1 00:00:00,000 --> 00:00:02,520 The following content is provided under a Creative 2 00:00:02,520 --> 00:00:03,970 Commons license. 3 00:00:03,970 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,660 continue to offer high quality educational resources for free. 5 00:00:10,660 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,190 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,190 --> 00:00:18,370 at ocw.mit.edu. 8 00:00:22,193 --> 00:00:23,610 IAIN STEWART: So last time we were 9 00:00:23,610 --> 00:00:28,677 talking about the subleading one of our MQ Lagrangian HQET. 10 00:00:28,677 --> 00:00:30,260 Hopefully more people remember that we 11 00:00:30,260 --> 00:00:33,620 have a makeup lecture today, or will not 12 00:00:33,620 --> 00:00:36,050 be stopped by the snow. 13 00:00:36,050 --> 00:00:38,660 So today we're going to continue this discussion of power 14 00:00:38,660 --> 00:00:40,250 suppressed Lagrangians. 15 00:00:40,250 --> 00:00:42,920 And I'll explain to you why we think 16 00:00:42,920 --> 00:00:46,195 of these as giving power corrections to observables. 17 00:00:46,195 --> 00:00:48,320 And we'll talk about a couple different observables 18 00:00:48,320 --> 00:00:52,650 where these specific operators are playing an important role. 19 00:00:52,650 --> 00:00:55,280 And then we're going to turn to the topic of renormalons, 20 00:00:55,280 --> 00:00:57,780 which is fun stuff. 21 00:00:57,780 --> 00:01:00,805 How many people here know what a renormalon is? 22 00:01:00,805 --> 00:01:01,680 AUDIENCE: [INAUDIBLE] 23 00:01:01,680 --> 00:01:03,090 IAIN STEWART: Yeah. 24 00:01:03,090 --> 00:01:07,420 By the end of the lecture, you all know what a renormalon is. 25 00:01:07,420 --> 00:01:09,650 OK, so this is where we left off. 26 00:01:09,650 --> 00:01:12,220 We were talking about this symmetry reparameterization 27 00:01:12,220 --> 00:01:12,940 invariance. 28 00:01:12,940 --> 00:01:17,110 And what we showed last time is that the Wilson coefficient 29 00:01:17,110 --> 00:01:20,410 of this operator is 1 to all orders in perturbation theory 30 00:01:20,410 --> 00:01:21,970 because of that symmetry. 31 00:01:21,970 --> 00:01:24,160 The Wilson coefficient of this operator is not. 32 00:01:24,160 --> 00:01:26,230 And I told you that at leading log order, 33 00:01:26,230 --> 00:01:29,080 it would be given by some expression like this. 34 00:01:29,080 --> 00:01:31,720 But in general, this is just the lowest order expression. 35 00:01:31,720 --> 00:01:35,532 And it gets perturbatively corrected. 36 00:01:35,532 --> 00:01:36,990 It's actually known at three loops. 37 00:01:40,430 --> 00:01:43,460 So let's continue. 38 00:01:43,460 --> 00:01:45,530 We've talked about reparameterization invariance 39 00:01:45,530 --> 00:01:47,060 for the Lagrangian. 40 00:01:47,060 --> 00:01:48,650 Reparameterization invariance also 41 00:01:48,650 --> 00:01:51,020 has important consequences for operators. 42 00:02:00,650 --> 00:02:04,040 So just like we wrote down sub-leading Lagrangians, 43 00:02:04,040 --> 00:02:06,110 we should also write sub-leading currents. 44 00:02:06,110 --> 00:02:10,009 So there's 1 over mq supressed currents. 45 00:02:10,009 --> 00:02:11,660 And if you look at those currents 46 00:02:11,660 --> 00:02:15,463 and you construct them, then you find, again, 47 00:02:15,463 --> 00:02:16,880 that reparameterization invariance 48 00:02:16,880 --> 00:02:20,080 plays an important role. 49 00:02:20,080 --> 00:02:23,768 And again, there's relations between Wilson coefficients. 50 00:02:27,260 --> 00:02:29,090 And what happens in this case is actually 51 00:02:29,090 --> 00:02:33,800 that the leading order operator had a Wilson coefficient, 52 00:02:33,800 --> 00:02:35,640 the lowest order operator. 53 00:02:35,640 --> 00:02:38,090 So the Wilson coefficients of the subleading operators 54 00:02:38,090 --> 00:02:40,629 get related to that of the leading operator. 55 00:02:50,120 --> 00:02:54,010 And I may give you a problem on that on your next problem set. 56 00:02:54,010 --> 00:02:58,650 But I haven't quite made up my mind yet. 57 00:02:58,650 --> 00:03:00,662 So rather than go through that, I 58 00:03:00,662 --> 00:03:02,370 want to talk about some observables where 59 00:03:02,370 --> 00:03:05,080 these operators played a role. 60 00:03:05,080 --> 00:03:07,563 So we've seen one example of reparameterization invariance. 61 00:03:07,563 --> 00:03:08,980 And I think from that one example, 62 00:03:08,980 --> 00:03:11,530 you would be able to do other examples. 63 00:03:11,530 --> 00:03:13,780 Let's talk about masses. 64 00:03:13,780 --> 00:03:15,720 These operators here actually have an impact 65 00:03:15,720 --> 00:03:18,080 on the masses of hadrons. 66 00:03:18,080 --> 00:03:24,120 If we think about the mass of a heavy meson, h, 67 00:03:24,120 --> 00:03:28,470 there's a heavy quark in that heavy meson. 68 00:03:28,470 --> 00:03:31,200 So we can pull out that heavy quark. 69 00:03:31,200 --> 00:03:33,720 And then there's some remainder. 70 00:03:33,720 --> 00:03:36,535 And we can characterize higher order terms in this formula. 71 00:03:41,360 --> 00:03:44,770 So let me first explain what this lambda bar is. 72 00:03:44,770 --> 00:03:49,060 Our Lagrangian here, if we're working at lowest order 73 00:03:49,060 --> 00:03:55,810 in the theory, our Lagrangian would be L0 of HQET plus, 74 00:03:55,810 --> 00:03:58,090 for the light quarks, they're basically 75 00:03:58,090 --> 00:04:03,475 just a QCD Lagrangian, so for the light quarks 76 00:04:03,475 --> 00:04:04,100 and the gluons. 77 00:04:07,148 --> 00:04:09,740 So for the heavy quark, we have HQET, 78 00:04:09,740 --> 00:04:11,390 and then there would be higher order 79 00:04:11,390 --> 00:04:13,973 terms that are 1 over m suppressed, 80 00:04:13,973 --> 00:04:15,140 which are the ones up there. 81 00:04:19,079 --> 00:04:21,079 So the question is, if we have this Lagrangian, 82 00:04:21,079 --> 00:04:24,870 what is the mass of a heavy state like the b meson 83 00:04:24,870 --> 00:04:26,810 or the d meson or the d star? 84 00:04:26,810 --> 00:04:29,370 And we can characterize that by knowing the structure 85 00:04:29,370 --> 00:04:31,430 of the Lagrangian. 86 00:04:31,430 --> 00:04:35,300 So first of all, what about this term, this lambda bar? 87 00:04:35,300 --> 00:04:39,257 That term actually comes from these parts of the Lagrangian. 88 00:04:39,257 --> 00:04:41,090 So if we take these parts of the Lagrangian, 89 00:04:41,090 --> 00:04:44,570 we can calculate the Hamiltonian. 90 00:04:44,570 --> 00:04:47,000 Let me call that H0. 91 00:04:47,000 --> 00:04:49,280 And the definition of this lambda bar 92 00:04:49,280 --> 00:04:58,537 is the thing that you get by taking that Hamiltonian 93 00:04:58,537 --> 00:04:59,870 and letting it act on the state. 94 00:05:03,390 --> 00:05:05,930 So you can think that this is-- 95 00:05:05,930 --> 00:05:08,390 if you like, you can think that this state is just 96 00:05:08,390 --> 00:05:10,760 an eigenstate of the Hamiltonian. 97 00:05:10,760 --> 00:05:13,310 You get the energy. 98 00:05:13,310 --> 00:05:17,360 But since the Hamiltonian has no heavy quark mass in it, 99 00:05:17,360 --> 00:05:19,260 what you're getting is some parameter, 100 00:05:19,260 --> 00:05:21,510 which is traditionally called lambda bar, that doesn't 101 00:05:21,510 --> 00:05:23,780 have a heavy quark mass in it. 102 00:05:23,780 --> 00:05:25,670 And so there's no heavy quark mass in here. 103 00:05:29,840 --> 00:05:32,960 So it's independent of mq. 104 00:05:32,960 --> 00:05:37,160 It's also independent of spin. 105 00:05:41,900 --> 00:05:50,120 So B versus B star and flavor B versus D. So 106 00:05:50,120 --> 00:05:52,548 whether it's a B meson, D meson, as long 107 00:05:52,548 --> 00:05:54,590 as we're in this limit where both the bottom mass 108 00:05:54,590 --> 00:06:01,880 and the char mass are heavy, if we're 109 00:06:01,880 --> 00:06:04,460 thinking of this expansion, then this 110 00:06:04,460 --> 00:06:06,740 is a universal thing just determined 111 00:06:06,740 --> 00:06:08,468 by these Lagrangians. 112 00:06:08,468 --> 00:06:10,010 It's traditionally called lambda bar. 113 00:06:10,010 --> 00:06:12,830 And it's telling you that if you think about this thing 114 00:06:12,830 --> 00:06:14,540 as having an order mq piece, this 115 00:06:14,540 --> 00:06:16,950 is the order mq to the 0 piece. 116 00:06:16,950 --> 00:06:18,740 The symmetry of the theory is telling us 117 00:06:18,740 --> 00:06:20,735 that this is a universal constant. 118 00:06:23,600 --> 00:06:29,660 It does depend on the state here, 119 00:06:29,660 --> 00:06:32,030 although that state could be connected to other states 120 00:06:32,030 --> 00:06:35,460 by spin symmetry and flavor symmetry. 121 00:06:35,460 --> 00:06:41,120 There was this quantum number we talked about last time, which 122 00:06:41,120 --> 00:06:44,330 we called SL for the spin of the light degrees of freedom, 123 00:06:44,330 --> 00:06:47,800 and then pi for the parity of the light degrees of freedom. 124 00:06:52,470 --> 00:06:54,650 So it's the same within every multiplet 125 00:06:54,650 --> 00:06:58,220 that we get from a fixed SL pi. 126 00:06:58,220 --> 00:07:00,722 But there's a different value for each multiplet. 127 00:07:04,852 --> 00:07:06,560 So you'd have a different value of lambda 128 00:07:06,560 --> 00:07:10,760 bar for the B, D, B star B than for the lambda B and the lambda 129 00:07:10,760 --> 00:07:11,570 C, for example. 130 00:07:15,100 --> 00:07:17,200 So that's this term. 131 00:07:17,200 --> 00:07:19,680 We can also characterize what's going on with this term 132 00:07:19,680 --> 00:07:21,720 by using this Lagrangian here, the L1. 133 00:07:25,750 --> 00:07:28,660 So at order 1 over MQ, we can figure out 134 00:07:28,660 --> 00:07:31,930 what kind of contributions to the mass we're getting. 135 00:07:31,930 --> 00:07:34,870 h1 is just minus L1. 136 00:07:34,870 --> 00:07:37,060 You can think of it now perturbatively. 137 00:07:37,060 --> 00:07:39,370 So that just flips the sign of what we had before. 138 00:08:00,880 --> 00:08:06,240 And we get two parameters from taking matrix elements here. 139 00:08:06,240 --> 00:08:07,780 They get the following names. 140 00:08:21,990 --> 00:08:24,000 And let me just do everything here in terms of-- 141 00:08:27,510 --> 00:08:30,068 it's not totally crucial, but it's 142 00:08:30,068 --> 00:08:31,860 a little easier to interpret certain things 143 00:08:31,860 --> 00:08:33,068 if we work in the rest frame. 144 00:08:33,068 --> 00:08:35,375 So I'll just work in the rest frame. 145 00:08:35,375 --> 00:08:36,750 So if you work in the rest frame, 146 00:08:36,750 --> 00:08:38,583 you should just think of this as like giving 147 00:08:38,583 --> 00:08:40,500 some kind of measure of the kinetic energy 148 00:08:40,500 --> 00:08:42,780 that the heavy quark gets wiggled by. 149 00:08:42,780 --> 00:08:44,760 And that's called lambda 1. 150 00:08:44,760 --> 00:08:47,040 This, again, is a matrix element that doesn't know 151 00:08:47,040 --> 00:08:48,870 about the heavy quark mass. 152 00:08:48,870 --> 00:08:52,260 And the heavy quark mass in an explicit MQ. 153 00:08:52,260 --> 00:08:56,275 So this guy has dimension 2 if you work out the dimensions. 154 00:09:01,500 --> 00:09:03,880 And then you can do the same for the other operator. 155 00:09:03,880 --> 00:09:08,980 And here you have to think a little bit harder. 156 00:09:08,980 --> 00:09:12,270 And if you think a little bit harder, 157 00:09:12,270 --> 00:09:15,510 you can characterize what the coefficients 158 00:09:15,510 --> 00:09:18,360 are for different spins. 159 00:09:18,360 --> 00:09:22,590 So there's some Wilson coefficient. 160 00:09:22,590 --> 00:09:23,940 And then we have this operator. 161 00:09:36,920 --> 00:09:44,235 And this here is like sigma dot B. I told you that before. 162 00:09:44,235 --> 00:09:45,610 And so you should think of what's 163 00:09:45,610 --> 00:09:49,300 going on here as you have sigma, which is giving the SQ, 164 00:09:49,300 --> 00:09:50,750 and then you have a B field. 165 00:09:50,750 --> 00:09:53,992 And then it's a question of what vector 166 00:09:53,992 --> 00:09:56,200 could be left over that the B field could know about. 167 00:09:56,200 --> 00:09:58,570 And it's basically SL. 168 00:09:58,570 --> 00:10:04,750 So this guy here is leading to the SQ. 169 00:10:04,750 --> 00:10:06,635 And this guy here is leading to the SL. 170 00:10:11,870 --> 00:10:13,960 And so that's where this SQ.SL comes from. 171 00:10:16,810 --> 00:10:20,040 So you can think about other possibilities, 172 00:10:20,040 --> 00:10:22,350 but you have to have the right symmetry 173 00:10:22,350 --> 00:10:24,270 structure under time reversal and parity 174 00:10:24,270 --> 00:10:25,650 and things like that. 175 00:10:25,650 --> 00:10:27,450 And that forces you to use a spin here 176 00:10:27,450 --> 00:10:31,230 for the B and not something like the V. 177 00:10:31,230 --> 00:10:34,230 So this SQ.SL is something that you can write out 178 00:10:34,230 --> 00:10:39,060 as, remembering the definition of it and its relation to J 179 00:10:39,060 --> 00:10:41,910 squared. 180 00:10:41,910 --> 00:10:45,270 You can write out that factor like that. 181 00:10:45,270 --> 00:10:48,540 And then we can derive here how this lambda 2 contributes 182 00:10:48,540 --> 00:10:52,410 to the differing states, because we 183 00:10:52,410 --> 00:10:54,750 have a different contribution for the B 184 00:10:54,750 --> 00:10:56,850 and the B star from the lambda 2 guy, 185 00:10:56,850 --> 00:11:00,450 because it was violating the spin symmetry. 186 00:11:00,450 --> 00:11:03,300 So the lambda 1 guy is going to violate the flavor symmetry. 187 00:11:03,300 --> 00:11:05,160 That's this 1 over MQ. 188 00:11:05,160 --> 00:11:06,840 And so charm quarks-- 189 00:11:06,840 --> 00:11:08,940 charm mesons and bottom mesons will 190 00:11:08,940 --> 00:11:11,010 get different contributions from that. 191 00:11:11,010 --> 00:11:14,250 They have the same lambda 1, but they 192 00:11:14,250 --> 00:11:16,530 have a different connection because it gets suppressed 193 00:11:16,530 --> 00:11:19,210 by a different factor. 194 00:11:19,210 --> 00:11:25,815 And then again for the lamba 2, Lambda 2 195 00:11:25,815 --> 00:11:27,690 is a little bit different because it actually 196 00:11:27,690 --> 00:11:29,072 has mu dependents. 197 00:11:32,852 --> 00:11:33,810 That's not quite right. 198 00:11:33,810 --> 00:11:39,240 So it actually has MQ dependents because the coefficient here 199 00:11:39,240 --> 00:11:41,160 has mu dependents. 200 00:11:41,160 --> 00:11:43,810 So this operator doesn't have any MQ dependence, 201 00:11:43,810 --> 00:11:46,350 but the coefficient does, remember. 202 00:11:46,350 --> 00:11:48,710 The mu dependence cancels and there's a left over MQ 203 00:11:48,710 --> 00:11:53,691 dependence to the lambda 2, but it's logarithmic MQ dependence. 204 00:11:56,780 --> 00:11:59,960 If I define it this way-- 205 00:11:59,960 --> 00:12:01,640 so that's why I wrote MQ there. 206 00:12:09,740 --> 00:12:11,990 So you could, as we talked about last time, 207 00:12:11,990 --> 00:12:15,962 you could sum up the logs of large logs and have this guy-- 208 00:12:15,962 --> 00:12:17,420 you could re-sum logs, if you want. 209 00:12:17,420 --> 00:12:19,295 So you could think of these as you can re-sum 210 00:12:19,295 --> 00:12:20,450 logs inside this lambda 2. 211 00:12:23,660 --> 00:12:28,280 So if we ignore those logs or we imagine that we re-sum them 212 00:12:28,280 --> 00:12:29,900 and we just look at the remainder, 213 00:12:29,900 --> 00:12:32,750 then we have the expectations based on power counting 214 00:12:32,750 --> 00:12:35,090 for how big these things are. 215 00:12:35,090 --> 00:12:36,740 And we just [? expect ?] that they're 216 00:12:36,740 --> 00:12:38,480 both given by the dimension. 217 00:12:38,480 --> 00:12:40,310 And the only [? dimensionful ?] parameter 218 00:12:40,310 --> 00:12:43,533 around is lambda QCD because the MQs can only 219 00:12:43,533 --> 00:12:45,950 occur in logarithms, and the Wilson coefficient in nowhere 220 00:12:45,950 --> 00:12:48,580 else. 221 00:12:48,580 --> 00:12:52,800 And these are non-perturbative parameters. 222 00:12:57,450 --> 00:13:02,162 Lambda bar is also a non-perturbative parameter. 223 00:13:02,162 --> 00:13:04,120 These ones have a little more dynamics in them. 224 00:13:06,780 --> 00:13:10,280 So then we could go out and write for our states 225 00:13:10,280 --> 00:13:13,170 the results by putting these things together. 226 00:13:13,170 --> 00:13:16,550 So we'd have M capital B plub M little B 227 00:13:16,550 --> 00:13:22,040 plus lambda bar lambda 1 over 2 and little B. 228 00:13:22,040 --> 00:13:24,890 And if we go through the spin structure there, 229 00:13:24,890 --> 00:13:30,410 we get a 3/2 and then lamba 2 logarithmic MB 230 00:13:30,410 --> 00:13:31,660 dependence over MB. 231 00:13:40,200 --> 00:13:43,410 So when we start with the B star, 232 00:13:43,410 --> 00:13:44,980 the first three terms are the same. 233 00:13:44,980 --> 00:13:48,030 And then this guy comes in with a plus. 234 00:13:48,030 --> 00:13:51,850 And it's lambda 2 over MB over 2MB. 235 00:13:55,600 --> 00:14:03,520 And then the D states are similar in turn. 236 00:14:03,520 --> 00:14:11,800 Lambda bar lambda 1 over 2M term, and then these two terms 237 00:14:11,800 --> 00:14:13,330 again. 238 00:14:13,330 --> 00:14:17,218 Same spins, so it's lambda 2. 239 00:14:17,218 --> 00:14:18,760 And the only difference is, now we're 240 00:14:18,760 --> 00:14:28,500 evaluating the lambda 2 as logarithm and charm dependence, 241 00:14:28,500 --> 00:14:29,000 OK. 242 00:14:29,000 --> 00:14:31,430 So there's some formulas that are correct, actually-- 243 00:14:31,430 --> 00:14:34,490 including all the way up to 1 over M corrections. 244 00:14:34,490 --> 00:14:36,140 And you can see that there's still sort 245 00:14:36,140 --> 00:14:37,580 of a structure to these things. 246 00:14:37,580 --> 00:14:40,310 This is a universal correction for all of them. 247 00:14:40,310 --> 00:14:43,730 These corrections are universal between B and B star. 248 00:14:43,730 --> 00:14:45,935 And so the splitting between B and B star 249 00:14:45,935 --> 00:14:48,660 is just given by this lambda 2. 250 00:14:48,660 --> 00:14:50,600 That's what causes the states to split. 251 00:14:50,600 --> 00:14:52,920 So up to that level, they're the same. 252 00:14:52,920 --> 00:14:55,095 And that's why the B and B star are very degenerate 253 00:14:55,095 --> 00:14:56,345 and the masses are very close. 254 00:14:59,010 --> 00:15:02,930 So you can form combinations to cancel things out. 255 00:15:02,930 --> 00:15:08,960 So you can take a kind of spin average mass where 256 00:15:08,960 --> 00:15:14,480 you take 3 times the vector plus the scalar divided by 4. 257 00:15:14,480 --> 00:15:17,750 And if you do that, then you're canceling out the lambda 2. 258 00:15:27,010 --> 00:15:29,360 And then you can also form differences. 259 00:15:29,360 --> 00:15:33,800 So if you looked at MV star squared minus MB squared, 260 00:15:33,800 --> 00:15:35,770 then the only thing that's causing a difference 261 00:15:35,770 --> 00:15:40,060 is lambda 2. 262 00:15:40,060 --> 00:15:44,860 Numerically, this is 0.49 GV squared. 263 00:15:44,860 --> 00:15:50,980 And from our formula up there, it's 4 lambda 2 of MB, 264 00:15:50,980 --> 00:16:00,020 so something like four times 0.12 GV squared. 265 00:16:00,020 --> 00:16:02,290 So that's the value of this lambda 266 00:16:02,290 --> 00:16:06,770 2 is 0.12 GV squared, which is about lambda QCD squared. 267 00:16:06,770 --> 00:16:09,555 So our dimensional analysis is working. 268 00:16:09,555 --> 00:16:11,680 And then you can do the same thing for the D meson. 269 00:16:15,917 --> 00:16:17,750 Everything here works beautifully, actually. 270 00:16:31,430 --> 00:16:33,100 Extract the value of lambda 2 and you 271 00:16:33,100 --> 00:16:39,130 see there's a slight difference because that difference can be 272 00:16:39,130 --> 00:16:40,834 attributed to the logarithms. 273 00:16:44,710 --> 00:16:48,662 So if you take lambda 2 MB over lambda 2 MC, 274 00:16:48,662 --> 00:16:51,120 then the only thing that differs is the Wilson coefficient. 275 00:16:51,120 --> 00:16:54,450 So that's something you can predict. 276 00:16:54,450 --> 00:17:00,890 So experimentally, if you take the ratio of those two things 277 00:17:00,890 --> 00:17:06,530 there, you get a prediction like this one. 278 00:17:06,530 --> 00:17:12,148 And if you plug in theory like leading log RGE, 279 00:17:12,148 --> 00:17:13,565 then you're getting alpha S of MB. 280 00:17:16,166 --> 00:17:17,290 Oh, sorry. 281 00:17:17,290 --> 00:17:18,109 [INAUDIBLE] 282 00:17:23,749 --> 00:17:28,220 And this is 1.17. 283 00:17:28,220 --> 00:17:30,790 So that's for three light flavors, 284 00:17:30,790 --> 00:17:33,500 which is the right thing to do for the beta function. 285 00:17:33,500 --> 00:17:37,150 So it's not so bad for a leading log prediction. 286 00:17:37,150 --> 00:17:40,090 That's kind of the accuracy you would expect. 287 00:17:44,090 --> 00:17:45,260 All right. 288 00:17:45,260 --> 00:17:47,020 So we're really understanding quite a bit 289 00:17:47,020 --> 00:17:48,603 from our [INAUDIBLE] [? instruction ?] 290 00:17:48,603 --> 00:17:51,280 of this effective theory about the states, 291 00:17:51,280 --> 00:17:55,570 and even make predictions for the ratios of the mass divided 292 00:17:55,570 --> 00:17:58,350 by that as something preservative, which 293 00:17:58,350 --> 00:18:01,165 is kind of non-trivial. 294 00:18:01,165 --> 00:18:03,040 Perturbative up to the order we were working. 295 00:18:03,040 --> 00:18:06,310 There would be corrections if we continued in our series here. 296 00:18:06,310 --> 00:18:08,470 There could be some non-perturbative corrections. 297 00:18:08,470 --> 00:18:11,530 But up to 1 over M, the ratio of this 298 00:18:11,530 --> 00:18:15,463 to that, which is what I'd wrote over there, is perturbative. 299 00:18:18,640 --> 00:18:20,580 All right. 300 00:18:20,580 --> 00:18:23,040 So that's some phenomenology. 301 00:18:23,040 --> 00:18:24,840 It's kind of baseline phenomenology 302 00:18:24,840 --> 00:18:27,330 that we can do with the Hamiltonian 303 00:18:27,330 --> 00:18:31,770 or with Hamiltonian derived from our subleading Lagrangian. 304 00:18:31,770 --> 00:18:35,010 There's other phenomenology that we can do. 305 00:18:35,010 --> 00:18:38,045 And I'll mention some of the most important phenomenology. 306 00:18:44,360 --> 00:18:46,407 There's another class of predictions 307 00:18:46,407 --> 00:18:48,740 that we can make where we have a lot of predictive power 308 00:18:48,740 --> 00:18:50,128 by using the effective theory. 309 00:18:50,128 --> 00:18:51,920 And I want to talk a little bit about that. 310 00:19:01,120 --> 00:19:03,420 So you can look at semi-leptonic decays. 311 00:19:03,420 --> 00:19:06,690 And there's two different types of semi-leptonic decays that 312 00:19:06,690 --> 00:19:08,430 you can look at-- 313 00:19:08,430 --> 00:19:12,430 so-called exclusive decays and inclusive decays. 314 00:19:12,430 --> 00:19:14,340 So exclusive is making transitions 315 00:19:14,340 --> 00:19:17,580 between the meson states and inclusive 316 00:19:17,580 --> 00:19:22,710 is making a transition where you allow any charm state, not just 317 00:19:22,710 --> 00:19:24,750 the lowest order ground state charm, 318 00:19:24,750 --> 00:19:28,530 but you could have in this state here, XE, you 319 00:19:28,530 --> 00:19:32,710 could have a D pi or a D star pi pi or other things like that. 320 00:19:32,710 --> 00:19:37,300 So it doesn't even have to be a single hadron state. 321 00:19:37,300 --> 00:19:39,600 So exclusive refers to a particular channel. 322 00:19:39,600 --> 00:19:43,350 Inclusive refers to some overall channels. 323 00:19:43,350 --> 00:19:46,360 So the theory in these two is quite different. 324 00:19:46,360 --> 00:19:51,960 In this one, you would have form factors for the current 325 00:19:51,960 --> 00:19:53,490 between the states. 326 00:19:53,490 --> 00:19:56,250 And I already mentioned to you that heavy quark symmetry 327 00:19:56,250 --> 00:19:57,810 reduces the number of form factors. 328 00:19:57,810 --> 00:20:00,210 You get this single Isgur-Wise function. 329 00:20:00,210 --> 00:20:06,270 And so heavy quark symmetry is very powerful here. 330 00:20:06,270 --> 00:20:11,760 And you can actually also work out 1 over MQ corrections. 331 00:20:11,760 --> 00:20:13,240 And people have done that. 332 00:20:13,240 --> 00:20:15,547 And it turns out there aren't any. 333 00:20:15,547 --> 00:20:18,270 So there's something called Luke's theorem, which actually 334 00:20:18,270 --> 00:20:20,160 was a result derived, I think, when 335 00:20:20,160 --> 00:20:24,390 Luke was a graduate student that proved there was no 1 over m Q 336 00:20:24,390 --> 00:20:26,400 corrections here. 337 00:20:26,400 --> 00:20:28,470 And then you can work out alpha s corrections. 338 00:20:28,470 --> 00:20:31,530 And so this kind of-- you can just keep going. 339 00:20:31,530 --> 00:20:34,350 People have worked out 1 over m Q squared corrections. 340 00:20:34,350 --> 00:20:36,360 And this kind of formalism is used 341 00:20:36,360 --> 00:20:41,010 to measure V c b because these decay rates would depend 342 00:20:41,010 --> 00:20:43,170 on V c b, and that's kind of what you're 343 00:20:43,170 --> 00:20:45,180 after if you're doing this. 344 00:20:45,180 --> 00:20:47,160 So you'd like to parameterize and figure out 345 00:20:47,160 --> 00:20:49,230 the hadronic physics as much as possible, 346 00:20:49,230 --> 00:20:52,110 and aid the experiments in getting V c b, 347 00:20:52,110 --> 00:20:54,480 and exactly this framework is used. 348 00:20:54,480 --> 00:20:56,790 People nowadays-- what they do is, the 1 over m Q 349 00:20:56,790 --> 00:20:59,950 squared corrections need to be computed on the lattice. 350 00:20:59,950 --> 00:21:02,195 So lattice QCD computes those corrections. 351 00:21:02,195 --> 00:21:03,570 The perturbative corrections need 352 00:21:03,570 --> 00:21:06,460 to be computed by continuum people like me, 353 00:21:06,460 --> 00:21:07,793 so those are computed. 354 00:21:07,793 --> 00:21:09,210 And they put these things together 355 00:21:09,210 --> 00:21:13,260 to get V c b from these decays. 356 00:21:13,260 --> 00:21:16,500 You could also do it with this inclusive guy. 357 00:21:20,430 --> 00:21:24,210 And here it's more interesting because you can basically do 358 00:21:24,210 --> 00:21:27,337 everything with pen and paper. 359 00:21:27,337 --> 00:21:29,670 So here, you can use an Operator Product Expansion, OPE. 360 00:21:34,770 --> 00:21:39,780 And actually, HQET constrains the form of the operator 361 00:21:39,780 --> 00:21:41,330 product expansion, as well. 362 00:21:52,540 --> 00:21:59,110 And indeed, when you work out the leading power corrections, 363 00:21:59,110 --> 00:22:03,520 they turn out to not enter until two orders down. 364 00:22:03,520 --> 00:22:06,370 So again, you're protected from first-order corrections, 365 00:22:06,370 --> 00:22:11,380 so you just get corrections that occur at 1 over m Q squared. 366 00:22:11,380 --> 00:22:15,403 And furthermore, they depend only on the two parameters 367 00:22:15,403 --> 00:22:16,195 we already defined. 368 00:22:19,560 --> 00:22:23,830 So it's not like you even get any new nonperturbative matrix 369 00:22:23,830 --> 00:22:24,330 comments. 370 00:22:24,330 --> 00:22:26,288 You just get the two we were talking over here, 371 00:22:26,288 --> 00:22:28,180 were the masses. 372 00:22:28,180 --> 00:22:28,680 OK? 373 00:22:28,680 --> 00:22:30,497 So there's perturbation corrections, 374 00:22:30,497 --> 00:22:32,580 and there's power corrections, but you're actually 375 00:22:32,580 --> 00:22:34,080 not even getting any new information 376 00:22:34,080 --> 00:22:38,230 relative to the masses, any new matrix elements. 377 00:22:41,680 --> 00:22:42,180 OK? 378 00:22:42,180 --> 00:22:44,100 So we'll talk a little bit more about this second one 379 00:22:44,100 --> 00:22:44,642 with the OPE. 380 00:22:54,260 --> 00:22:56,060 There's a lot of work that's gone 381 00:22:56,060 --> 00:22:58,850 into this one, both from the theory side, 382 00:22:58,850 --> 00:23:00,397 as well as from experiment. 383 00:23:04,970 --> 00:23:08,060 So when you think about the decay rate for this B 384 00:23:08,060 --> 00:23:11,780 to X sub c l nu bar, it's a doubly differential decay rate 385 00:23:11,780 --> 00:23:17,540 in Q squared over the lepton pair, so that's Q mu, E l-- 386 00:23:17,540 --> 00:23:19,520 because you can pick it as a second variable-- 387 00:23:19,520 --> 00:23:21,260 and then you don't know the mass of the state 388 00:23:21,260 --> 00:23:23,130 X sub c because it could be different things. 389 00:23:23,130 --> 00:23:24,650 It could be a D. It could be a D star. 390 00:23:24,650 --> 00:23:25,442 It could be a D pi. 391 00:23:32,750 --> 00:23:34,798 So that's another variable. 392 00:23:34,798 --> 00:23:36,590 So you have a triply differential spectrum, 393 00:23:36,590 --> 00:23:40,223 and you can compute it with an operator product expansion. 394 00:23:49,638 --> 00:23:51,930 I'm not going to go through the details of carrying out 395 00:23:51,930 --> 00:23:54,180 the operator product expansion. 396 00:23:54,180 --> 00:23:56,555 If you want to do reading about that, 397 00:23:56,555 --> 00:23:57,930 there's some supplemental reading 398 00:23:57,930 --> 00:23:59,712 that I haven't assigned to you where 399 00:23:59,712 --> 00:24:01,170 you can read about that in the book 400 00:24:01,170 --> 00:24:03,990 Heavy Quark Physics by Manohar and Wise. 401 00:24:08,730 --> 00:24:13,470 So for our purposes here, let's think of what operator product 402 00:24:13,470 --> 00:24:15,817 expansion means. 403 00:24:15,817 --> 00:24:17,400 It is simply that we want to carry out 404 00:24:17,400 --> 00:24:35,370 an expansion in lambda QCD over m Q. 405 00:24:35,370 --> 00:24:38,130 And if you go into the details, there's 406 00:24:38,130 --> 00:24:39,630 an important role played by the fact 407 00:24:39,630 --> 00:24:41,130 that you're summing over all states, 408 00:24:41,130 --> 00:24:43,110 and that's allowing you to connect 409 00:24:43,110 --> 00:24:46,350 the partonic calculations to the hadronic calculations, 410 00:24:46,350 --> 00:24:48,740 basically by probability conservation, 411 00:24:48,740 --> 00:24:52,380 but I'm not going to go so much into that. 412 00:24:52,380 --> 00:24:56,370 So when you do this operator product expansion, 413 00:24:56,370 --> 00:24:58,470 you can think of it in terms of diagrams. 414 00:24:58,470 --> 00:25:02,100 And usually, you draw these diagrams as forward diagrams. 415 00:25:02,100 --> 00:25:04,380 So here's kind of the matrix elements squared. 416 00:25:07,740 --> 00:25:08,970 Here is the final state. 417 00:25:13,350 --> 00:25:16,543 There's the b quark going in and another b quark going out, 418 00:25:16,543 --> 00:25:18,210 so that's like a matrix element squared. 419 00:25:22,194 --> 00:25:24,698 The amplitude squared. 420 00:25:24,698 --> 00:25:26,990 And you think about doing an operator product expansion 421 00:25:26,990 --> 00:25:27,490 for that. 422 00:25:30,210 --> 00:25:32,420 And you match onto operators in the effective theory. 423 00:25:38,790 --> 00:25:41,790 Just to give you a schematic structure for those operators. 424 00:25:52,770 --> 00:25:55,860 So there's a leading-order operator. 425 00:25:55,860 --> 00:25:57,660 There's a subleading-order operator 426 00:25:57,660 --> 00:26:00,000 that has an extra covariant derivative in it. 427 00:26:00,000 --> 00:26:01,560 That's kind of what you would do. 428 00:26:01,560 --> 00:26:03,780 And then you have Wilson coefficients. 429 00:26:03,780 --> 00:26:07,600 The leading-order operator here is just b bar b. 430 00:26:07,600 --> 00:26:09,540 That's what it turns out to be. 431 00:26:09,540 --> 00:26:12,540 This operator here would look familiar 432 00:26:12,540 --> 00:26:16,360 because it's D transverse squared, 433 00:26:16,360 --> 00:26:18,040 so that's like our lambda 1 operator, 434 00:26:18,040 --> 00:26:20,520 and that's where the lambda 1's are going to come from. 435 00:26:20,520 --> 00:26:22,187 And then there would be a lambda 2 term, 436 00:26:22,187 --> 00:26:24,770 and I just didn't write it. 437 00:26:24,770 --> 00:26:26,852 So there's a magnetic guy here, too. 438 00:26:34,570 --> 00:26:36,910 So when you look at the leading-order operator, 439 00:26:36,910 --> 00:26:40,570 b bar b, that counts the number of b quarks, OK? 440 00:26:40,570 --> 00:26:42,930 That's just a number operator. 441 00:26:42,930 --> 00:26:47,680 So to all orders in perturbation theory, b bar b is 1. 442 00:26:50,420 --> 00:26:50,920 Yeah? 443 00:26:53,990 --> 00:26:56,960 AUDIENCE: How do you cut these-- 444 00:26:56,960 --> 00:26:58,567 like, you're actually calculating 445 00:26:58,567 --> 00:26:59,900 [? B to A, right? ?] [INAUDIBLE] 446 00:26:59,900 --> 00:27:00,733 IAIN STEWART: Right. 447 00:27:00,733 --> 00:27:02,660 So you should think of actually looking-- 448 00:27:02,660 --> 00:27:04,920 so this is a charm quark. 449 00:27:04,920 --> 00:27:07,160 And so you would write down that propagator, 450 00:27:07,160 --> 00:27:09,770 and then you have a large injection of momentum 451 00:27:09,770 --> 00:27:11,600 from the b quark, right? 452 00:27:11,600 --> 00:27:14,780 Use momentum conservation and expand the propagator. 453 00:27:14,780 --> 00:27:16,700 Thinking about the propagator, if you like, 454 00:27:16,700 --> 00:27:21,410 thinking about this guy here having-- 455 00:27:21,410 --> 00:27:27,110 so you could write this guy's momentum as m b v plus k, 456 00:27:27,110 --> 00:27:27,650 all right? 457 00:27:27,650 --> 00:27:30,177 And you would expand in k. 458 00:27:30,177 --> 00:27:32,510 So at first order, you just drop all the k's, and that's 459 00:27:32,510 --> 00:27:34,520 basically giving you this term. 460 00:27:34,520 --> 00:27:36,230 At second order, you'd keep the k's, 461 00:27:36,230 --> 00:27:37,970 so the k squared would be identified 462 00:27:37,970 --> 00:27:40,130 with this D t squared. 463 00:27:40,130 --> 00:27:43,880 AUDIENCE: Right, but how do you cut the effective [INAUDIBLE]?? 464 00:27:43,880 --> 00:27:46,850 IAIN STEWART: The effective theory diagram, it's 465 00:27:46,850 --> 00:27:48,830 already been cut, if you like. 466 00:27:48,830 --> 00:27:51,260 So this is just a real thing, and I've already 467 00:27:51,260 --> 00:27:53,190 taken the imaginary part. 468 00:27:53,190 --> 00:27:55,760 So this is all real. 469 00:27:55,760 --> 00:27:58,140 There's no cut to make in the effective theory. 470 00:27:58,140 --> 00:27:59,970 The thing I'm integrating out here-- 471 00:27:59,970 --> 00:28:01,280 so this is a good comment. 472 00:28:01,280 --> 00:28:04,580 The thing I'm integrating out here is hard and off-shell. 473 00:28:04,580 --> 00:28:06,680 So when I go over to the effective theory 474 00:28:06,680 --> 00:28:12,080 and I get rid of the off-shell stuff, I have a real part here. 475 00:28:12,080 --> 00:28:16,210 There's nothing to cut in the effective theory. 476 00:28:16,210 --> 00:28:17,657 Sometimes, as you know, there are 477 00:28:17,657 --> 00:28:19,240 things to cut in the effective theory, 478 00:28:19,240 --> 00:28:21,640 but here there's nothing to cut on this side. 479 00:28:30,550 --> 00:28:33,790 OK, so this is one tall order. 480 00:28:33,790 --> 00:28:35,770 So that actually means that we don't even 481 00:28:35,770 --> 00:28:37,750 have a nontrivial matrix element here. 482 00:28:37,750 --> 00:28:39,790 We just have a Wilson coefficient. 483 00:28:39,790 --> 00:28:44,640 So everything in the first-order term is completely calculable. 484 00:28:44,640 --> 00:28:54,540 No nonperturbative parameter because of symmetry. 485 00:28:54,540 --> 00:28:56,997 And this guy here is the power correction. 486 00:29:03,680 --> 00:29:06,630 And if you look at what the Wilson coefficient is-- 487 00:29:06,630 --> 00:29:08,465 so it's a function of alpha s. 488 00:29:12,150 --> 00:29:14,440 You could think of it like this, if you want. 489 00:29:24,040 --> 00:29:28,090 And all the kinematic variables, like kinematic variables 490 00:29:28,090 --> 00:29:30,070 like these guys-- 491 00:29:30,070 --> 00:29:41,890 and it is equal to the free b quark decay, including the loop 492 00:29:41,890 --> 00:29:43,750 corrections, of course. 493 00:29:43,750 --> 00:29:45,490 So not only the tree-level result, 494 00:29:45,490 --> 00:29:50,170 but adding loop corrections to the right-hand side here. 495 00:29:50,170 --> 00:29:52,960 If you add the loop corrections and you 496 00:29:52,960 --> 00:29:58,300 drop all the k squared's, then you get just the free b quark 497 00:29:58,300 --> 00:30:00,923 decay, no nonperturbative matrix elements, 498 00:30:00,923 --> 00:30:03,340 so you can just calculate that first term, order by order. 499 00:30:03,340 --> 00:30:05,380 Just doing a [? partonic ?] calculation 500 00:30:05,380 --> 00:30:07,840 gives you the right thing to describe this decay. 501 00:30:10,670 --> 00:30:13,295 And the OPE is telling you that that formally 502 00:30:13,295 --> 00:30:14,420 is exactly the right thing. 503 00:30:14,420 --> 00:30:16,070 Even if you just started to do it, 504 00:30:16,070 --> 00:30:19,130 it's actually technically the right thing to do here. 505 00:30:19,130 --> 00:30:23,360 It wouldn't be the right thing to do for the exclusive decays, 506 00:30:23,360 --> 00:30:26,510 but it is the right thing to do here for the inclusive. 507 00:30:37,280 --> 00:30:40,130 So what is the range of validity of this sort of-- 508 00:30:40,130 --> 00:30:42,740 is there any restrictions on these kinematic variables? 509 00:30:42,740 --> 00:30:44,000 And there is. 510 00:30:44,000 --> 00:30:46,505 We're treating the kinematic variables as if they're hard. 511 00:30:51,060 --> 00:30:52,770 And that means basically that we're 512 00:30:52,770 --> 00:30:55,740 thinking that they scale like powers of the heavy quark mass. 513 00:31:00,560 --> 00:31:02,540 So there can be regions of phase space 514 00:31:02,540 --> 00:31:04,820 where that wouldn't be true, but as long 515 00:31:04,820 --> 00:31:10,210 as we stay away from those corners, edges, then 516 00:31:10,210 --> 00:31:11,130 what I said is right. 517 00:31:15,790 --> 00:31:18,980 And it's a little more powerful than that. 518 00:31:18,980 --> 00:31:22,690 So stay away from edges. 519 00:31:27,340 --> 00:31:31,180 Or there's another way of thinking about it, 520 00:31:31,180 --> 00:31:32,890 and that is that you can integrate over 521 00:31:32,890 --> 00:31:37,045 regions of the Dalitz plane for the phase-space variables. 522 00:31:43,480 --> 00:31:46,670 That includes all the way up to the edges. 523 00:31:46,670 --> 00:31:53,770 And as long as the size of your integration region 524 00:31:53,770 --> 00:31:57,600 is order m Q, then you're also fine. 525 00:31:57,600 --> 00:32:00,540 So think of, like, something with, 526 00:32:00,540 --> 00:32:05,777 like, m B squared up here, d m x squared, for example. 527 00:32:05,777 --> 00:32:08,110 And you can-- it doesn't have to be exactly m B squared. 528 00:32:08,110 --> 00:32:10,652 It could be m B squared over 2, m B squared over 4, something 529 00:32:10,652 --> 00:32:12,880 that you're counting as order m B squared. 530 00:32:12,880 --> 00:32:16,410 And again, you would be fine with doing this type 531 00:32:16,410 --> 00:32:20,207 of operator product expansion. 532 00:32:20,207 --> 00:32:22,040 So what happens is, if you restrict yourself 533 00:32:22,040 --> 00:32:24,347 to be close to the edges, then restricting 534 00:32:24,347 --> 00:32:26,180 yourself to be close to the edges introduces 535 00:32:26,180 --> 00:32:27,952 new scales in the problem. 536 00:32:27,952 --> 00:32:29,660 And if there's new scales in the problem, 537 00:32:29,660 --> 00:32:32,160 just having a power counting that separates out lambda, QCD, 538 00:32:32,160 --> 00:32:34,070 and m B would not be enough. 539 00:32:34,070 --> 00:32:37,670 You would have to do more detail. 540 00:32:37,670 --> 00:32:39,860 And there is actually some interesting things 541 00:32:39,860 --> 00:32:41,737 that happen there, and we probably 542 00:32:41,737 --> 00:32:43,820 will talk about at least one example later on when 543 00:32:43,820 --> 00:32:47,840 we talk about [? SCT. ?] 544 00:32:47,840 --> 00:32:50,870 OK, so I already said it, that there's a nontrivial fact 545 00:32:50,870 --> 00:32:53,570 happening at NLO. 546 00:32:53,570 --> 00:32:56,840 Well, I already implied that there's 547 00:32:56,840 --> 00:33:00,800 no 1 over m b corrections, and so that's 548 00:33:00,800 --> 00:33:04,950 an important outcome from this result. 549 00:33:04,950 --> 00:33:07,430 If you want to derive this, you need the effective theory. 550 00:33:10,010 --> 00:33:13,280 You basically use the equation of motion 551 00:33:13,280 --> 00:33:14,420 to get that to be true. 552 00:33:21,360 --> 00:33:23,610 So this is NLO in the power corrections, 553 00:33:23,610 --> 00:33:34,520 and then NNLO as I said, is just lambda 1 and lambda 2 554 00:33:34,520 --> 00:33:37,880 at order lambda QCD squared over m b squared. 555 00:33:45,800 --> 00:33:49,180 So phenomenologically, this is actually wildly successful. 556 00:33:49,180 --> 00:33:51,670 This is-- as far as I know, this is the case 557 00:33:51,670 --> 00:33:54,490 in QCD or in any theory where people have actually 558 00:33:54,490 --> 00:33:59,140 carried out the operator product expansion in the most detail. 559 00:33:59,140 --> 00:34:02,170 So people have gone up to 1 over m Q to the 4th. 560 00:34:02,170 --> 00:34:05,440 So they calculated at least two loops 561 00:34:05,440 --> 00:34:07,870 for everything and maybe even three loops for the [? NF ?] 562 00:34:07,870 --> 00:34:09,070 pieces. 563 00:34:09,070 --> 00:34:13,969 And experimentally it's been explored to death. 564 00:34:13,969 --> 00:34:17,050 You have these three variables, and they've 565 00:34:17,050 --> 00:34:20,050 constructed of order 80 moments from these three 566 00:34:20,050 --> 00:34:22,690 different kinematic variables, sliced 567 00:34:22,690 --> 00:34:25,900 and diced the decay rate in all sorts 568 00:34:25,900 --> 00:34:28,742 of imaginative and crazy ways. 569 00:34:28,742 --> 00:34:30,909 And they've really tested that this OPE [? always ?] 570 00:34:30,909 --> 00:34:33,219 works beautifully, OK? 571 00:34:33,219 --> 00:34:36,850 So you can think of that you're getting a consistent picture, 572 00:34:36,850 --> 00:34:41,270 80 different observables, just from a few simple predictions 573 00:34:41,270 --> 00:34:42,520 and your perturbative results. 574 00:35:02,070 --> 00:35:03,570 You also get v c b. 575 00:35:14,430 --> 00:35:17,070 So everything fits together with the framework agreement 576 00:35:17,070 --> 00:35:19,840 I've discussed with you, and you get a result for v c b, 577 00:35:19,840 --> 00:35:23,070 which I'm not going to write on the board. 578 00:35:23,070 --> 00:35:23,580 OK? 579 00:35:23,580 --> 00:35:24,570 So that gives you-- 580 00:35:24,570 --> 00:35:26,130 I didn't go through the details. 581 00:35:26,130 --> 00:35:28,890 I will give you some reading to learn more about this OPE 582 00:35:28,890 --> 00:35:32,090 if you're interested, and sort of [? seize ?] how 583 00:35:32,090 --> 00:35:33,840 some of the things I mentioned, like using 584 00:35:33,840 --> 00:35:36,390 the equation of motion, how that actually works out. 585 00:35:36,390 --> 00:35:39,840 It's all done very nicely in this book, Heavy Quark Physics. 586 00:35:39,840 --> 00:35:42,150 But I just wanted to give you a flavor 587 00:35:42,150 --> 00:35:44,010 with this effective theory that there's 588 00:35:44,010 --> 00:35:47,230 something very useful you can do with it once you have it. 589 00:35:47,230 --> 00:35:50,280 And in this case, kind of the main phenomenological thing 590 00:35:50,280 --> 00:35:53,580 that you're after is v c b and what I've described to you, 591 00:35:53,580 --> 00:35:55,560 but you could also think about b decays, 592 00:35:55,560 --> 00:35:57,210 where you're looking for new physics. 593 00:35:57,210 --> 00:36:00,900 And again, if you can construct how the decay rate is looking, 594 00:36:00,900 --> 00:36:03,660 then you can have a hope of finding 595 00:36:03,660 --> 00:36:06,540 new physics in the coefficient of those decay rates, 596 00:36:06,540 --> 00:36:09,830 effectively in the Wilson coefficients of decay rates. 597 00:36:09,830 --> 00:36:10,330 OK? 598 00:36:10,330 --> 00:36:12,760 So we're going to turn to something else, 599 00:36:12,760 --> 00:36:15,780 but let me pause and see if there's any questions. 600 00:36:15,780 --> 00:36:18,660 AUDIENCE: Is this the best way to [? get ?] [? v c b? ?] 601 00:36:18,660 --> 00:36:20,910 IAIN STEWART: The two are actually pretty competitive, 602 00:36:20,910 --> 00:36:22,770 the exclusive and inclusive. 603 00:36:22,770 --> 00:36:25,320 Basically because the lattice is doing 604 00:36:25,320 --> 00:36:27,270 a good job of the matrix elements 605 00:36:27,270 --> 00:36:30,210 that you need in the exclusive. 606 00:36:30,210 --> 00:36:32,640 And again, you have a lot of kinematic information 607 00:36:32,640 --> 00:36:35,113 you can use from the experiment on the shape. 608 00:36:35,113 --> 00:36:36,780 So they're pretty competitive, actually. 609 00:36:36,780 --> 00:36:38,067 AUDIENCE: Do they agree? 610 00:36:38,067 --> 00:36:39,150 IAIN STEWART: They agree-- 611 00:36:39,150 --> 00:36:42,690 I think the disagreement is at, like, kind of the 1.8 sigma 612 00:36:42,690 --> 00:36:43,980 level, so they agree. 613 00:36:47,580 --> 00:36:49,650 There's an interesting story in v u b. 614 00:36:49,650 --> 00:36:52,590 So you could do the same kind of thing, not for a charm quark, 615 00:36:52,590 --> 00:36:54,990 as we did here, but for a light quark. 616 00:36:54,990 --> 00:36:57,183 And then actually, if you do b to pi l nu 617 00:36:57,183 --> 00:36:59,100 and you look at the inclusive version of that, 618 00:36:59,100 --> 00:37:00,785 and you can do the same type of OPE-- 619 00:37:00,785 --> 00:37:02,160 a little bit different, actually, 620 00:37:02,160 --> 00:37:04,248 but you can do it in OPE there, as well. 621 00:37:04,248 --> 00:37:05,790 Then you go through and you get a v u 622 00:37:05,790 --> 00:37:07,915 b in the two different ways, and the disagreement's 623 00:37:07,915 --> 00:37:11,760 at the 2 and 1/2 sigma level, maybe even-- yeah. 624 00:37:11,760 --> 00:37:15,180 It fluctuates with time, but on average it's 2 and 1/2 sigma. 625 00:37:15,180 --> 00:37:16,290 [CHUCKLES] 626 00:37:16,290 --> 00:37:17,320 So that's interesting. 627 00:37:17,320 --> 00:37:21,900 There's not an understanding of what's going on there. 628 00:37:21,900 --> 00:37:23,220 That's more interesting than-- 629 00:37:23,220 --> 00:37:27,000 this one basically agrees. 630 00:37:27,000 --> 00:37:29,610 OK, so rather than go further into this, 631 00:37:29,610 --> 00:37:32,300 I want to turn now to my promised topic to you, 632 00:37:32,300 --> 00:37:35,760 to tell you about renormalons. 633 00:37:35,760 --> 00:37:37,260 Actually, renormalons have something 634 00:37:37,260 --> 00:37:38,640 to do with power correction, so it's not 635 00:37:38,640 --> 00:37:40,820 disconnected from what we've been talking about. 636 00:37:48,160 --> 00:37:49,560 So what are the ideas here? 637 00:38:02,523 --> 00:38:04,940 So we've already seen in our discussion of renormalization 638 00:38:04,940 --> 00:38:07,340 that there is a freedom in defining 639 00:38:07,340 --> 00:38:09,120 the perturbative series. 640 00:38:09,120 --> 00:38:11,570 And we kind of focused on the m s bar scheme 641 00:38:11,570 --> 00:38:12,740 as being the simplest thing. 642 00:38:16,180 --> 00:38:19,603 But I told you about some subtleties with m s bar, 643 00:38:19,603 --> 00:38:21,020 and now we're going to address one 644 00:38:21,020 --> 00:38:24,170 of them that's related to this thing called renormalons. 645 00:38:36,590 --> 00:38:41,090 So we have some freedom, if you like, in adjusting the cutoffs. 646 00:38:41,090 --> 00:38:42,712 And we had this cutoff, mu, and it 647 00:38:42,712 --> 00:38:44,420 was dividing up what was perturbative and 648 00:38:44,420 --> 00:38:46,695 nonperturbative in m s bar. 649 00:38:49,400 --> 00:38:51,863 But we could have done something else. 650 00:38:51,863 --> 00:38:53,780 We could have used a different type of cutoff. 651 00:38:53,780 --> 00:38:55,987 We could have used a Wilsonian cutoff, 652 00:38:55,987 --> 00:38:58,070 and that would have divided up things a little bit 653 00:38:58,070 --> 00:38:59,490 differently. 654 00:38:59,490 --> 00:39:01,940 And you should ask the question, are any possible way 655 00:39:01,940 --> 00:39:03,803 of dividing things up equivalent? 656 00:39:07,430 --> 00:39:10,790 We've already saw when we were doing calculations 657 00:39:10,790 --> 00:39:14,300 that, for the logarithms, they were equivalent. 658 00:39:14,300 --> 00:39:19,790 But it turns out that, if you don't-- 659 00:39:19,790 --> 00:39:23,270 that the powers, sort of the power separation 660 00:39:23,270 --> 00:39:27,080 of power divergences can actually have an impact. 661 00:39:31,470 --> 00:39:33,830 And that's related to what renormalons are. 662 00:39:41,580 --> 00:39:44,780 And m s bar and dim reg basically 663 00:39:44,780 --> 00:39:46,800 avoids thinking about power corrections. 664 00:39:46,800 --> 00:39:48,710 You just set them to zero. 665 00:39:48,710 --> 00:39:51,650 And so what can happen is, in m s bar, 666 00:39:51,650 --> 00:39:54,500 your matrix elements are having a little bit too much UV 667 00:39:54,500 --> 00:39:56,690 physics, and your Wilson coefficients are actually 668 00:39:56,690 --> 00:39:57,753 sensitive to the IR. 669 00:39:57,753 --> 00:39:59,420 They're not sensitive-- you don't really 670 00:39:59,420 --> 00:40:01,087 see it when you look at the coefficient. 671 00:40:01,087 --> 00:40:03,230 You just see a number, and it looks pretty good, 672 00:40:03,230 --> 00:40:05,455 but there's actually an asymptotic structure 673 00:40:05,455 --> 00:40:06,830 to those Wilson coefficients that 674 00:40:06,830 --> 00:40:09,050 comes from higher orders of perturbation theory. 675 00:40:09,050 --> 00:40:10,590 There's things hiding there. 676 00:40:10,590 --> 00:40:12,980 That's what this discussion is going to be about. 677 00:40:23,620 --> 00:40:29,490 So if you do actually not make a good choice, 678 00:40:29,490 --> 00:40:31,080 then it can be the case that when 679 00:40:31,080 --> 00:40:34,470 you go to higher orders in the perturbative expansion, 680 00:40:34,470 --> 00:40:36,210 you get lousy convergence. 681 00:40:40,830 --> 00:40:44,280 And that goes hand-in-hand, actually, 682 00:40:44,280 --> 00:40:50,550 with kind of another thing, which is harder to visualize, 683 00:40:50,550 --> 00:40:54,180 but I'll explain it. 684 00:40:54,180 --> 00:40:56,640 And that is that you have trouble extracting 685 00:40:56,640 --> 00:40:58,476 the nonperturbative parameters. 686 00:41:15,168 --> 00:41:16,960 And the reason that you're having trouble-- 687 00:41:16,960 --> 00:41:19,000 so you could think about doing some calculation 688 00:41:19,000 --> 00:41:20,710 like that OPE over there. 689 00:41:20,710 --> 00:41:23,650 And you extract a value for lambda 1 from all your fits 690 00:41:23,650 --> 00:41:25,652 to all these moments I was describing to you. 691 00:41:25,652 --> 00:41:27,610 And then you go to one higher order in alpha s, 692 00:41:27,610 --> 00:41:29,420 and you extract another value, and all of a sudden, 693 00:41:29,420 --> 00:41:30,795 this guy changes by a factor of 2 694 00:41:30,795 --> 00:41:33,050 and you wonder what's going on. 695 00:41:33,050 --> 00:41:35,170 And you wonder, well, what's nature telling you 696 00:41:35,170 --> 00:41:37,180 about the kinetic energy if I change 697 00:41:37,180 --> 00:41:39,638 the order of my perturbation theory and all of a sudden I'm 698 00:41:39,638 --> 00:41:43,060 extracting a factor of 2 different value for that matrix 699 00:41:43,060 --> 00:41:44,170 element. 700 00:41:44,170 --> 00:41:47,150 And that's related actually to this poor convergence. 701 00:41:47,150 --> 00:41:48,520 These things go hand in hand. 702 00:41:48,520 --> 00:41:51,010 If there's pure convergence in the series, 703 00:41:51,010 --> 00:41:54,940 then when you extract matrix elements 704 00:41:54,940 --> 00:41:56,722 you can get different values. 705 00:41:56,722 --> 00:41:58,180 Phenomenologically, you can imagine 706 00:41:58,180 --> 00:42:00,790 how that would be related, but it's physically related, too. 707 00:42:00,790 --> 00:42:03,730 It has to do with the fact that there's a poor convergence here 708 00:42:03,730 --> 00:42:05,740 because you haven't divided up the IR physics 709 00:42:05,740 --> 00:42:07,820 and the UV physics fully correctly, 710 00:42:07,820 --> 00:42:09,898 and that's reflected in this inability 711 00:42:09,898 --> 00:42:11,815 to extract these guys in a convergent fashion. 712 00:42:15,220 --> 00:42:18,700 And we can actually quantify that using something 713 00:42:18,700 --> 00:42:22,670 called renormalon techniques. 714 00:42:22,670 --> 00:42:30,010 So one way of saying it is that, if you make a poor choice, 715 00:42:30,010 --> 00:42:32,050 you're plagued by something called renormalons, 716 00:42:32,050 --> 00:42:34,467 so they're actually something bad and you don't want them. 717 00:42:47,932 --> 00:43:02,510 So what goes wrong is that the short distance of coefficients 718 00:43:02,510 --> 00:43:14,390 have hidden power law sensitivity to the IR. 719 00:43:14,390 --> 00:43:16,880 You wanted these to be UV things, 720 00:43:16,880 --> 00:43:19,520 but they are sensitive to the IR. 721 00:43:19,520 --> 00:43:22,940 And it comes in dimensional regularization in the powers. 722 00:43:22,940 --> 00:43:26,030 We've been very careful about separating out logarithms, 723 00:43:26,030 --> 00:43:28,190 but we weren't as careful about powers, 724 00:43:28,190 --> 00:43:33,860 and it comes back to haunt us if we look carefully 725 00:43:33,860 --> 00:43:34,490 at the theory. 726 00:43:40,970 --> 00:43:44,040 And there's a corresponding sensitivity 727 00:43:44,040 --> 00:43:48,020 to the UV in the matrix elements. 728 00:43:48,020 --> 00:43:49,520 So let me give you an example that's 729 00:43:49,520 --> 00:43:51,620 not working through formalism, but just numerics. 730 00:43:55,550 --> 00:44:01,040 So let's look at b to u e bar nu at lowest order. 731 00:44:01,040 --> 00:44:04,460 The up quark is massless. 732 00:44:04,460 --> 00:44:07,522 And we'll think about this-- like we 733 00:44:07,522 --> 00:44:09,230 were talking about for the OPE for charm, 734 00:44:09,230 --> 00:44:11,772 we'll think about it inclusively so that we can actually just 735 00:44:11,772 --> 00:44:13,615 look at this and it makes physical sense. 736 00:44:13,615 --> 00:44:15,740 The same thing that I told you about the charm case 737 00:44:15,740 --> 00:44:17,240 applies for the up quark case. 738 00:44:17,240 --> 00:44:19,790 The lowest-order prediction is just calculate this guy 739 00:44:19,790 --> 00:44:21,470 and include loop corrections. 740 00:44:21,470 --> 00:44:23,870 The leading power prediction is the same story 741 00:44:23,870 --> 00:44:25,730 as for the charm. 742 00:44:25,730 --> 00:44:28,130 So that means that physically it's 743 00:44:28,130 --> 00:44:31,490 relevant to think about this decay rate. 744 00:44:31,490 --> 00:44:34,580 So what does it look like? 745 00:44:34,580 --> 00:44:37,220 There's some G Fermi. 746 00:44:37,220 --> 00:44:40,310 Integrating out the W boson. 747 00:44:40,310 --> 00:44:43,220 There's some factors of pi. 748 00:44:43,220 --> 00:44:46,940 The mass dimensions of this guy are 1. 749 00:44:46,940 --> 00:44:50,780 G Fermi squared is minus 4, so you need 5 powers, 750 00:44:50,780 --> 00:44:53,300 and the thing that's the setting the mass dimensions 751 00:44:53,300 --> 00:44:55,310 is m b, so you get 5 powers of m b. 752 00:44:58,370 --> 00:45:02,480 And if you look at the perturbative series 753 00:45:02,480 --> 00:45:06,830 and you said set mu to equal m b, this is what it looks like. 754 00:45:17,790 --> 00:45:19,340 So epsilon-- I'm just introducing 755 00:45:19,340 --> 00:45:23,960 something which is a counting parameter, and it's 1. 756 00:45:23,960 --> 00:45:26,297 I could also make alpha s the counting parameter, 757 00:45:26,297 --> 00:45:28,505 but I actually want to stick a number in for alpha s, 758 00:45:28,505 --> 00:45:31,767 so alpha s is going to disappear in the next line, 759 00:45:31,767 --> 00:45:34,100 and I'll just keep the epsilon, which is just telling me 760 00:45:34,100 --> 00:45:37,310 the contribution is coming from this order. 761 00:45:37,310 --> 00:45:41,090 So you have some choice here for how you define the b quark 762 00:45:41,090 --> 00:45:43,370 mass, OK? 763 00:45:43,370 --> 00:45:46,820 You could use the pole mass, pole in the b quark propagator. 764 00:45:46,820 --> 00:45:48,650 Or you could use the m s bar mass. 765 00:45:48,650 --> 00:45:50,660 Or you could use some other definition. 766 00:45:50,660 --> 00:45:53,040 And it's raised to the 5th power. 767 00:45:53,040 --> 00:45:55,040 So whatever definition you pick, it 768 00:45:55,040 --> 00:45:57,243 can be pretty sensitive to that. 769 00:45:57,243 --> 00:45:58,910 So let me tell you what the results look 770 00:45:58,910 --> 00:46:04,142 like in three different schemes for that mass. 771 00:46:04,142 --> 00:46:06,530 So let's first do the pole scheme. 772 00:46:06,530 --> 00:46:09,540 This is the same. 773 00:46:09,540 --> 00:46:12,155 We have m b pole to the 5th power, 774 00:46:12,155 --> 00:46:14,030 and then we'd write out what the series looks 775 00:46:14,030 --> 00:46:15,386 like in that scheme. 776 00:46:23,427 --> 00:46:25,010 You look like you're doing pretty good 777 00:46:25,010 --> 00:46:26,010 when you're at one loop. 778 00:46:26,010 --> 00:46:27,555 You went down by a factor of 5. 779 00:46:27,555 --> 00:46:29,180 But then you go into two loops, and you 780 00:46:29,180 --> 00:46:31,040 find that your correction at two loops 781 00:46:31,040 --> 00:46:33,540 is pretty much the same size as your correction at one loop. 782 00:46:39,230 --> 00:46:40,730 So you say, well, let's use m s bar. 783 00:46:44,703 --> 00:46:46,245 That changes the perturbative series. 784 00:46:59,993 --> 00:47:02,160 And maybe you think you're doing a little bit better 785 00:47:02,160 --> 00:47:03,510 because at least, well-- 786 00:47:03,510 --> 00:47:05,010 (CHUCKLING) but this guy got bigger, 787 00:47:05,010 --> 00:47:06,850 and this guy is bigger than that guy, 788 00:47:06,850 --> 00:47:09,030 so it's also not really working. 789 00:47:18,590 --> 00:47:23,860 Well, if the phenomenology is as accurate as I told you, 790 00:47:23,860 --> 00:47:25,840 you must imagine that there is something that 791 00:47:25,840 --> 00:47:27,722 does work better than that. 792 00:47:27,722 --> 00:47:30,190 Indeed there is. 793 00:47:30,190 --> 00:47:37,150 So there's other mass games, and I'll talk about one of them, 794 00:47:37,150 --> 00:47:54,360 continuing this table, where we switch to something 795 00:47:54,360 --> 00:47:55,950 called the 1S mass. 796 00:47:55,950 --> 00:47:57,840 Now [INAUDIBLE] looks like that. 797 00:47:57,840 --> 00:47:59,310 We're very happy. 798 00:47:59,310 --> 00:48:01,560 Or at least we're much happier than we were over here. 799 00:48:06,570 --> 00:48:09,960 So what is the 1S mass? 800 00:48:09,960 --> 00:48:11,880 The 1S mass is basically that you 801 00:48:11,880 --> 00:48:16,470 take half the perturbative mass for the epsilon system. 802 00:48:16,470 --> 00:48:25,350 So the 1S mass is m b b bar system calculated 803 00:48:25,350 --> 00:48:28,170 perturbatively, divided by 2. 804 00:48:34,870 --> 00:48:39,260 OK, we'll talk more about why this is working in a minute. 805 00:48:39,260 --> 00:48:42,760 Let me also write down-- 806 00:48:42,760 --> 00:48:46,360 these are conversion formulas because you calculate the decay 807 00:48:46,360 --> 00:48:48,955 rate once and for all, so you calculate the first line. 808 00:48:48,955 --> 00:48:50,830 Then you know how to convert between schemes, 809 00:48:50,830 --> 00:48:54,190 and that's what's causing the perturbative serious to differ. 810 00:48:54,190 --> 00:49:02,800 So m b pole is equal to m b m s bar, 811 00:49:02,800 --> 00:49:04,210 and there's a series that relates 812 00:49:04,210 --> 00:49:18,220 them that looks like that. 813 00:49:23,530 --> 00:49:24,970 And I could put in numbers here. 814 00:49:33,290 --> 00:49:35,290 And so that's what's changing the numbers when I 815 00:49:35,290 --> 00:49:37,040 go from this line to this line. 816 00:49:37,040 --> 00:49:41,353 Basically, kind of the 5 hours of 0.09 that you get 817 00:49:41,353 --> 00:49:42,520 takes you from here to here. 818 00:49:50,327 --> 00:49:52,660 We could also think about switching from the pole scheme 819 00:49:52,660 --> 00:49:56,680 to this m b 1S scheme. 820 00:49:56,680 --> 00:49:57,805 We have a different series. 821 00:50:02,920 --> 00:50:04,963 It still doesn't look very convergent. 822 00:50:10,280 --> 00:50:12,170 The numbers look small, but-- 823 00:50:12,170 --> 00:50:14,040 this is supposed to be a 6-- 824 00:50:14,040 --> 00:50:16,040 but this number here is bigger than that number. 825 00:50:18,770 --> 00:50:21,550 So that when I stick this series and combine it together 826 00:50:21,550 --> 00:50:24,130 with this series for the pole mass, 827 00:50:24,130 --> 00:50:27,430 then I get that series at the top of the board. 828 00:50:27,430 --> 00:50:30,160 And the problem is not in the m b 1S. 829 00:50:30,160 --> 00:50:32,073 The problem is in the m b pole. 830 00:50:32,073 --> 00:50:33,490 And the problem in the m b pole is 831 00:50:33,490 --> 00:50:35,282 being reflected in this series that relates 832 00:50:35,282 --> 00:50:38,781 them and its poor convergence. 833 00:50:38,781 --> 00:50:40,948 AUDIENCE: Can you explain [? what ?] [? m ?] [? b ?] 834 00:50:40,948 --> 00:50:43,156 [? bar ?] [? being ?] [? perturbative ?] [? means? ?] 835 00:50:43,156 --> 00:50:44,740 IAIN STEWART: Yeah, so you calculate-- 836 00:50:44,740 --> 00:50:48,520 so you can think of calculating the Coulomb potential between b 837 00:50:48,520 --> 00:50:53,020 and b bar, and that will give you alpha s corrections 838 00:50:53,020 --> 00:50:54,610 to just the mass. 839 00:50:54,610 --> 00:50:57,700 So you have 2 m b plus Coulomb potential plus, you know, 840 00:50:57,700 --> 00:50:59,010 radiative corrections to that. 841 00:50:59,010 --> 00:51:01,750 And you keep dressing it up and just perturbatively 842 00:51:01,750 --> 00:51:06,910 calculate as if it was a Coulomb problem, a QED problem, 843 00:51:06,910 --> 00:51:11,942 the mass of the b b bar state, and then you divide by 2. 844 00:51:11,942 --> 00:51:15,295 AUDIENCE: [INAUDIBLE] 845 00:51:15,295 --> 00:51:16,270 IAIN STEWART: Yeah 846 00:51:16,270 --> 00:51:18,060 AUDIENCE: But you just pretend? 847 00:51:18,060 --> 00:51:20,102 IAIN STEWART: There's nonperturbative corrections 848 00:51:20,102 --> 00:51:23,960 to it, but what we're extracting from it is the series. 849 00:51:23,960 --> 00:51:24,930 OK? 850 00:51:24,930 --> 00:51:26,990 I'll explain more why it's going to work, 851 00:51:26,990 --> 00:51:31,330 why this is, like, a reasonable choice, and we'll come to that. 852 00:51:31,330 --> 00:51:37,270 OK, so the lesson here is simply that the choice of mass scheme 853 00:51:37,270 --> 00:51:41,680 has a big impact on the perturbative series. 854 00:51:47,550 --> 00:51:50,075 And we haven't yet figured out why some things work 855 00:51:50,075 --> 00:51:51,700 and why some things don't, but we will. 856 00:51:56,800 --> 00:51:58,867 It's absolutely crucial to use the right one. 857 00:51:58,867 --> 00:52:00,700 Otherwise, the predictions will not be good. 858 00:52:03,540 --> 00:52:05,730 Well, physically, we can actually 859 00:52:05,730 --> 00:52:09,250 argue right away why m b pole is not so good. 860 00:52:12,928 --> 00:52:14,970 And that's because, physically, there is no pole. 861 00:52:18,870 --> 00:52:20,670 There's no pole in the quark propagator. 862 00:52:29,790 --> 00:52:32,026 And that is because of confinement. 863 00:52:42,450 --> 00:52:44,750 So we use this notion of pole and a quark propagator 864 00:52:44,750 --> 00:52:46,620 when we're doing perturbation theory. 865 00:52:46,620 --> 00:52:50,330 But nonperturbatively, it's not a well-defined notion, 866 00:52:50,330 --> 00:52:52,340 or at least there doesn't really exist poles 867 00:52:52,340 --> 00:52:53,825 in quark propagators. 868 00:52:59,280 --> 00:53:02,460 So it's only perturbatively a good notion. 869 00:53:24,240 --> 00:53:28,320 So I'll say it's only perturbatively meaningful. 870 00:53:28,320 --> 00:53:39,290 And in reality, it's ambiguous, it's an ambiguous notion. 871 00:53:39,290 --> 00:53:41,690 And you can think that the amount by which it's ambiguous 872 00:53:41,690 --> 00:53:43,460 is related to hadronization, which 873 00:53:43,460 --> 00:53:47,090 is set by the scale lambda QCD. 874 00:53:47,090 --> 00:53:48,890 So there's an ambiguity in what you 875 00:53:48,890 --> 00:53:54,230 mean by a pole mass physically due to nonperturbative effects. 876 00:53:54,230 --> 00:53:56,960 Now, when we set up HQET, we actually used m pole. 877 00:54:01,920 --> 00:54:10,800 So we're using a questionable physical parameter. 878 00:54:10,800 --> 00:54:15,230 So when we did these phase redefinitions, 879 00:54:15,230 --> 00:54:18,050 we were actually using the pole mass scheme 880 00:54:18,050 --> 00:54:20,249 because we were expanding about mass shell. 881 00:54:33,960 --> 00:54:36,580 So if you were to do other things, 882 00:54:36,580 --> 00:54:40,230 you can think about getting to other choices 883 00:54:40,230 --> 00:54:42,230 and implementing them in the HQET. 884 00:54:45,060 --> 00:54:48,790 No problem with doing that. 885 00:54:48,790 --> 00:54:54,240 But there does exist another operator 886 00:54:54,240 --> 00:54:59,470 that we sort of dropped without thinking too hard about it. 887 00:54:59,470 --> 00:55:02,580 So if we switch to using a different mass scheme, 888 00:55:02,580 --> 00:55:04,860 there's some delta m, which you can 889 00:55:04,860 --> 00:55:10,560 think of as a series in alpha s, which we wrote 890 00:55:10,560 --> 00:55:12,150 on the board a moment ago. 891 00:55:17,310 --> 00:55:20,370 And where that delta m would show up 892 00:55:20,370 --> 00:55:25,410 is that you would have kind of an L delta m operator, which 893 00:55:25,410 --> 00:55:32,580 would just be exactly delta m, and then Q v bar Q v. 894 00:55:32,580 --> 00:55:35,550 That would be left over when we cancel the masses 895 00:55:35,550 --> 00:55:37,670 in the Lagrangian, if you like. 896 00:55:37,670 --> 00:55:39,170 That's one way of thinking about it. 897 00:55:44,040 --> 00:55:47,810 So we have an extra term in the Lagrangian in HQET. 898 00:55:47,810 --> 00:55:50,360 And now you can ask, well, if we have an extra term, 899 00:55:50,360 --> 00:55:52,940 we'd better worry about power counting. 900 00:55:52,940 --> 00:55:54,440 And it turns out that the way we can 901 00:55:54,440 --> 00:55:56,630 understand why m s bar wasn't working 902 00:55:56,630 --> 00:55:57,890 is related to power counting. 903 00:56:02,770 --> 00:56:08,260 So in m s bar, if you look at what delta m is, 904 00:56:08,260 --> 00:56:13,120 delta m is m bar itself times alpha. 905 00:56:13,120 --> 00:56:15,100 But by power counting, you don't want 906 00:56:15,100 --> 00:56:17,860 something that's growing with m b in the Lagrangian. 907 00:56:21,050 --> 00:56:25,395 m b is suppressed by alpha, but it's alpha at m b that is 908 00:56:25,395 --> 00:56:27,020 providing [? some ?] [? suppression, ?] 909 00:56:27,020 --> 00:56:29,840 but that's only 0.2, and that's actually just not enough 910 00:56:29,840 --> 00:56:32,600 suppression to get rid of the big m b here. 911 00:56:32,600 --> 00:56:35,955 You don't get-- you go from 5 GV-- 912 00:56:35,955 --> 00:56:37,580 you know, there's some number in front. 913 00:56:37,580 --> 00:56:40,280 Maybe you get down to 1 and 1/2, 2 GV, 914 00:56:40,280 --> 00:56:43,130 but you're not getting down to a small enough value. 915 00:56:46,500 --> 00:56:51,830 So parametrically, this is just not good, both parametrically, 916 00:56:51,830 --> 00:56:57,810 because it grows with m b, and numerically it's too big. 917 00:57:01,160 --> 00:57:03,470 It's not order lambda QCD, which would be OK. 918 00:57:07,510 --> 00:57:12,420 So parametrically and numerically, 919 00:57:12,420 --> 00:57:20,040 delta m is just too big for HQET power counting in this m 920 00:57:20,040 --> 00:57:20,700 s bar scheme. 921 00:57:24,268 --> 00:57:27,930 You'd include an operator that effectively is on average 922 00:57:27,930 --> 00:57:32,720 larger and more important than your kinetic term because the v 923 00:57:32,720 --> 00:57:36,930 dot D operator-- v dot D is counting like lambda QCD, 924 00:57:36,930 --> 00:57:39,690 and it had v dot D with two Q's. 925 00:57:39,690 --> 00:57:42,360 Now you have something delta m with two Q's, and if it's 926 00:57:42,360 --> 00:57:44,790 on average larger, then you're messed up. 927 00:57:51,640 --> 00:57:54,480 So that's why actually physically the m b bar 928 00:57:54,480 --> 00:57:55,440 is not a good choice. 929 00:57:55,440 --> 00:57:58,200 You can think about it from the effective theory from that way. 930 00:57:58,200 --> 00:58:01,213 m b bar is actually a mass that's-- 931 00:58:01,213 --> 00:58:02,880 you're supposed to think of it as a mass 932 00:58:02,880 --> 00:58:04,422 that you use for high-energy physics, 933 00:58:04,422 --> 00:58:07,740 for physics really high energy, above the b quark mass scale. 934 00:58:07,740 --> 00:58:10,210 Then it's a good parameter. 935 00:58:10,210 --> 00:58:13,510 We're now doing physics below the b quark mass scale, 936 00:58:13,510 --> 00:58:15,880 and this is just one way of seeing 937 00:58:15,880 --> 00:58:18,490 why it's not such a good parameter there, 938 00:58:18,490 --> 00:58:20,740 because the perturbative corrections are just too big. 939 00:58:23,960 --> 00:58:26,494 So what about this 1S mass? 940 00:58:31,240 --> 00:58:36,370 So here, it turns out that if you calculate delta m, 941 00:58:36,370 --> 00:58:37,320 you get-- 942 00:58:37,320 --> 00:58:39,820 in the same way that you think about corrections to hydrogen 943 00:58:39,820 --> 00:58:44,320 being order alpha squared, you get m b 1s alpha squared. 944 00:58:44,320 --> 00:58:47,290 And numerically, that's small enough 945 00:58:47,290 --> 00:58:49,840 when you put the coefficient in. 946 00:58:49,840 --> 00:58:54,340 So it still doesn't make us feel very good because it grows 947 00:58:54,340 --> 00:58:56,655 with m b, but if I just care about numerics 948 00:58:56,655 --> 00:58:58,030 and I'm doing b quark physics, it 949 00:58:58,030 --> 00:59:01,870 works because the alpha squared is enough suppression 950 00:59:01,870 --> 00:59:03,820 that this is a good mass scheme, and that's 951 00:59:03,820 --> 00:59:06,520 why the perturbation theory treating as order lambda 952 00:59:06,520 --> 00:59:11,050 QCD numerically, if not parametically. 953 00:59:14,830 --> 00:59:15,580 Is OK. 954 00:59:18,610 --> 00:59:22,330 Now, if that doesn't sit pretty with you, 955 00:59:22,330 --> 00:59:25,008 you can do something even more fancy, 956 00:59:25,008 --> 00:59:27,550 and we'll talk about one example of more fancy a little later 957 00:59:27,550 --> 00:59:31,480 on, where you basically, instead of having m show up there, 958 00:59:31,480 --> 00:59:36,132 you have some parameter R, and you have something 959 00:59:36,132 --> 00:59:37,090 that you can just pick. 960 00:59:41,410 --> 00:59:43,000 And so you could pick a scheme here 961 00:59:43,000 --> 00:59:46,990 where R is of order lambda QCD-- by however, 962 00:59:46,990 --> 00:59:52,600 maybe you pick it be 1 GV or 500 MeV or whatever you like. 963 00:59:52,600 --> 00:59:54,880 So you can make up a scheme where this is true, 964 00:59:54,880 --> 00:59:58,330 and then both parametrically and numerically you're OK. 965 00:59:58,330 --> 01:00:01,300 And these schemes also work just as well as the 1S scheme. 966 01:00:04,740 --> 01:00:06,750 So we can get more fancy and also satisfy 967 01:00:06,750 --> 01:00:10,290 our formal requirement. 968 01:00:10,290 --> 01:00:14,190 All right, so that's kind of phenomenological numbers 969 01:00:14,190 --> 01:00:16,300 and a bit of physics. 970 01:00:16,300 --> 01:00:19,140 Let's come back to some mathematics and talk about what 971 01:00:19,140 --> 01:00:20,985 renormalons are mathematically. 972 01:00:24,450 --> 01:00:26,418 Is there a mathematical way of characterizing 973 01:00:26,418 --> 01:00:27,210 what's going wrong? 974 01:00:27,210 --> 01:00:29,730 What if I didn't think about this physics? 975 01:00:29,730 --> 01:00:33,550 Could I do a calculation and see that something's going wrong? 976 01:00:33,550 --> 01:00:36,900 And the answer is yes. 977 01:00:36,900 --> 01:00:39,030 So first I have to teach you a few things, 978 01:00:39,030 --> 01:00:43,500 if you don't know them already, about the asymptotics 979 01:00:43,500 --> 01:00:48,820 of perturbative series in quantum field theory. 980 01:00:48,820 --> 01:00:52,335 And these are actually not convergent series. 981 01:01:01,930 --> 01:01:05,580 So they are what are called asymptotic series. 982 01:01:12,470 --> 01:01:16,020 So what's the definition of an asymptotic series? 983 01:01:16,020 --> 01:01:22,310 So we say a function has an asymmetric series, which 984 01:01:22,310 --> 01:01:24,170 I'll write as follows. 985 01:01:24,170 --> 01:01:26,000 Some coefficients. 986 01:01:26,000 --> 01:01:28,100 And we'll just call the expansion parameter alpha. 987 01:01:28,100 --> 01:01:30,600 You can think of it as alpha s, if you like, 988 01:01:30,600 --> 01:01:32,533 but this is just math, so you don't 989 01:01:32,533 --> 01:01:34,700 have to think about it as anything but the expansion 990 01:01:34,700 --> 01:01:35,720 parameter. 991 01:01:35,720 --> 01:01:39,860 And we say that a function has an asymptotic series if 992 01:01:39,860 --> 01:01:43,820 and only if the following is true. 993 01:01:43,820 --> 01:01:52,730 f of alpha minus the partial sum up to some level n 994 01:01:52,730 --> 01:01:57,110 is less than kind of the scaling that you 995 01:01:57,110 --> 01:02:04,610 would get from the next term in the series, which is n plus 2, 996 01:02:04,610 --> 01:02:11,300 for some numbers K N plus 2. 997 01:02:14,660 --> 01:02:18,050 So that's actually a quite different definition 998 01:02:18,050 --> 01:02:20,593 than what you would have for a convergent series. 999 01:02:20,593 --> 01:02:22,010 For a convergence areas, you would 1000 01:02:22,010 --> 01:02:24,820 say that you could pick any epsilon you like here. 1001 01:02:24,820 --> 01:02:26,660 You could make the N big enough that this 1002 01:02:26,660 --> 01:02:28,730 would get close to that. 1003 01:02:28,730 --> 01:02:31,520 Here, I'm just saying that it's less than this 1004 01:02:31,520 --> 01:02:33,480 with some power of alpha. 1005 01:02:33,480 --> 01:02:36,100 So imagine that I pick alpha to be 0.1. 1006 01:02:36,100 --> 01:02:40,130 The thing is that this K here could still grow with N, 1007 01:02:40,130 --> 01:02:42,470 and that actually will happen. 1008 01:02:42,470 --> 01:02:43,970 So the truncation area in some sense 1009 01:02:43,970 --> 01:02:46,400 is not being bounded in these asymptotic series. 1010 01:02:50,620 --> 01:02:52,780 And perturbation theory and quantum field theory 1011 01:02:52,780 --> 01:02:54,370 is generically of this structure, 1012 01:02:54,370 --> 01:02:56,610 and I'll show you some examples of that. 1013 01:03:00,230 --> 01:03:02,378 It's asymptotic rather than convergent. 1014 01:03:10,360 --> 01:03:13,615 So when you do a QFT calculation, 1015 01:03:13,615 --> 01:03:19,810 a kind of typical result if you work out some asymptotics 1016 01:03:19,810 --> 01:03:23,080 is that you would have these fn coefficients that 1017 01:03:23,080 --> 01:03:28,720 are some power of a, and then times an n factorial. 1018 01:03:31,300 --> 01:03:34,510 And that's kind of how they're scaling as n is getting large. 1019 01:03:34,510 --> 01:03:37,930 You can think about the number of diagrams as just growing, 1020 01:03:37,930 --> 01:03:40,180 but even when diagrams are not growing 1021 01:03:40,180 --> 01:03:41,490 you can get these n factorials. 1022 01:03:44,590 --> 01:03:49,630 So that means if you're fixing alpha as some value 1023 01:03:49,630 --> 01:03:53,710 of the parameter, and no matter how small you take it, 1024 01:03:53,710 --> 01:03:56,230 at some order in perturbation theory, 1025 01:03:56,230 --> 01:04:00,340 you're basically running out of gas, and your truncation-- 1026 01:04:00,340 --> 01:04:02,890 you start to-- 1027 01:04:02,890 --> 01:04:08,810 your predictions start to grow and diverge. 1028 01:04:08,810 --> 01:04:11,050 So the corresponding values of these K's, 1029 01:04:11,050 --> 01:04:13,030 if you had these f's that were of that form, 1030 01:04:13,030 --> 01:04:15,700 would be that the K's are basically growing 1031 01:04:15,700 --> 01:04:18,770 in the same way. 1032 01:04:18,770 --> 01:04:20,860 So you'd need larger and larger numbers in order 1033 01:04:20,860 --> 01:04:23,680 to satisfy the definition of asymptotic over there, 1034 01:04:23,680 --> 01:04:25,660 and having these large numbers you're never 1035 01:04:25,660 --> 01:04:29,320 able to satisfy convergent. 1036 01:04:29,320 --> 01:04:32,830 So the series has zero radius of convergence. 1037 01:04:40,063 --> 01:04:41,980 And then you wonder, why have we been teaching 1038 01:04:41,980 --> 01:04:43,022 you quantum field theory? 1039 01:04:43,022 --> 01:04:44,920 [CHUCKLES] That's true. 1040 01:04:47,540 --> 01:04:48,400 So what do we know? 1041 01:04:48,400 --> 01:04:50,440 Even though these series are asymptotic, 1042 01:04:50,440 --> 01:04:55,500 we can still make use of them. 1043 01:04:55,500 --> 01:05:00,500 So typically what happens is that the series will decrease. 1044 01:05:00,500 --> 01:05:02,390 For a while, it'll look like it's convergent, 1045 01:05:02,390 --> 01:05:04,760 and then it'll start diverging. 1046 01:05:04,760 --> 01:05:10,770 And you can characterize where it starts to diverge. 1047 01:05:10,770 --> 01:05:12,770 And I'm not going to go through all the algebra, 1048 01:05:12,770 --> 01:05:15,075 but I'll just tell you some facts. 1049 01:05:18,720 --> 01:05:20,220 And so you could imagine that you're 1050 01:05:20,220 --> 01:05:21,803 doing some perturbation series, and it 1051 01:05:21,803 --> 01:05:24,240 looks like it's going well. 1052 01:05:24,240 --> 01:05:26,050 Things are converging. 1053 01:05:26,050 --> 01:05:29,340 But then, [? some ?] [? things ?] don't start to go 1054 01:05:29,340 --> 01:05:30,520 so well. 1055 01:05:30,520 --> 01:05:32,130 So think of what I'm plotting here 1056 01:05:32,130 --> 01:05:34,440 as the partial sum of your perturbation 1057 01:05:34,440 --> 01:05:40,180 theory at N-th order. 1058 01:05:40,180 --> 01:05:43,860 I sum up all the connections up to N, so it's like the series 1059 01:05:43,860 --> 01:05:45,620 that I was writing over here. 1060 01:05:45,620 --> 01:05:47,730 It's this thing. 1061 01:05:47,730 --> 01:05:53,760 Now, in perturbative QED, perturbative QCD, 1062 01:05:53,760 --> 01:05:57,150 we're never getting more than a few terms. 1063 01:05:57,150 --> 01:05:58,920 So you could say, well, OK, maybe I 1064 01:05:58,920 --> 01:06:00,840 don't care so much about this. 1065 01:06:00,840 --> 01:06:02,460 And in QED, that's actually the case. 1066 01:06:02,460 --> 01:06:05,310 You don't really care so much about that. 1067 01:06:05,310 --> 01:06:07,020 But in QCD, actually, the turnover 1068 01:06:07,020 --> 01:06:10,538 happens already around three loops, and so as soon 1069 01:06:10,538 --> 01:06:13,080 as you start including two loop corrections this is something 1070 01:06:13,080 --> 01:06:14,038 you have to care about. 1071 01:06:16,590 --> 01:06:18,720 Now, you can say, well, the best thing I can do 1072 01:06:18,720 --> 01:06:21,990 is stop doing perturbation theory at some N star 1073 01:06:21,990 --> 01:06:24,270 because that's kind of where it looks the best. 1074 01:06:24,270 --> 01:06:27,430 Like, I've gone toward something in it. 1075 01:06:27,430 --> 01:06:30,410 And that actually is not such a bad thing. 1076 01:06:30,410 --> 01:06:32,490 And it turns out you can characterize actually 1077 01:06:32,490 --> 01:06:35,160 the mistake you're making by stopping there, 1078 01:06:35,160 --> 01:06:37,050 and the mistake you're making by stopping 1079 01:06:37,050 --> 01:06:40,140 there is of the following form. 1080 01:06:44,790 --> 01:06:51,670 So it's an exponential in 1 over alpha. 1081 01:06:55,690 --> 01:07:00,360 So you would never be able to see kind of the correction 1082 01:07:00,360 --> 01:07:03,420 that you'd need to get to the correct value, which I'm 1083 01:07:03,420 --> 01:07:06,690 imagining is on the axis here, in perturbation theory 1084 01:07:06,690 --> 01:07:09,450 because this doesn't have a perturbative expansion. 1085 01:07:09,450 --> 01:07:12,570 And I'll actually show you that kind of the bad behavior 1086 01:07:12,570 --> 01:07:15,990 here and this gap are related to power corrections. 1087 01:07:15,990 --> 01:07:18,390 And they're exactly related to this connection 1088 01:07:18,390 --> 01:07:21,180 that I was telling you between perturbative corrections 1089 01:07:21,180 --> 01:07:24,510 and large-order asymptotics and power corrections. 1090 01:07:24,510 --> 01:07:27,073 This type of exponential is something 1091 01:07:27,073 --> 01:07:28,740 that we can relate to power corrections, 1092 01:07:28,740 --> 01:07:31,830 and we'll see how that pans out, although we might-- 1093 01:07:31,830 --> 01:07:35,110 probably won't get there until next class. 1094 01:07:35,110 --> 01:07:37,950 OK, so this is kind of a prelude, 1095 01:07:37,950 --> 01:07:40,230 and now we're going to go into more detail. 1096 01:07:40,230 --> 01:07:43,050 And it turns out, in order to go into more detail here, 1097 01:07:43,050 --> 01:07:46,050 that, much as you make Fourier transforms 1098 01:07:46,050 --> 01:07:48,240 to explore another space, we're going 1099 01:07:48,240 --> 01:07:51,045 to do a transform to something called Borel space. 1100 01:08:01,420 --> 01:08:03,940 So it turns out that, when you have 1101 01:08:03,940 --> 01:08:07,870 a divergent series or an asymptotic series, 1102 01:08:07,870 --> 01:08:10,930 there's still degrees of divergence. 1103 01:08:10,930 --> 01:08:12,940 And we can classify how divergent 1104 01:08:12,940 --> 01:08:16,479 it is by using something called the Borel transform. 1105 01:08:29,529 --> 01:08:30,340 Yeah? 1106 01:08:30,340 --> 01:08:36,130 AUDIENCE: [INAUDIBLE] will the series still be asymptotic? 1107 01:08:36,130 --> 01:08:39,580 IAIN STEWART: So no, it's not. 1108 01:08:39,580 --> 01:08:41,950 Yeah, so 1S mass-- 1109 01:08:41,950 --> 01:08:43,420 let's see. 1110 01:08:43,420 --> 01:08:49,902 Yeah, it is still asymptotic, but it's-- 1111 01:08:49,902 --> 01:08:51,819 in a way that I'll describe once we understand 1112 01:08:51,819 --> 01:08:55,960 what this Borel is, it's much less divergent 1113 01:08:55,960 --> 01:08:58,623 than the other series, than the pole series. 1114 01:08:58,623 --> 01:09:00,040 If you think about the pole series 1115 01:09:00,040 --> 01:09:04,479 as being degree-0 divergent, then this is like a few orders 1116 01:09:04,479 --> 01:09:06,880 down less divergent. 1117 01:09:06,880 --> 01:09:10,976 So once we define less divergent of a divergent series, 1118 01:09:10,976 --> 01:09:12,809 then I'll be able to make that more precise. 1119 01:09:18,477 --> 01:09:20,560 It's a question of when power corrections come in. 1120 01:09:20,560 --> 01:09:22,420 And if you look at, like-- so physically, 1121 01:09:22,420 --> 01:09:26,800 the 1S mass is a physical thing for an epsilon state. 1122 01:09:26,800 --> 01:09:28,970 If you look at how power corrections come in, 1123 01:09:28,970 --> 01:09:31,330 then they're suppressed because there's 1124 01:09:31,330 --> 01:09:33,580 an ambiguity between power corrections in perturbation 1125 01:09:33,580 --> 01:09:33,850 theory. 1126 01:09:33,850 --> 01:09:35,260 If the power corrections are suppressed, 1127 01:09:35,260 --> 01:09:37,069 you have less ambiguity in the perturbation theory. 1128 01:09:37,069 --> 01:09:38,930 That's another way of thinking about it. 1129 01:09:38,930 --> 01:09:41,359 But the answer to your question. 1130 01:09:41,359 --> 01:09:45,960 AUDIENCE: [INAUDIBLE] 1131 01:09:45,960 --> 01:09:48,430 IAIN STEWART: It basically affects this a. 1132 01:09:54,530 --> 01:09:56,070 All right, lots of good questions. 1133 01:09:56,070 --> 01:09:58,860 Let's see how we get there. 1134 01:09:58,860 --> 01:10:03,200 So when I transform, I'm going to call the transform function 1135 01:10:03,200 --> 01:10:06,920 capital F, and I'm going to call its argument B for Borel. 1136 01:10:06,920 --> 01:10:10,760 And so what's the definition of the Borel transform? 1137 01:10:10,760 --> 01:10:13,740 We're going to define it as follows. 1138 01:10:13,740 --> 01:10:16,790 There's a first-order term, and I'm just going 1139 01:10:16,790 --> 01:10:19,220 to put a delta function there. 1140 01:10:19,220 --> 01:10:21,080 The real thing that matters is the series 1141 01:10:21,080 --> 01:10:23,510 of terms that come next. 1142 01:10:23,510 --> 01:10:27,080 And instead of having F to the n alpha to the n, 1143 01:10:27,080 --> 01:10:29,970 I'm going to divide by an extra n factorial, 1144 01:10:29,970 --> 01:10:31,910 and that's the definition of, given 1145 01:10:31,910 --> 01:10:34,310 a set of F's, how I construct what's 1146 01:10:34,310 --> 01:10:36,410 called the Borel transform. 1147 01:10:36,410 --> 01:10:44,980 And because of the n factorial, I get improved convergence, 1148 01:10:44,980 --> 01:10:46,980 so the n factorial is making it converge better. 1149 01:10:50,570 --> 01:10:54,610 So if there's a transform, there should be an inverse transform. 1150 01:10:54,610 --> 01:10:57,820 So here's the inverse transform. 1151 01:10:57,820 --> 01:11:04,325 It's an integral from 0 to infinity, b over alpha F of b. 1152 01:11:16,546 --> 01:11:19,930 And that would get me back to the F. 1153 01:11:19,930 --> 01:11:22,990 So if you have a convergent series 1154 01:11:22,990 --> 01:11:24,650 and you think of this transform, then 1155 01:11:24,650 --> 01:11:26,650 you just can go back and forth, and you get back 1156 01:11:26,650 --> 01:11:28,930 to the original-- the function that you started with. 1157 01:11:34,150 --> 01:11:38,000 So if you have some series that's like that, 1158 01:11:38,000 --> 01:11:41,090 you get back-- 1159 01:11:41,090 --> 01:11:42,920 you could calculate that series, and you 1160 01:11:42,920 --> 01:11:46,400 get some F of alpha just before Borel transforming. 1161 01:11:46,400 --> 01:11:47,930 If you do this Borel transform, then 1162 01:11:47,930 --> 01:11:51,110 you calculate this integral, you get back 1163 01:11:51,110 --> 01:11:56,510 the same F of alpha from the inverse transform. 1164 01:12:05,090 --> 01:12:08,990 Let me give you an example that shows you 1165 01:12:08,990 --> 01:12:12,710 that, if you have a divergent series, 1166 01:12:12,710 --> 01:12:15,890 it's not too crazy to define the sum of the divergent series 1167 01:12:15,890 --> 01:12:19,160 by using these transforms. 1168 01:12:19,160 --> 01:12:22,430 So I claim that, for a divergent series, 1169 01:12:22,430 --> 01:12:27,825 where F of b and the inverse transform exist-- 1170 01:12:27,825 --> 01:12:29,450 so if we start with a divergent series, 1171 01:12:29,450 --> 01:12:35,480 but these things exist, then we could just 1172 01:12:35,480 --> 01:12:39,110 use this transform as a way of defining F of alpha. 1173 01:12:53,820 --> 01:12:55,520 So let me give you a simple example 1174 01:12:55,520 --> 01:12:57,350 of that to convince you that it's not 1175 01:12:57,350 --> 01:12:59,160 such a bad thing to do. 1176 01:12:59,160 --> 01:13:00,785 So let's consider the following series. 1177 01:13:09,260 --> 01:13:12,570 Just an alternating series, but alpha 1178 01:13:12,570 --> 01:13:13,820 is a parameter greater than 1. 1179 01:13:16,370 --> 01:13:19,438 That series doesn't converge. 1180 01:13:19,438 --> 01:13:21,605 Of course, for alpha less than 1, it would converge. 1181 01:13:27,368 --> 01:13:28,910 And if I asked you what it should be, 1182 01:13:28,910 --> 01:13:31,202 you would say, well, I calculate for alpha less than 1, 1183 01:13:31,202 --> 01:13:34,970 I analytically continue, and it's 1 plus alpha. 1184 01:13:34,970 --> 01:13:36,840 Well, that's one way of getting there. 1185 01:13:36,840 --> 01:13:40,890 Another way of getting there is using this Borel transform. 1186 01:13:40,890 --> 01:13:47,600 So if you calculate it here, F of b, 1187 01:13:47,600 --> 01:13:52,585 you sum up 0 to infinity minus b to the n-- 1188 01:13:52,585 --> 01:13:53,960 once you put the minus 1 in there 1189 01:13:53,960 --> 01:13:56,820 and there's an extra n factorial. 1190 01:13:56,820 --> 01:14:00,020 So that's e to the minus b, perfectly well-behaved 1191 01:14:00,020 --> 01:14:01,460 function. 1192 01:14:01,460 --> 01:14:05,380 And then you have to do an interval, 0 to infinity, db. 1193 01:14:05,380 --> 01:14:09,500 You could have e to the minus b over alpha times the F of b, 1194 01:14:09,500 --> 01:14:12,860 which is e to the minus b, and that just gives 1195 01:14:12,860 --> 01:14:15,020 you alpha over 1 plus alpha. 1196 01:14:18,705 --> 01:14:20,580 And this is a perfectly well-defined interval 1197 01:14:20,580 --> 01:14:23,600 for large alpha. 1198 01:14:23,600 --> 01:14:25,670 Large alpha is even making it better. 1199 01:14:31,850 --> 01:14:33,600 So the integral is perfectly well-defined, 1200 01:14:33,600 --> 01:14:36,240 and we get an answer that we like. 1201 01:14:45,630 --> 01:14:48,960 OK, so this Borel transform is a useful way 1202 01:14:48,960 --> 01:14:52,095 of dealing with divergent series. 1203 01:15:01,033 --> 01:15:02,450 The real question is, what happens 1204 01:15:02,450 --> 01:15:04,310 if the inverse transform doesn't exist? 1205 01:15:13,202 --> 01:15:16,100 And that's what's going to be the thing that we're 1206 01:15:16,100 --> 01:15:16,910 most interested in. 1207 01:15:34,122 --> 01:15:36,330 So if the F of the integral, integral over the F of b 1208 01:15:36,330 --> 01:15:43,620 doesn't exist, then the integrand 1209 01:15:43,620 --> 01:16:02,820 can tell us about the severity of the singularities, I.e., 1210 01:16:02,820 --> 01:16:04,697 the severity of the divergence. 1211 01:16:08,680 --> 01:16:12,020 And that's the real power of this Borel method. 1212 01:16:12,020 --> 01:16:14,390 So let's do another example, where actually things 1213 01:16:14,390 --> 01:16:15,830 will diverge. 1214 01:16:15,830 --> 01:16:21,580 So let me take F n to be a to the minus n, n plus k 1215 01:16:21,580 --> 01:16:24,050 factorial. 1216 01:16:24,050 --> 01:16:26,140 And if we do that Borel transform of that-- 1217 01:16:28,660 --> 01:16:32,653 if you have a series, you can do it. 1218 01:16:32,653 --> 01:16:34,820 And there's only one piece of it that we care about, 1219 01:16:34,820 --> 01:16:36,403 so I'm just going to write that piece. 1220 01:16:40,660 --> 01:16:52,130 So there's this piece that has a kind of pole-like structure, 1221 01:16:52,130 --> 01:16:53,840 and it has a pole at b equals a. 1222 01:16:56,940 --> 01:16:59,760 And what you say is you call this pole a renormalon. 1223 01:16:59,760 --> 01:17:03,450 This is the renormalon. 1224 01:17:03,450 --> 01:17:05,778 You call it a b equals a renromalon. 1225 01:17:09,235 --> 01:17:10,860 So you're characterizing the renormalon 1226 01:17:10,860 --> 01:17:13,120 by where that pole is. 1227 01:17:13,120 --> 01:17:14,940 A renormalon is a flavor of a particle. 1228 01:17:14,940 --> 01:17:16,998 This has nothing to do with a particle. 1229 01:17:16,998 --> 01:17:18,540 Usually, you think of poles as having 1230 01:17:18,540 --> 01:17:19,890 something to do with particles, and that's 1231 01:17:19,890 --> 01:17:21,000 where the name comes from. 1232 01:17:21,000 --> 01:17:21,780 It's due to 't Hooft. 1233 01:17:21,780 --> 01:17:22,290 Blame him. 1234 01:17:22,290 --> 01:17:26,360 [CHUCKLES] 1235 01:17:26,360 --> 01:17:29,780 So if a is less than 0, your integration contour 1236 01:17:29,780 --> 01:17:32,550 is positive, and the pole is on the other side, 1237 01:17:32,550 --> 01:17:34,340 so you don't have a problem. 1238 01:17:34,340 --> 01:17:35,840 The inverse transform exists. 1239 01:17:42,710 --> 01:17:44,945 These are called UV renormalons. 1240 01:17:44,945 --> 01:17:46,820 They still can guide the perturbation theory, 1241 01:17:46,820 --> 01:17:48,650 and they can be important for thinking 1242 01:17:48,650 --> 01:17:50,150 about why the perturbation theory is 1243 01:17:50,150 --> 01:17:53,420 behaving the way it is, but you can do the inverse transform. 1244 01:17:53,420 --> 01:17:56,300 The real ones that are kind of problematic 1245 01:17:56,300 --> 01:18:00,080 are a greater than 0 because then the pole 1246 01:18:00,080 --> 01:18:04,505 is on the integration contour, and we can't do the transform. 1247 01:18:12,930 --> 01:18:14,987 So we're integrating d b from 0 to infinity, 1248 01:18:14,987 --> 01:18:17,570 and we just have a pole sitting on the axis, so what do we do? 1249 01:18:42,175 --> 01:18:43,800 So you could think about characterizing 1250 01:18:43,800 --> 01:18:46,200 how poorly behaved a perturbation theory is 1251 01:18:46,200 --> 01:18:48,330 by looking at these poles, and that's 1252 01:18:48,330 --> 01:18:51,570 kind of what I meant by having some kind of notion of how 1253 01:18:51,570 --> 01:18:53,730 divergent a series is. 1254 01:18:53,730 --> 01:19:01,215 So you look in this Borel space, and you look at the real axis, 1255 01:19:01,215 --> 01:19:06,300 and those poles actually can be on both sides. 1256 01:19:06,300 --> 01:19:11,490 The most severe pole is the one that's closest to the origin, 1257 01:19:11,490 --> 01:19:14,130 and that has the smallest value of b. 1258 01:19:14,130 --> 01:19:17,310 And you can see if you have a small value of b, 1259 01:19:17,310 --> 01:19:20,200 then you're multiplying here with some large numbers. 1260 01:19:20,200 --> 01:19:24,180 So if a was 0.1, you'd be multiplying by 10 to the n. 1261 01:19:24,180 --> 01:19:25,770 So these guys are more severe. 1262 01:19:25,770 --> 01:19:29,100 As you go to larger values, you get additional suppression 1263 01:19:29,100 --> 01:19:31,860 from this a to the n, so if a was-- 1264 01:19:31,860 --> 01:19:34,860 if you went all the way out to 100, then you'd have 1 over 100 1265 01:19:34,860 --> 01:19:35,550 to the n. 1266 01:19:35,550 --> 01:19:39,100 That's good, but factorial eventually wins. 1267 01:19:39,100 --> 01:19:42,640 So it's still divergent, but we look like it was a lot better. 1268 01:19:42,640 --> 01:19:43,140 OK? 1269 01:19:43,140 --> 01:19:46,530 So you can characterize how severe the series is diverging 1270 01:19:46,530 --> 01:19:49,540 by the location of these poles. 1271 01:19:49,540 --> 01:19:55,770 And you can characterize the ambiguity 1272 01:19:55,770 --> 01:19:58,650 in doing an integration by-- 1273 01:19:58,650 --> 01:20:00,972 if you just think about one pole, 1274 01:20:00,972 --> 01:20:02,430 you can characterize the ambiguity, 1275 01:20:02,430 --> 01:20:05,010 so let's just imagine there's one pole. 1276 01:20:05,010 --> 01:20:06,570 You can characterize the ambiguity 1277 01:20:06,570 --> 01:20:09,420 by going above or below. 1278 01:20:09,420 --> 01:20:13,770 So you have two possible ways of defining the interval. 1279 01:20:13,770 --> 01:20:15,720 You don't know which to pick. 1280 01:20:15,720 --> 01:20:16,710 One is to go above. 1281 01:20:19,530 --> 01:20:21,270 The other is to go below. 1282 01:20:27,600 --> 01:20:31,140 And then you can think about what the ambiguity is. 1283 01:20:31,140 --> 01:20:36,990 And the ambiguity is the contour 1. 1284 01:20:36,990 --> 01:20:38,620 Call this c1. 1285 01:20:38,620 --> 01:20:39,972 This is c2. 1286 01:20:39,972 --> 01:20:45,570 The ambiguity is c1 minus c2, and that is circling the pole. 1287 01:20:49,340 --> 01:20:52,220 So you can look at the residue of the pole, 1288 01:20:52,220 --> 01:20:53,750 and that gives you the ambiguity. 1289 01:21:02,100 --> 01:21:04,440 So what we'll do next time is we'll 1290 01:21:04,440 --> 01:21:06,900 look at an example in perturbative QCD. 1291 01:21:06,900 --> 01:21:09,780 We'll see that there's a series that has a renormalon, 1292 01:21:09,780 --> 01:21:10,752 and we'll find it. 1293 01:21:10,752 --> 01:21:12,210 And then we'll look at the residue, 1294 01:21:12,210 --> 01:21:16,710 and we'll find actually that out will pop lambda QCD. 1295 01:21:16,710 --> 01:21:20,490 So we'll see, we'll calculate, for some perturbative series, 1296 01:21:20,490 --> 01:21:23,100 that the pole mass really has an ambiguity of order lambda 1297 01:21:23,100 --> 01:21:26,460 QCD by using this technique of going around 1298 01:21:26,460 --> 01:21:27,360 a residue of a pole. 1299 01:21:32,410 --> 01:21:33,258 OK? 1300 01:21:33,258 --> 01:21:35,800 AUDIENCE: [INAUDIBLE] when we say that the b quark has a 4 GV 1301 01:21:35,800 --> 01:21:37,940 mass, what's the mass? 1302 01:21:37,940 --> 01:21:38,440 [INAUDIBLE] 1303 01:21:38,440 --> 01:21:40,810 IAIN STEWART: Yeah, so if you say the b quark has, 1304 01:21:40,810 --> 01:21:43,600 like, 4.2 GV mass, that's the m s bar mass. 1305 01:21:43,600 --> 01:21:47,770 If you say it has a 4.73 GV mass, 1306 01:21:47,770 --> 01:21:49,660 that's like the m b 1S mass. 1307 01:21:49,660 --> 01:21:52,325 And if you look at how the PDG measures the b quark mass, 1308 01:21:52,325 --> 01:21:53,950 they measure it in one of these schemes 1309 01:21:53,950 --> 01:21:55,690 that I'm telling you about, like the 1S scheme. 1310 01:21:55,690 --> 01:21:57,190 They [INAUDIBLE] result for that. 1311 01:21:57,190 --> 01:22:00,970 You can convert between the 1S and the m s bar mass. 1312 01:22:00,970 --> 01:22:03,550 The m s bar mass is a perfectly well-defined quantity 1313 01:22:03,550 --> 01:22:07,390 it doesn't actually have a renormalon, OK? 1314 01:22:07,390 --> 01:22:08,530 That's not its problem. 1315 01:22:08,530 --> 01:22:10,420 The pole mass has a renormalon. 1316 01:22:10,420 --> 01:22:13,960 The m s bar mass, its problem is related to power counting 1317 01:22:13,960 --> 01:22:15,730 in the low-energy theory. 1318 01:22:15,730 --> 01:22:17,560 It's a good mass for high-energy physics. 1319 01:22:17,560 --> 01:22:19,257 It's a bad mass for low-energy physics. 1320 01:22:19,257 --> 01:22:20,590 That's the problem with m s bar. 1321 01:22:20,590 --> 01:22:23,050 It's not it's not a technical problem, that it has a-- 1322 01:22:23,050 --> 01:22:27,100 [INAUDIBLE] severity of renormalon. 1323 01:22:27,100 --> 01:22:28,870 It can have a higher-order renormalon, 1324 01:22:28,870 --> 01:22:30,995 but it doesn't have the same severity of renormalon 1325 01:22:30,995 --> 01:22:32,230 as the pole mass. 1326 01:22:32,230 --> 01:22:34,180 The pole mass basically has the most severe 1327 01:22:34,180 --> 01:22:39,240 possible renormalon, which is, in some units, 1/2, 1328 01:22:39,240 --> 01:22:40,707 so there's an extra 2 to the n. 1329 01:22:40,707 --> 01:22:42,040 We'll talk about that next time. 1330 01:22:45,972 --> 01:22:48,430 So basically, it's the u equals 1/2 renormalons that really 1331 01:22:48,430 --> 01:22:50,677 are causing a problem in perturbative [? QCD ?] 1332 01:22:50,677 --> 01:22:52,510 because they're the most important-- there's 1333 01:22:52,510 --> 01:22:53,990 higher-order ones. 1334 01:22:53,990 --> 01:22:58,060 Even alpha s actually has a normalon, 1335 01:22:58,060 --> 01:23:00,760 alpha s in the m s bar scheme, I suspect. 1336 01:23:00,760 --> 01:23:05,420 [CHUCKLES] But even up to five loops you won't see it, 1337 01:23:05,420 --> 01:23:07,710 or it's very hard to see. 1338 01:23:07,710 --> 01:23:09,500 Let's stop there.