to you since maybe you'll 1179 01:18:43,270 --> 01:18:44,395 encounter the word someday. 1180 01:18:57,270 --> 01:19:01,050 So we've sketched the physics and the basic stuff that 1181 01:19:01,050 --> 01:19:02,590 would be involved in the analysis, 1182 01:19:02,590 --> 01:19:04,715 but we haven't written down the full operator basis 1183 01:19:04,715 --> 01:19:06,750 with the full set of mixing and dozens 1184 01:19:06,750 --> 01:19:08,820 and dozens of diagrams, which people have done. 1185 01:19:12,450 --> 01:19:15,730 Mostly what you should be thinking of this is as a user. 1186 01:19:15,730 --> 01:19:17,520 So I'm teaching you the things that you 1187 01:19:17,520 --> 01:19:19,505 need to be able to use results like that. 1188 01:19:19,505 --> 01:19:20,880 In an effective theory, if you're 1189 01:19:20,880 --> 01:19:22,422 using a higher order result, you have 1190 01:19:22,422 --> 01:19:25,060 to worry about scheme dependent. 1191 01:19:25,060 --> 01:19:26,370 So what are the subtleties? 1192 01:19:26,370 --> 01:19:28,530 Well, one of them is that there's gamma 5s. 1193 01:19:31,265 --> 01:19:34,200 This theory is chiral. 1194 01:19:34,200 --> 01:19:37,076 And gamma 5 is inherently four-dimensional. 1195 01:19:44,700 --> 01:19:46,680 And you have to worry about that. 1196 01:19:46,680 --> 01:19:48,930 And you have to create that carefully in dim reg. 1197 01:19:52,120 --> 01:19:54,550 And when people originally did these calculations, 1198 01:19:54,550 --> 01:19:57,880 that caused some confusion. 1199 01:19:57,880 --> 01:20:00,820 Be careful enough. 1200 01:20:00,820 --> 01:20:03,850 Obviously, dim reg is a powerful way of doing the calculation, 1201 01:20:03,850 --> 01:20:06,513 but you do have to be careful about gamma 5. 1202 01:20:06,513 --> 01:20:07,930 And there's another thing you have 1203 01:20:07,930 --> 01:20:10,553 to be careful about in dim reg. 1204 01:20:10,553 --> 01:20:12,970 And that's something that are called evanescent operators. 1205 01:20:18,513 --> 01:20:20,430 You see, part of our arguments, and originally 1206 01:20:20,430 --> 01:20:22,430 when we were writing down the basis of operators 1207 01:20:22,430 --> 01:20:24,210 for our calculation, were actually 1208 01:20:24,210 --> 01:20:25,881 inherently four-dimensional. 1209 01:20:29,460 --> 01:20:32,280 When we wrote down the operators, 1210 01:20:32,280 --> 01:20:40,920 we said we effectively used that these Dirac structures, which 1211 01:20:40,920 --> 01:20:47,430 are 16 of them, we used completeness over those 16. 1212 01:20:47,430 --> 01:20:51,090 And the problem is that, in d dimensions, 1213 01:20:51,090 --> 01:20:53,220 that's not a complete set. 1214 01:21:05,860 --> 01:21:09,130 And any opinions that are outside that 1215 01:21:09,130 --> 01:21:10,900 set that are additional operators that you 1216 01:21:10,900 --> 01:21:16,434 need in d dimensions are called evanescent operators. 1217 01:21:21,080 --> 01:21:23,120 So they involve Dirac structures that 1218 01:21:23,120 --> 01:21:33,330 vanish as epsilon goes to 0, but are technically needed 1219 01:21:33,330 --> 01:21:36,150 to get some calculations right. 1220 01:21:36,150 --> 01:21:37,680 OK, so those are two subtle things 1221 01:21:37,680 --> 01:21:41,610 to be aware of in the full calculation. 1222 01:21:41,610 --> 01:21:43,448 And I think we'll stop there for today. 1223 01:21:43,448 --> 01:21:45,240 And we'll do something different next time. 1224 01:21:47,860 --> 01:21:51,460 So homework is due next Tuesday. 1225 01:21:51,460 --> 01:21:54,160 And as I said in my original handout, 1226 01:21:54,160 --> 01:21:56,230 you should talk to each other about the homework. 1227 01:21:56,230 --> 01:21:58,680 That's how you learn.