1 00:00:00,000 --> 00:00:02,520 The following content is provided under a Creative 2 00:00:02,520 --> 00:00:03,970 Commons license. 3 00:00:03,970 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,660 continue to offer high quality educational resources for free. 5 00:00:10,660 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,160 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,160 --> 00:00:18,370 at ocw.mit.edu. 8 00:00:21,890 --> 00:00:24,680 IAIN STEWART: [INAUDIBLE] effective theory workshop. 9 00:00:24,680 --> 00:00:26,660 So last time we were talking about renormalons, 10 00:00:26,660 --> 00:00:29,030 and we're going to continue to do that today. 11 00:00:29,030 --> 00:00:31,850 So we said that typically in quantum field theory 12 00:00:31,850 --> 00:00:34,060 you have perturbation theories, but actually they're 13 00:00:34,060 --> 00:00:34,970 asymptotic. 14 00:00:34,970 --> 00:00:38,420 And the kind of growth that you might expect as some power. 15 00:00:38,420 --> 00:00:41,840 I think I use two different conventions for this power, 16 00:00:41,840 --> 00:00:43,590 whether it's plus n or minus n, but that's 17 00:00:43,590 --> 00:00:48,390 just a goes to 1 over a, and there's a factorial growth. 18 00:00:48,390 --> 00:00:52,460 So we said we could characterize those series 19 00:00:52,460 --> 00:00:53,990 by thinking about doing something 20 00:00:53,990 --> 00:00:55,490 called a Borel transform, and that 21 00:00:55,490 --> 00:00:56,690 makes them more convergent. 22 00:00:56,690 --> 00:00:58,523 Because, basically, we take the coefficients 23 00:00:58,523 --> 00:01:00,650 and we divide the coefficients by an n factorial. 24 00:01:00,650 --> 00:01:03,110 So we get more conversion series. 25 00:01:03,110 --> 00:01:05,810 And it may be that this thing exists, 26 00:01:05,810 --> 00:01:09,260 or it may be that this thing exists and has poles. 27 00:01:09,260 --> 00:01:12,440 And if it has poles, as I've indicated here 28 00:01:12,440 --> 00:01:15,350 on the real axis, then the inverse transform 29 00:01:15,350 --> 00:01:16,342 might not exist. 30 00:01:16,342 --> 00:01:18,800 So even though this thing is something that you can define, 31 00:01:18,800 --> 00:01:20,720 you can't get back to f. 32 00:01:20,720 --> 00:01:22,730 And if that's the case, you can characterize 33 00:01:22,730 --> 00:01:26,150 how the original series was diverging based on those poles. 34 00:01:26,150 --> 00:01:27,890 And those poles are called renormalons. 35 00:01:33,020 --> 00:01:36,690 So if we could calculate to all orders in perturbation theory 36 00:01:36,690 --> 00:01:40,100 and some in quantum field theory, like a gauge theory, 37 00:01:40,100 --> 00:01:41,870 then we could just calculate these f's. 38 00:01:41,870 --> 00:01:44,360 We would find that actually there is this factorial growth, 39 00:01:44,360 --> 00:01:46,190 and we'd be able to do this. 40 00:01:46,190 --> 00:01:49,190 But we can't do that, because we don't know 41 00:01:49,190 --> 00:01:51,120 how to do all the diagrams. 42 00:01:51,120 --> 00:01:53,720 So in order to give you an explicit example of what 43 00:01:53,720 --> 00:01:55,730 we talked about last time, instead 44 00:01:55,730 --> 00:01:58,910 of looking at all the diagrams, we pick a particular subset 45 00:01:58,910 --> 00:01:59,900 that we can identify. 46 00:01:59,900 --> 00:02:02,200 And that's what we're going to do today, 47 00:02:02,200 --> 00:02:03,680 at least start by doing today. 48 00:02:10,370 --> 00:02:12,260 Part of our discussion of motivating 49 00:02:12,260 --> 00:02:15,110 why one might want to think about renormalons 50 00:02:15,110 --> 00:02:18,920 was thinking about the pole mass versus the MS bar mass 51 00:02:18,920 --> 00:02:22,650 and showing you that phenomenology 52 00:02:22,650 --> 00:02:24,810 starts to work better in terms of the MS bar mass. 53 00:02:24,810 --> 00:02:26,310 And then we even further than that 54 00:02:26,310 --> 00:02:28,810 and said we should talk about this thing called the 1S mass, 55 00:02:28,810 --> 00:02:31,080 and we'll come back to that discussion today. 56 00:02:31,080 --> 00:02:33,840 But I'd like to consider just the explicit calculation 57 00:02:33,840 --> 00:02:36,330 of the pole versus the MS bar mass 58 00:02:36,330 --> 00:02:38,749 and show you that you can see this renormalon. 59 00:02:47,240 --> 00:02:52,640 So if we want to think about the definition of these masses, 60 00:02:52,640 --> 00:02:56,510 then we have to implement those definitions order by order 61 00:02:56,510 --> 00:02:59,600 in perturbation theory. 62 00:02:59,600 --> 00:03:01,770 And order by order in perturbation theory, 63 00:03:01,770 --> 00:03:02,900 we could calculate the f's. 64 00:03:02,900 --> 00:03:05,840 But we need to know something about the entire series, 65 00:03:05,840 --> 00:03:10,430 and that is too hard to get the complete series. 66 00:03:18,100 --> 00:03:22,180 So we'll pick out an infinite subset of that complete series 67 00:03:22,180 --> 00:03:23,940 that we could identify which is unique. 68 00:03:37,080 --> 00:03:40,080 And of course we should pick an infinite subset that's 69 00:03:40,080 --> 00:03:41,980 the easiest to calculate. 70 00:03:41,980 --> 00:03:44,640 And the one that we can pick out that's easy to calculate 71 00:03:44,640 --> 00:03:48,170 is the following-- 72 00:03:48,170 --> 00:03:49,870 we look at so-called bubble sum. 73 00:03:56,570 --> 00:04:00,170 So graphs with a bubble, and then more and more of them. 74 00:04:09,810 --> 00:04:10,310 OK. 75 00:04:10,310 --> 00:04:12,060 So the reason that these graphs are unique 76 00:04:12,060 --> 00:04:14,690 is because if you look at any order in perturbation theory, 77 00:04:14,690 --> 00:04:16,940 they're the grass with the most factors 78 00:04:16,940 --> 00:04:19,022 of the number of fermions running around 79 00:04:19,022 --> 00:04:19,730 in these bubbles. 80 00:04:19,730 --> 00:04:21,897 So if you call the number of fermions running around 81 00:04:21,897 --> 00:04:25,760 in these bubbles nf, this graph has the most powers of nf. 82 00:04:33,690 --> 00:04:37,110 So at each order in perturbation theory, it's unique. 83 00:04:37,110 --> 00:04:38,670 It's a unique contribution. 84 00:04:38,670 --> 00:04:40,110 And it's pretty easy to calculate, 85 00:04:40,110 --> 00:04:42,360 because there's only one type of diagram contributing. 86 00:04:42,360 --> 00:04:43,693 AUDIENCE: What does unique mean? 87 00:04:43,693 --> 00:04:44,730 [INAUDIBLE] 88 00:04:44,730 --> 00:04:47,560 IAIN STEWART: Unique means that there's no other diagram. 89 00:04:47,560 --> 00:04:49,560 So you might think at each other in perturbation 90 00:04:49,560 --> 00:04:53,400 theory there's lots and lots of diagrams. 91 00:04:53,400 --> 00:04:55,290 There's only this diagram that contributes 92 00:04:55,290 --> 00:04:56,850 to the highest power nf. 93 00:04:56,850 --> 00:04:59,310 It's also gauge invariant, but it's really 94 00:04:59,310 --> 00:05:02,580 a unique diagram that contributes to that color 95 00:05:02,580 --> 00:05:04,290 slash flavor structure. 96 00:05:08,360 --> 00:05:08,860 OK. 97 00:05:08,860 --> 00:05:11,650 So obviously an ingredient in this is the bubble itself. 98 00:05:14,450 --> 00:05:21,790 So if we go back to the bubble plus it's counter term, 99 00:05:21,790 --> 00:05:35,110 and we say there's a momentum p going through it, 100 00:05:35,110 --> 00:05:41,310 it's diagonal in color, transverse in momentum. 101 00:05:41,310 --> 00:05:48,510 It has a factor of an ns nf, some other numerical factors. 102 00:05:48,510 --> 00:05:54,485 There's a logarithm, there' constant, 103 00:05:54,485 --> 00:05:55,860 and then there's some divergence, 104 00:05:55,860 --> 00:05:58,030 but we cancelled the divergence with a counter. 105 00:05:58,030 --> 00:06:01,590 So the graph would look like that. 106 00:06:05,120 --> 00:06:07,220 And I'm going to absorb all these things 107 00:06:07,220 --> 00:06:09,828 into a single logarithm just for convenience. 108 00:06:15,090 --> 00:06:19,200 So I define the constant c bar, which is this 5/3, 109 00:06:19,200 --> 00:06:22,250 and then I just put it into the logarithm. 110 00:06:25,730 --> 00:06:26,230 OK. 111 00:06:26,230 --> 00:06:28,520 So then this is a geometric series, 112 00:06:28,520 --> 00:06:31,070 so it's easy to deal with. 113 00:06:31,070 --> 00:06:40,240 So G mu nu, which is as a function of p and alpha 114 00:06:40,240 --> 00:06:51,490 s which sums up this chain plus counterterm 115 00:06:51,490 --> 00:06:55,990 diagrams where there's n bubbles here, 116 00:06:55,990 --> 00:06:58,150 and we start with no bubbles. 117 00:07:08,230 --> 00:07:09,400 This n is a projector. 118 00:07:38,280 --> 00:07:39,090 OK. 119 00:07:39,090 --> 00:07:41,940 So this is explicitly writing out the sum. 120 00:07:41,940 --> 00:07:44,220 And what I've done here is I've used the fact 121 00:07:44,220 --> 00:07:46,740 that even though I wrote up nf up there, 122 00:07:46,740 --> 00:07:48,950 I can convert it to beta 0. 123 00:07:48,950 --> 00:07:55,740 So beta 0 was 11/3 C a minus 2/3 nf. 124 00:07:55,740 --> 00:08:00,300 I can solve that equation for nf and substitute it back in. 125 00:08:00,300 --> 00:08:02,580 And that is a convenient thing to do, 126 00:08:02,580 --> 00:08:04,800 just because you can think that these terms are then 127 00:08:04,800 --> 00:08:06,912 related to the beta 0, which has to do 128 00:08:06,912 --> 00:08:07,995 with the running coupling. 129 00:08:10,883 --> 00:08:12,300 In some sense, what we're doing is 130 00:08:12,300 --> 00:08:15,600 we're saying that even though we just calculated these nf 131 00:08:15,600 --> 00:08:17,100 contributions, we know there's going 132 00:08:17,100 --> 00:08:20,070 to be some non-Abelian contribution that comes along 133 00:08:20,070 --> 00:08:22,800 with it that makes this really into a beta 0. 134 00:08:22,800 --> 00:08:24,840 Or another way of saying it is that if you think 135 00:08:24,840 --> 00:08:27,780 about characterizing color structures, 136 00:08:27,780 --> 00:08:32,909 you could write nf C a, nf f squared. 137 00:08:32,909 --> 00:08:37,900 Or you could write beta 0 C a, beta 0 squared, 138 00:08:37,900 --> 00:08:41,520 and these are equivalent ways of representing 139 00:08:41,520 --> 00:08:43,200 the different kinds of color structures 140 00:08:43,200 --> 00:08:45,150 that can come in at higher order. 141 00:08:45,150 --> 00:08:48,660 And we're just choosing to do it this way with the beta 0, 142 00:08:48,660 --> 00:08:51,030 rather than the nf's. 143 00:08:51,030 --> 00:08:52,800 So we convert our nf to a beta 0. 144 00:08:56,910 --> 00:09:00,480 And it's still a unique subset of the entire set of terms. 145 00:09:00,480 --> 00:09:03,240 The terms we're not including our terms like this-- nf C a 146 00:09:03,240 --> 00:09:03,975 or C a squared. 147 00:09:06,600 --> 00:09:09,240 And we're just keeping these guys, which 148 00:09:09,240 --> 00:09:10,500 is equivalent to this guy. 149 00:09:13,113 --> 00:09:14,030 Foes that makes sense? 150 00:09:16,740 --> 00:09:17,800 Everybody happy? 151 00:09:17,800 --> 00:09:19,550 AUDIENCE: [INAUDIBLE] photon [INAUDIBLE].. 152 00:09:19,550 --> 00:09:20,240 IAIN STEWART: Sorry. 153 00:09:20,240 --> 00:09:21,270 AUDIENCE: Where are the photon propagators? 154 00:09:21,270 --> 00:09:22,812 IAIN STEWART: The photon propagators. 155 00:09:22,812 --> 00:09:26,958 So the photon propagators all cancel each other out. 156 00:09:26,958 --> 00:09:29,000 So you get a piece greater than 1 over p squared, 157 00:09:29,000 --> 00:09:31,840 p squared 1 over p squared. 158 00:09:31,840 --> 00:09:34,117 So the whole thing looks like a photon propagator, 159 00:09:34,117 --> 00:09:35,200 just with a bunch of logs. 160 00:09:40,440 --> 00:09:41,490 Any other questions? 161 00:09:45,440 --> 00:09:47,500 All right. 162 00:09:47,500 --> 00:09:49,810 So this is already of the structure 163 00:09:49,810 --> 00:09:53,980 where we could think about doing the Borel transform. 164 00:09:53,980 --> 00:09:55,630 Because we have a series in alpha s, 165 00:09:55,630 --> 00:09:58,210 and what the Borel transport meant was taking that series 166 00:09:58,210 --> 00:10:01,330 and converting it to a series in the Borel variable, 167 00:10:01,330 --> 00:10:03,600 just dividing by n factorial and then doing a sum. 168 00:10:07,130 --> 00:10:15,030 So previously, the way I wrote that, 169 00:10:15,030 --> 00:10:17,010 for the conjugate variables, we wrote that we 170 00:10:17,010 --> 00:10:18,930 had alpha to the n plus 1. 171 00:10:18,930 --> 00:10:23,400 And if you think about what we do, we just went to b to the n 172 00:10:23,400 --> 00:10:26,250 over n factorial. 173 00:10:26,250 --> 00:10:29,280 So basically, to convert from this guy 174 00:10:29,280 --> 00:10:30,930 to this guy, that's what we did. 175 00:10:30,930 --> 00:10:34,747 And we had to take special care at the first term. 176 00:10:34,747 --> 00:10:36,330 But other than the first term, we just 177 00:10:36,330 --> 00:10:37,413 had this replacement rule. 178 00:10:45,300 --> 00:10:49,290 Now, it's a little convenient actually not to do that 179 00:10:49,290 --> 00:10:51,990 with exactly the same variable but a slightly different 180 00:10:51,990 --> 00:10:55,260 variable that's just rescaled. 181 00:10:55,260 --> 00:11:03,790 So I'm going to use a slightly different variable here, 182 00:11:03,790 --> 00:11:06,970 which is typical. 183 00:11:09,860 --> 00:11:12,770 So what we'll do is we'll say that instead of just taking 184 00:11:12,770 --> 00:11:16,640 along the alpha s, we'll take along with it the beta 0. 185 00:11:19,580 --> 00:11:23,870 And we'll say that the conjugate variable of that 186 00:11:23,870 --> 00:11:27,355 is u instead of b, but it's just rescaled. 187 00:11:27,355 --> 00:11:28,730 We're just taking something along 188 00:11:28,730 --> 00:11:31,543 with the coupling constant. 189 00:11:31,543 --> 00:11:33,210 Which is always we're always free to do. 190 00:11:33,210 --> 00:11:36,180 It's just a multiplicative factor. 191 00:11:36,180 --> 00:11:38,470 Now, if we look at this diagram here, 192 00:11:38,470 --> 00:11:40,470 the factors of the coupling that we've included 193 00:11:40,470 --> 00:11:43,080 are all the ones that are attached to the bubbles. 194 00:11:43,080 --> 00:11:47,578 But there's two more-- there's one here and one there. 195 00:11:47,578 --> 00:11:49,620 And if we're going to make this replacement rule, 196 00:11:49,620 --> 00:11:51,840 we have to identify all the factors of alpha s, 197 00:11:51,840 --> 00:11:55,500 because they're all going to disappear and become u's. So we 198 00:11:55,500 --> 00:11:57,300 have to take these guys into account too. 199 00:12:07,230 --> 00:12:13,460 We have an extra G squared, which 200 00:12:13,460 --> 00:12:15,335 I'm going to write in terms of this variable. 201 00:12:46,800 --> 00:12:50,065 So there's an extra factor of G squared from the ends. 202 00:12:50,065 --> 00:12:51,690 And if you think about what that means, 203 00:12:51,690 --> 00:12:54,630 it means that every diagram that we're considering here 204 00:12:54,630 --> 00:12:56,220 starts with n equals 1, so we don't 205 00:12:56,220 --> 00:12:59,590 have to worry about this delta function term. 206 00:12:59,590 --> 00:13:02,820 It starts with n equals 0. 207 00:13:02,820 --> 00:13:05,490 There's no term with no couplings here. 208 00:13:08,250 --> 00:13:09,755 So then it is literally what I said, 209 00:13:09,755 --> 00:13:12,255 that we just take this factor and replace it by that factor. 210 00:13:22,990 --> 00:13:25,330 So a Borel transform, remember, is a transform 211 00:13:25,330 --> 00:13:27,040 of the variable alpha s. 212 00:13:27,040 --> 00:13:30,440 Alpha s goes over to a different space. 213 00:13:30,440 --> 00:13:32,138 So it's not a kinematic variable. 214 00:13:32,138 --> 00:13:34,180 It's not like what we're used to transforming in, 215 00:13:34,180 --> 00:13:39,090 but it's perfectly fine. 216 00:13:39,090 --> 00:13:41,548 For those people that came in after I said this, 217 00:13:41,548 --> 00:13:42,840 there's no lecture on Thursday. 218 00:13:56,900 --> 00:13:59,360 So let me use the following notation, 219 00:13:59,360 --> 00:14:05,950 taking that G squared along with our geometric sum, 220 00:14:05,950 --> 00:14:08,680 and now expressing that whole thing as a function of momentum 221 00:14:08,680 --> 00:14:10,660 still, and u instead of alpha s. 222 00:14:34,660 --> 00:14:46,080 So I replaced the alpha s's, and I 223 00:14:46,080 --> 00:14:49,290 get that series, which we can see 224 00:14:49,290 --> 00:14:52,950 is just an exponential series. 225 00:14:52,950 --> 00:14:54,548 So I can just calculate the sum. 226 00:15:12,000 --> 00:15:14,212 Borel variable u log-- 227 00:15:23,060 --> 00:15:26,820 which if you have an exponential of a log, it's just a power. 228 00:15:26,820 --> 00:15:28,990 So let's organize it that way. 229 00:15:41,780 --> 00:16:04,710 So let me write it this way So just writing 230 00:16:04,710 --> 00:16:08,490 the exponential of a log is that thing raised to the power u, 231 00:16:08,490 --> 00:16:12,885 taking the mu squared to the c bar u, that's a constant. 232 00:16:12,885 --> 00:16:14,760 And then taking all the factors of p squared, 233 00:16:14,760 --> 00:16:17,890 there is one from here, and then there's u of them from there. 234 00:16:17,890 --> 00:16:19,530 but I took an extra one from here, 235 00:16:19,530 --> 00:16:21,840 multiplied on top and bottom by an extra p squared. 236 00:16:24,428 --> 00:16:25,720 Hopefully I got the sign right. 237 00:16:29,066 --> 00:16:30,440 Looks like it should be plus. 238 00:16:38,070 --> 00:16:38,570 OK. 239 00:16:38,570 --> 00:16:41,090 So this thing is kind of nice for computations, 240 00:16:41,090 --> 00:16:44,480 because when you stick it back into our bubble sum-- 241 00:16:44,480 --> 00:16:47,690 when you stick this bubble sum back 242 00:16:47,690 --> 00:16:51,290 into the diagram we want to compute-- 243 00:16:51,290 --> 00:16:54,140 we just have effectively a modified gluon propagator. 244 00:16:57,960 --> 00:17:04,895 So think of it as some kind of modified gluon propagator. 245 00:17:04,895 --> 00:17:06,770 Maybe I should write it in a different color. 246 00:17:10,157 --> 00:17:12,490 So it's just like this, where we use a different Feynman 247 00:17:12,490 --> 00:17:14,053 rule for that propagator. 248 00:17:28,099 --> 00:17:29,630 So it's not actually-- 249 00:17:29,630 --> 00:17:33,230 doing these types of bubble sum calculations 250 00:17:33,230 --> 00:17:35,570 is not any more hard than doing a one loop calculation, 251 00:17:35,570 --> 00:17:38,210 you just have to use a different power for the propagator. 252 00:17:38,210 --> 00:17:44,090 But your Feynman integral tricks can handle that very easily. 253 00:17:48,280 --> 00:17:48,780 OK. 254 00:17:48,780 --> 00:17:51,072 So it's about as hard as computing this one with graph. 255 00:17:53,830 --> 00:18:02,350 So let's call this thing sigma bubble, 256 00:18:02,350 --> 00:18:07,150 and let's think about calculating it 257 00:18:07,150 --> 00:18:08,890 in terms of the MS bar mass. 258 00:18:21,380 --> 00:18:24,290 And then we also cancel any one of our epsilon poles, which 259 00:18:24,290 --> 00:18:26,600 we're not really interested in here. 260 00:18:36,530 --> 00:18:39,644 The 1 over epsilon poles actually come from-- 261 00:18:39,644 --> 00:18:42,790 if we think about what happened with the variables-- 262 00:18:42,790 --> 00:18:48,090 the case u equal 0, which is the log divergent piece. 263 00:18:56,620 --> 00:18:59,260 And if you think about what we want 264 00:18:59,260 --> 00:19:02,470 to do in order to look for poles, 265 00:19:02,470 --> 00:19:06,220 u equals 0 is actually related to the ultraviolet divergence, 266 00:19:06,220 --> 00:19:09,123 while values u greater than 0 and u less than 0, 267 00:19:09,123 --> 00:19:10,540 those are probing these poles that 268 00:19:10,540 --> 00:19:12,580 are off the axis in the Borel plane. 269 00:19:28,100 --> 00:19:30,380 And it's those power law divergences that 270 00:19:30,380 --> 00:19:33,050 are probing for renormalons. 271 00:19:33,050 --> 00:19:36,860 And we'll see that they lead to poles in the Borel plane 272 00:19:36,860 --> 00:19:38,390 through this modified propagator. 273 00:19:46,510 --> 00:19:48,103 So let's calculate-- the goal here 274 00:19:48,103 --> 00:19:50,020 is going to be to calculate an expression that 275 00:19:50,020 --> 00:19:52,653 relates to the MS bar mass and the pole mass. 276 00:19:52,653 --> 00:19:54,070 And that'll be an infinite series, 277 00:19:54,070 --> 00:19:57,738 and it will be a function in Borel space. 278 00:19:57,738 --> 00:20:00,530 The alpha s space, that would be an infinite series in alpha s, 279 00:20:00,530 --> 00:20:02,780 because we've included an infinite number of alpha s's 280 00:20:02,780 --> 00:20:04,625 in our diagrams. 281 00:20:04,625 --> 00:20:06,000 And if we go over to Borel space, 282 00:20:06,000 --> 00:20:07,850 there'll be some functional relationship between those two 283 00:20:07,850 --> 00:20:08,350 masses. 284 00:20:14,370 --> 00:20:17,520 OK, so what's the definition of a pole mass. 285 00:20:17,520 --> 00:20:21,510 If you go back to the propagator and you write it 286 00:20:21,510 --> 00:20:22,860 in terms of the MS bar mass-- 287 00:20:26,928 --> 00:20:28,970 so you formulated your whole quantum field theory 288 00:20:28,970 --> 00:20:30,720 in terms of the MS bar mass, you calculate 289 00:20:30,720 --> 00:20:33,260 that bubble some, called it sigma, 290 00:20:33,260 --> 00:20:35,690 wrote it in terms of MS bar mass-- 291 00:20:35,690 --> 00:20:39,230 the general structure of sigma would 292 00:20:39,230 --> 00:20:42,020 be that there is some constant piece, which I'll call sigma 1. 293 00:20:49,780 --> 00:21:06,680 And then there's some sigma 2 piece. 294 00:21:06,680 --> 00:21:10,490 And the pole mass is the place where you demand-- 295 00:21:10,490 --> 00:21:14,150 you just have p slash minus m pole. 296 00:21:14,150 --> 00:21:32,355 So another way of saying that is that if p squared is 297 00:21:32,355 --> 00:21:49,028 equal to m pole squared, you set that to be 0, which I can 298 00:21:49,028 --> 00:21:50,320 write as an equation like that. 299 00:21:50,320 --> 00:21:52,400 And then I can square it. 300 00:21:52,400 --> 00:21:57,690 So m pole squared, which is p slash squared, 301 00:21:57,690 --> 00:22:11,360 m bar squared, this thing squared. 302 00:22:11,360 --> 00:22:14,740 And if you just think about what that gives, 303 00:22:14,740 --> 00:22:16,600 this is giving some equation that 304 00:22:16,600 --> 00:22:23,230 looks like m pole is m bar, m 1 plus sigma 305 00:22:23,230 --> 00:22:25,630 1 plus the extra stuff. 306 00:22:38,690 --> 00:22:43,690 So let's do that and figure out what 307 00:22:43,690 --> 00:22:45,820 we get from this diagram for the sigma. 308 00:22:56,690 --> 00:23:07,380 So it's a usual one loop type calculation, 309 00:23:07,380 --> 00:23:09,210 except instead of the gluon propagator, 310 00:23:09,210 --> 00:23:19,270 I have this G mu nu thing, which at this point 311 00:23:19,270 --> 00:23:20,770 is in alpha space, let's say. 312 00:23:24,260 --> 00:23:27,040 And then I have the fermion propagator. 313 00:23:34,360 --> 00:23:38,750 And I've taken into account all the g squareds and the alpha 314 00:23:38,750 --> 00:23:40,930 s's in here. 315 00:23:40,930 --> 00:23:42,880 And so to go to the Borel space, I 316 00:23:42,880 --> 00:23:44,320 can do that under the integrand. 317 00:23:51,690 --> 00:23:59,230 So I just take g squared G mu nu bubble k and alpha 318 00:23:59,230 --> 00:24:02,040 s, k being the loop momenta, and just 319 00:24:02,040 --> 00:24:06,450 send that over to this thing that I'm calling g squared G mu 320 00:24:06,450 --> 00:24:11,370 nu bubble, which is not a function of g squared anymore, 321 00:24:11,370 --> 00:24:15,060 but just a function of k and u. 322 00:24:19,710 --> 00:24:23,160 And as long as you're staying away from u equals 0, 323 00:24:23,160 --> 00:24:24,840 there's no issues related to the order 324 00:24:24,840 --> 00:24:27,045 of doing those operations. 325 00:24:27,045 --> 00:24:29,670 So at u equals 0, you have to be actually careful if you really 326 00:24:29,670 --> 00:24:31,980 want to go figure out what's happening at u equals 0. 327 00:24:31,980 --> 00:24:34,390 But that's actually not what we're interested in, 328 00:24:34,390 --> 00:24:37,320 so we'll just ignore what's happening at u equals 0. 329 00:24:41,810 --> 00:24:42,310 OK. 330 00:24:42,310 --> 00:24:44,740 So combine-- stick in the expression 331 00:24:44,740 --> 00:24:47,553 that we had over here for this guy, 332 00:24:47,553 --> 00:24:49,220 and it's just some p squared to a power. 333 00:24:49,220 --> 00:24:51,285 So it becomes a k squared to a power. 334 00:24:51,285 --> 00:24:52,660 So you have k squared to a power, 335 00:24:52,660 --> 00:24:54,190 you have this guy to a power-- 336 00:24:54,190 --> 00:24:56,300 use the usual tricks. 337 00:24:56,300 --> 00:25:10,810 Combine denominators, do algebra, 338 00:25:10,810 --> 00:25:23,135 and you get some expression for sigma 1 in Borel space. 339 00:25:30,680 --> 00:25:33,830 And then, from that, you can construct 340 00:25:33,830 --> 00:25:37,040 the relation between the pole mass and the MS bar mass. 341 00:25:49,190 --> 00:25:53,330 And it might not surprise, having done some one loop 342 00:25:53,330 --> 00:26:00,320 calculations that you're getting some gamma functions, 343 00:26:00,320 --> 00:26:04,610 after the dust settles, this is what it looks like. 344 00:26:18,130 --> 00:26:27,970 6, 1 minus u, gamma of u, gamma of 1 minus 2u over gamma of 3 345 00:26:27,970 --> 00:26:28,570 minus u. 346 00:26:32,310 --> 00:26:34,230 So the piece that we just calculated, 347 00:26:34,230 --> 00:26:37,680 or sketched how you would calculate, is this. 348 00:26:37,680 --> 00:26:39,450 And then there's the tree level relation 349 00:26:39,450 --> 00:26:42,570 between these two things, which in alpha s space 350 00:26:42,570 --> 00:26:44,850 was just that m pole was equal to m bar. 351 00:26:44,850 --> 00:26:47,310 But when you transform one to the Borel space, 352 00:26:47,310 --> 00:26:49,300 you get a delta function. 353 00:26:49,300 --> 00:26:50,160 So this comes from-- 354 00:27:01,190 --> 00:27:02,720 just that equality at lowest order 355 00:27:02,720 --> 00:27:04,640 becomes a delta function in the Borel space. 356 00:27:07,600 --> 00:27:10,930 So now, if you look at this and you ignore u equals 0, 357 00:27:10,930 --> 00:27:14,140 then the closest pole to u equals 0 is equals 1/2 358 00:27:14,140 --> 00:27:15,490 coming from this gamma function. 359 00:27:22,530 --> 00:27:26,300 And remember that we had a kind of notation 360 00:27:26,300 --> 00:27:27,740 for the strength of a renormalon. 361 00:27:27,740 --> 00:27:33,410 Those closest to the origin were the strongest. 362 00:27:33,410 --> 00:27:37,322 So the strongest renormalon here is a u equals 1/2 renormalon. 363 00:27:49,630 --> 00:27:51,370 That strength was related to the fact 364 00:27:51,370 --> 00:27:55,780 that basically you got something that would be, in this case, 2 365 00:27:55,780 --> 00:27:58,910 to the n, n factorial. 366 00:27:58,910 --> 00:28:03,320 So 1.2 to the minus n. 367 00:28:03,320 --> 00:28:07,220 And so the guy that's closest gets the strongest prefactor 368 00:28:07,220 --> 00:28:10,070 through the n factorial growth. 369 00:28:10,070 --> 00:28:12,470 That's where that naming came from. 370 00:28:12,470 --> 00:28:13,850 Yeah. 371 00:28:13,850 --> 00:28:15,860 AUDIENCE: So there weren't [? mu poles ?] in-- 372 00:28:15,860 --> 00:28:19,700 the only pole in the modified gluon propagator, was it-- 373 00:28:19,700 --> 00:28:23,657 Oh, like, the one that made the denominator vanish. 374 00:28:23,657 --> 00:28:25,490 But the rest of the poles came from actually 375 00:28:25,490 --> 00:28:26,570 doing the one with the diagram. 376 00:28:26,570 --> 00:28:26,990 IAIN STEWART: That's right. 377 00:28:26,990 --> 00:28:27,830 [INTERPOSING VOICES] 378 00:28:27,830 --> 00:28:28,430 AUDIENCE: --in the series. 379 00:28:28,430 --> 00:28:29,222 IAIN STEWART: Yeah. 380 00:28:29,222 --> 00:28:32,030 So what you're doing is you're actually 381 00:28:32,030 --> 00:28:35,330 thinking of probing the infrared structure in this loop. 382 00:28:35,330 --> 00:28:37,110 That's what the gluon is doing. 383 00:28:37,110 --> 00:28:40,575 And the modified propagator p squared to the u 384 00:28:40,575 --> 00:28:43,273 is just giving you a kind of a regulator if you like, 385 00:28:43,273 --> 00:28:44,690 which is the u that's allowing you 386 00:28:44,690 --> 00:28:48,680 to probe what's the infrastructure in powers 387 00:28:48,680 --> 00:28:51,680 in this loop diagram, not just kind of logarithmic divergences 388 00:28:51,680 --> 00:28:55,310 which we usually focus on but also in power law sensitivity 389 00:28:55,310 --> 00:28:56,370 to the infrared. 390 00:28:56,370 --> 00:28:58,760 So it's like probing low momentum gluons 391 00:28:58,760 --> 00:29:01,790 with this modified gluon propagator. 392 00:29:01,790 --> 00:29:07,190 And that's what's resulting in these poles. 393 00:29:07,190 --> 00:29:09,620 You're probing the infrared structure of the loop. 394 00:29:12,850 --> 00:29:16,810 So there would be some other terms that I haven't written. 395 00:29:16,810 --> 00:29:23,990 So let me just comment that there's actually 396 00:29:23,990 --> 00:29:26,450 a term that's proportional to 1 over u that would make 397 00:29:26,450 --> 00:29:34,122 it finite at u equals 0 if we're more careful, 398 00:29:34,122 --> 00:29:36,080 when we carry out the renormalization properly. 399 00:29:40,970 --> 00:29:50,600 And I also wasn't so worried about anything analytic in u. 400 00:29:50,600 --> 00:29:53,970 So I only was looking at these gamma functions. 401 00:29:57,720 --> 00:29:58,220 OK. 402 00:29:58,220 --> 00:30:03,050 So if we actually just specify ourselves to that u equals 1/2, 403 00:30:03,050 --> 00:30:04,660 we can simplify things even further. 404 00:30:15,210 --> 00:30:18,560 So stick in u equals 1/2. 405 00:30:18,560 --> 00:30:24,170 And if I stick in u to the 1/2, this becomes mu over m bar 406 00:30:24,170 --> 00:30:26,785 e to the c bar over 2. 407 00:30:26,785 --> 00:30:31,460 If I just put this 1/2 in there, all this stuff 408 00:30:31,460 --> 00:30:34,490 with the gamma functions becomes a minus 2 over u minus 1/2. 409 00:30:38,480 --> 00:30:39,410 So there's a pole. 410 00:30:39,410 --> 00:30:44,280 It's sitting on the real axis, and we can't do the transform. 411 00:30:44,280 --> 00:30:47,795 AUDIENCE: At this point can you drop the delta u? 412 00:30:47,795 --> 00:30:49,010 Because you know it's not-- 413 00:30:49,010 --> 00:30:51,800 IAIN STEWART: Yeah, I don't really need it. 414 00:30:51,800 --> 00:30:55,290 But just sort of for completeness I kept it. 415 00:30:55,290 --> 00:30:55,790 Yeah. 416 00:30:55,790 --> 00:30:59,640 If I just wanted to think about this term, I could drop it. 417 00:30:59,640 --> 00:31:00,140 Yeah. 418 00:31:05,610 --> 00:31:07,350 So let's think about the inverse Borel. 419 00:31:12,090 --> 00:31:15,240 So we wanted to do an integral which in this u space 420 00:31:15,240 --> 00:31:17,560 would look like this-- 421 00:31:17,560 --> 00:31:23,510 so u is the conjugate variable to 4 pi over beta 0 alpha s 422 00:31:23,510 --> 00:31:25,140 of mu. 423 00:31:25,140 --> 00:31:28,410 That was what we transformed. 424 00:31:28,410 --> 00:31:31,653 So that's the difference between our old notation 425 00:31:31,653 --> 00:31:32,320 and our new one. 426 00:31:32,320 --> 00:31:36,310 We just have a conjugate variable to this thing. 427 00:31:36,310 --> 00:31:39,038 And so this has an ambiguity, because on the real axis 428 00:31:39,038 --> 00:31:40,330 we have a pole at u equals 1/2. 429 00:31:50,800 --> 00:31:53,920 So there's that pole at 1/2, and you 430 00:31:53,920 --> 00:31:57,025 have to decide do we want to go around that above or below. 431 00:32:00,315 --> 00:32:01,690 So I mentioned this last time, we 432 00:32:01,690 --> 00:32:07,450 could think about going past the pole that way 433 00:32:07,450 --> 00:32:09,360 or going past the pole this way. 434 00:32:09,360 --> 00:32:11,110 And the fact that we have to make a choice 435 00:32:11,110 --> 00:32:14,650 means there's an ambiguity. 436 00:32:14,650 --> 00:32:16,420 And if you like, you can say, well, 437 00:32:16,420 --> 00:32:20,170 it's the average above and below or that the ambiguity would 438 00:32:20,170 --> 00:32:22,450 then be something like 1/2 going around the pole. 439 00:32:32,130 --> 00:32:34,290 So let's actually look at what the ambiguity is. 440 00:32:54,670 --> 00:33:00,930 So the ambiguity, well, if we go around the pole, 441 00:33:00,930 --> 00:33:03,708 we get a 2 pi i. 442 00:33:03,708 --> 00:33:15,375 There's 1/2, integral around the pole, this factor. 443 00:33:19,980 --> 00:33:26,390 So mu c bar over 2, bar. 444 00:33:26,390 --> 00:33:27,940 And if we put everything together, 445 00:33:27,940 --> 00:33:34,010 the rest of the prefactor, there's a C f over 3 pi beta 0, 446 00:33:34,010 --> 00:33:37,300 and there's also another m bar, which 447 00:33:37,300 --> 00:33:40,810 is this m bar that's out front. 448 00:33:40,810 --> 00:33:47,996 And I combined the minus 2 with the six together, 3. 449 00:33:47,996 --> 00:33:50,472 So the m bars actually are cancelling. 450 00:34:01,920 --> 00:34:10,426 We close on the pole, stick that value in everywhere else, 451 00:34:10,426 --> 00:34:11,560 and we get that. 452 00:34:16,503 --> 00:34:17,670 Does anybody recognize this? 453 00:34:22,810 --> 00:34:25,389 AUDIENCE: It's like an instanton. 454 00:34:25,389 --> 00:34:26,639 IAIN STEWART: It's lambda QCD. 455 00:34:26,639 --> 00:34:27,139 Yeah. 456 00:34:29,937 --> 00:34:32,520 Yeah, it looks kind of like an instanton with the exponential, 457 00:34:32,520 --> 00:34:36,130 but this particular thing is actually exactly lambda QCD. 458 00:34:40,960 --> 00:34:42,835 So if we solve this, and we invert it, 459 00:34:42,835 --> 00:34:45,210 and we solve for alpha s, that would be the usual formula 460 00:34:45,210 --> 00:34:46,290 for lambda QCD-- 461 00:34:46,290 --> 00:34:47,790 alpha in terms of lambda QCD. 462 00:34:52,719 --> 00:34:55,469 So we've just shown that the ambiguity 463 00:34:55,469 --> 00:35:03,180 up to some constant prefactor is exactly lambda QCD. 464 00:35:03,180 --> 00:35:14,890 And that is exactly our physical idea 465 00:35:14,890 --> 00:35:19,720 that we said last time, that the pole mass has a delta m 466 00:35:19,720 --> 00:35:24,990 of order lambda QCD ambiguity. 467 00:35:24,990 --> 00:35:26,820 And we've just seen that explicitly 468 00:35:26,820 --> 00:35:29,475 arise from calculations with this Borel space. 469 00:35:35,330 --> 00:35:37,930 So that's kind of neat Yeah. 470 00:35:37,930 --> 00:35:42,530 AUDIENCE: So Is there a way to be careful about the pi epsilon 471 00:35:42,530 --> 00:35:44,750 description when we do these Borel transforms? 472 00:35:44,750 --> 00:35:45,110 IAIN STEWART: Yeah. 473 00:35:45,110 --> 00:35:46,478 AUDIENCE: Because it seems like you just multiplied 474 00:35:46,478 --> 00:35:48,500 a bunch of p squareds plus i epsilons 475 00:35:48,500 --> 00:35:50,678 together, and then just ignored the i epsilon. 476 00:35:50,678 --> 00:35:51,470 IAIN STEWART: Yeah. 477 00:35:51,470 --> 00:35:53,900 I didn't ignore the i epsilon. 478 00:35:53,900 --> 00:35:56,320 The reason I wrote it as a minus p squared to the power 479 00:35:56,320 --> 00:35:58,440 was because of that i epsilon. 480 00:35:58,440 --> 00:36:01,110 I didn't write the i epsilons on the board. 481 00:36:01,110 --> 00:36:04,590 But if I had, it wouldn't change anything that I wrote. 482 00:36:04,590 --> 00:36:07,160 So the one way I can say it is that when I actually 483 00:36:07,160 --> 00:36:09,020 did these loop calculations, I was actually 484 00:36:09,020 --> 00:36:12,260 careful about the i epsilon. 485 00:36:12,260 --> 00:36:14,510 AUDIENCE: I'm not worried about getting the sign-- 486 00:36:14,510 --> 00:36:14,875 [INTERPOSING VOICES] 487 00:36:14,875 --> 00:36:15,667 IAIN STEWART: Yeah. 488 00:36:15,667 --> 00:36:16,750 AUDIENCE: --like of logs. 489 00:36:16,750 --> 00:36:19,400 I'm worried about actually defining products 490 00:36:19,400 --> 00:36:20,830 of distributions right. 491 00:36:20,830 --> 00:36:22,955 Like when you interchange the order of integration, 492 00:36:22,955 --> 00:36:25,428 presumably you're doing something [INAUDIBLE]---- 493 00:36:25,428 --> 00:36:27,470 IAIN STEWART: You mean the order of the summation 494 00:36:27,470 --> 00:36:28,460 and the integration? 495 00:36:28,460 --> 00:36:29,127 AUDIENCE: Right. 496 00:36:29,127 --> 00:36:30,350 IAIN STEWART: Yeah. 497 00:36:30,350 --> 00:36:33,350 When you interchange that order, the only thing that you-- 498 00:36:33,350 --> 00:36:34,942 so you can do it in the other order. 499 00:36:34,942 --> 00:36:36,650 The only thing that actually goes wrong-- 500 00:36:36,650 --> 00:36:38,692 and it's much easier to present it in this order. 501 00:36:38,692 --> 00:36:39,920 This is why I did. 502 00:36:39,920 --> 00:36:41,520 If you do do it in the other order, 503 00:36:41,520 --> 00:36:44,905 the only thing that will change is what happens at u equals 0. 504 00:36:44,905 --> 00:36:46,280 So you'll actually see everything 505 00:36:46,280 --> 00:36:48,560 that we saw, except that u equals zero, 506 00:36:48,560 --> 00:36:49,820 it'll be slightly different. 507 00:36:49,820 --> 00:36:52,093 And that 1 over u term that I was talking about, 508 00:36:52,093 --> 00:36:53,510 if you want to get that right, you 509 00:36:53,510 --> 00:36:55,610 have to think about the issue of changing 510 00:36:55,610 --> 00:36:57,740 the order of integration more carefully than I did. 511 00:36:57,740 --> 00:36:59,407 AUDIENCE: But if it's near u equals 1/2. 512 00:36:59,407 --> 00:37:01,550 IAIN STEWART: Near u equals 1/2, it's exactly-- 513 00:37:01,550 --> 00:37:04,960 this is the right result. And near any other u except u 514 00:37:04,960 --> 00:37:08,385 equals 0 we're OK. 515 00:37:08,385 --> 00:37:09,010 AUDIENCE: Iain? 516 00:37:09,010 --> 00:37:09,802 IAIN STEWART: Yeah. 517 00:37:09,802 --> 00:37:11,096 AUDIENCE: So which [INAUDIBLE]? 518 00:37:11,096 --> 00:37:12,010 IAIN STEWART: Sorry. 519 00:37:12,010 --> 00:37:14,360 AUDIENCE: Which [INAUDIBLE] would it be to define-- 520 00:37:14,360 --> 00:37:16,330 IAIN STEWART: We don't know. 521 00:37:16,330 --> 00:37:20,440 So the ambiguity-- you have to make a choice, right? 522 00:37:20,440 --> 00:37:22,690 If you make this choice, you've picked one definition. 523 00:37:22,690 --> 00:37:24,648 If you make this choice, you've picked another. 524 00:37:24,648 --> 00:37:27,370 And I said, we can quantify the ambiguity in our choice 525 00:37:27,370 --> 00:37:31,660 by thinking about it as 1/2 going around the pole. 526 00:37:31,660 --> 00:37:34,510 That's kind of a measure of the ambiguity 527 00:37:34,510 --> 00:37:37,390 that we have in which way to close. 528 00:37:37,390 --> 00:37:38,672 It's an ambiguity. 529 00:37:38,672 --> 00:37:39,380 AUDIENCE: I know. 530 00:37:39,380 --> 00:37:42,760 But when [INAUDIBLE] on how we calculate it, right? 531 00:37:42,760 --> 00:37:48,610 So when we can m pole, what do we mean by-- 532 00:37:48,610 --> 00:37:51,637 IAIN STEWART: Different people mean different things. 533 00:37:51,637 --> 00:37:53,470 I mean, you're not going to see this, right, 534 00:37:53,470 --> 00:37:55,512 unless you got all orders in perturbation theory. 535 00:37:55,512 --> 00:37:57,640 So order by order in perturbation theory, 536 00:37:57,640 --> 00:38:01,280 you will never see this pole. 537 00:38:01,280 --> 00:38:03,403 So unless you're working to all orders-- 538 00:38:03,403 --> 00:38:04,820 as soon as you work to all orders, 539 00:38:04,820 --> 00:38:07,170 you have to make a choice because of what we just did. 540 00:38:07,170 --> 00:38:09,420 If you work order by order, there's no choice to make, 541 00:38:09,420 --> 00:38:11,000 and you just order by order define it 542 00:38:11,000 --> 00:38:12,692 with the usual definition. 543 00:38:19,590 --> 00:38:22,040 So the moral is not how do we define m pole, 544 00:38:22,040 --> 00:38:23,540 the moral is we should use something 545 00:38:23,540 --> 00:38:24,620 else other than m pole. 546 00:38:24,620 --> 00:38:26,240 Because m pole has an ambiguity. 547 00:38:42,930 --> 00:38:47,340 So we use the MS bar mass m bar here for our calculation, 548 00:38:47,340 --> 00:38:50,580 but actually the ambiguity wasn't depending on that. 549 00:38:50,580 --> 00:38:52,470 It really will show up-- 550 00:38:52,470 --> 00:38:54,900 even if we thought of some other short distance mass 551 00:38:54,900 --> 00:38:57,600 beside the MS bar mass, we would still 552 00:38:57,600 --> 00:38:59,100 see the same ambiguity if we tried 553 00:38:59,100 --> 00:39:01,710 to associated it to the m pole. 554 00:39:01,710 --> 00:39:05,940 And actually, this thing is a unique property 555 00:39:05,940 --> 00:39:06,840 of the pole mass. 556 00:39:06,840 --> 00:39:09,960 It's the pole mass that has this problem, not the MS bar mass. 557 00:39:20,220 --> 00:39:22,740 Another thing you'll notice is that when 558 00:39:22,740 --> 00:39:25,620 we went back and looked at the ambiguity back 559 00:39:25,620 --> 00:39:29,970 in the original space, in the coupling 560 00:39:29,970 --> 00:39:37,060 space, which is where we live, then the mu dependence 561 00:39:37,060 --> 00:39:39,100 went away. 562 00:39:39,100 --> 00:39:41,680 We just got something that was a constant times lambda QCD. 563 00:39:41,680 --> 00:39:43,236 There was no mu dependence. 564 00:39:48,472 --> 00:39:52,950 So the ambiguity is mu independent. 565 00:40:00,030 --> 00:40:01,490 It's proportional to lambda QCD. 566 00:40:04,310 --> 00:40:08,840 If we looked, on the other hand, in the Borel space, 567 00:40:08,840 --> 00:40:13,471 at the residue of the pole, it was mu dependent. 568 00:40:22,252 --> 00:40:23,960 So before we did the integral, if we just 569 00:40:23,960 --> 00:40:26,840 look at even the pole as u minus 1/2, 570 00:40:26,840 --> 00:40:30,530 the residue has mu power in it in the numerator. 571 00:40:30,530 --> 00:40:33,620 And that just went along for the ride and became this mu. 572 00:40:33,620 --> 00:40:37,580 But when we transformed back, we had alpha mu there, 573 00:40:37,580 --> 00:40:42,030 and the mu dependence was cancelling. 574 00:40:42,030 --> 00:40:43,670 And this actually is-- 575 00:40:43,670 --> 00:40:46,620 there's one important fact about this. 576 00:40:46,620 --> 00:40:58,800 So when we express something like a decay rate, which 577 00:40:58,800 --> 00:41:03,240 we might originally calculate in terms of the pole mass 578 00:41:03,240 --> 00:41:14,320 since maybe that's simpler, and we get some series, if we want 579 00:41:14,320 --> 00:41:22,180 to express that, say, in terms of the MS bar mass, 580 00:41:22,180 --> 00:41:24,040 then what we're after doing is we're 581 00:41:24,040 --> 00:41:28,690 after actually canceling a pole in the Borel plane 582 00:41:28,690 --> 00:41:30,670 that occurs in kind of the series that relates 583 00:41:30,670 --> 00:41:32,590 this guy to the MS bar mass. 584 00:41:32,590 --> 00:41:34,240 And then there's a corresponding pole 585 00:41:34,240 --> 00:41:36,307 in this series in the decay rate. 586 00:41:36,307 --> 00:41:38,140 And those have to cancel against each other. 587 00:41:54,660 --> 00:41:59,668 So you should think that there's a 1 over u minus 1/2 pole 588 00:41:59,668 --> 00:42:01,460 in the relation of this thing to the MS bar 589 00:42:01,460 --> 00:42:03,570 mass that we just computed. 590 00:42:03,570 --> 00:42:05,240 And one could also look at, for example, 591 00:42:05,240 --> 00:42:08,010 bubble chains for this particular observable. 592 00:42:08,010 --> 00:42:09,180 People have done that. 593 00:42:09,180 --> 00:42:11,930 And if we did that, you'd also find exactly the same, 594 00:42:11,930 --> 00:42:16,890 u equals 1/2 pole in that series. 595 00:42:16,890 --> 00:42:21,060 And these are cancelling if we make that change of variable 596 00:42:21,060 --> 00:42:21,560 here. 597 00:42:36,498 --> 00:42:37,790 I guess I've already said that. 598 00:42:43,210 --> 00:42:45,370 When you do that-- 599 00:42:45,370 --> 00:42:47,350 in practice of course, what you do is 600 00:42:47,350 --> 00:42:49,300 you don't have the whole series. 601 00:42:49,300 --> 00:42:52,930 So what you do is you want the cancellation to take place 602 00:42:52,930 --> 00:42:54,550 at the order you've worked. 603 00:42:54,550 --> 00:42:57,050 So say I know alpha squared in this decay rate-- 604 00:42:57,050 --> 00:42:59,530 which is true, people do know that-- 605 00:42:59,530 --> 00:43:01,210 and the alpha squared relation. 606 00:43:01,210 --> 00:43:02,710 What you then want to do is you want 607 00:43:02,710 --> 00:43:05,562 to make sure that the terms, any large numerical factors 608 00:43:05,562 --> 00:43:07,270 in front of the series, are canceling out 609 00:43:07,270 --> 00:43:09,640 to order alpha squared. 610 00:43:09,640 --> 00:43:11,930 And there's only one little caveat 611 00:43:11,930 --> 00:43:15,440 which you have to be careful about, 612 00:43:15,440 --> 00:43:17,815 and that is that you have to expand in the same coupling. 613 00:43:28,750 --> 00:43:33,082 So when we do this cancellation order by order, 614 00:43:33,082 --> 00:43:35,665 you should use the same coupling constant with the same scale. 615 00:43:40,915 --> 00:43:42,040 So as long as both series-- 616 00:43:55,780 --> 00:43:57,820 so you can't, for example, make a conversion 617 00:43:57,820 --> 00:44:00,340 with the M b pole with alpha s evaluated 618 00:44:00,340 --> 00:44:01,900 at the b scale or some other scale. 619 00:44:01,900 --> 00:44:03,650 You have to really do it at the same scale 620 00:44:03,650 --> 00:44:06,610 that whatever this series is expressed in terms of. 621 00:44:09,650 --> 00:44:12,800 And you can also think of that as being related to the fact 622 00:44:12,800 --> 00:44:16,760 that the Borel variable u itself was defined 623 00:44:16,760 --> 00:44:20,080 as the conjugate variable to something 624 00:44:20,080 --> 00:44:22,210 that was mu dependent. 625 00:44:22,210 --> 00:44:26,290 So even the definition of u involves alpha 626 00:44:26,290 --> 00:44:27,730 at it's particular scale mu. 627 00:44:31,460 --> 00:44:32,310 OK. 628 00:44:32,310 --> 00:44:35,310 So I said that in a bit of a convoluted way. 629 00:44:35,310 --> 00:44:38,670 But hopefully what I'm saying is clear, even if what I'm writing 630 00:44:38,670 --> 00:44:39,380 is not so clear. 631 00:44:42,560 --> 00:44:43,060 OK. 632 00:44:43,060 --> 00:44:48,490 So same coupling in this series and the conversion series. 633 00:44:48,490 --> 00:44:51,940 And this, what I've just said, is also a general fact, 634 00:44:51,940 --> 00:44:57,287 that these poles are artifacts of splitting up physics 635 00:44:57,287 --> 00:44:58,120 at different scales. 636 00:45:02,260 --> 00:45:04,330 And they will always be canceling out 637 00:45:04,330 --> 00:45:05,080 observable things. 638 00:45:05,080 --> 00:45:10,405 They're remnants of splitting up physics in particular ways. 639 00:45:12,910 --> 00:45:14,685 They always cancel out of observables. 640 00:45:14,685 --> 00:45:16,810 AUDIENCE: When you say physics at different scales, 641 00:45:16,810 --> 00:45:19,240 is that a proxy for taking a certain infinite 642 00:45:19,240 --> 00:45:20,188 subset of diagrams-- 643 00:45:20,188 --> 00:45:20,980 IAIN STEWART: Yeah. 644 00:45:20,980 --> 00:45:23,650 So if you like, we introduced a cutoff in our diagrams 645 00:45:23,650 --> 00:45:25,050 when we did them. 646 00:45:25,050 --> 00:45:25,550 Right. 647 00:45:25,550 --> 00:45:27,010 AUDIENCE: Like the number of gluon propagators [INAUDIBLE]?? 648 00:45:27,010 --> 00:45:27,802 IAIN STEWART: Yeah. 649 00:45:27,802 --> 00:45:28,923 I mean-- right. 650 00:45:28,923 --> 00:45:30,340 Effectively, you could think of it 651 00:45:30,340 --> 00:45:33,390 like that or the number of bubbles. 652 00:45:33,390 --> 00:45:34,330 Yeah. 653 00:45:34,330 --> 00:45:38,930 And that was a proxy that became this u variable. 654 00:45:38,930 --> 00:45:41,840 And that was becoming kind of a probe for the cutoff 655 00:45:41,840 --> 00:45:43,203 dependence. 656 00:45:43,203 --> 00:45:44,870 So we'll talk more about that right now. 657 00:45:54,340 --> 00:45:55,390 So what is this? 658 00:45:55,390 --> 00:45:57,015 How should you think about this cutoff? 659 00:46:19,340 --> 00:46:23,030 So when we remove the ambiguity, we actually introduce a scale. 660 00:46:23,030 --> 00:46:27,990 And the way you can think of that is as follows-- 661 00:46:27,990 --> 00:46:31,770 you should think that, in general, 662 00:46:31,770 --> 00:46:35,510 the relation between the pole mass and some other mass 663 00:46:35,510 --> 00:46:41,380 is some infinite series which has 664 00:46:41,380 --> 00:46:49,420 the general structure of a bunch of logarithms perhaps, 665 00:46:49,420 --> 00:46:55,530 and coupling constants, and some coefficients. 666 00:46:55,530 --> 00:46:57,420 So it's a double series in n and k. 667 00:46:57,420 --> 00:46:58,980 It starts at order alpha. 668 00:46:58,980 --> 00:47:02,225 So that's why n is starting at order one. 669 00:47:02,225 --> 00:47:04,850 And just because of dimensions-- if this is dimension 1, that's 670 00:47:04,850 --> 00:47:06,920 dimension 1-- there has to be something of a dimension 1 671 00:47:06,920 --> 00:47:07,753 that we stick there. 672 00:47:07,753 --> 00:47:11,930 And let's be flexible about we put what we put there 673 00:47:11,930 --> 00:47:25,990 and call it R. 674 00:47:25,990 --> 00:47:28,480 So this is a general scheme conversion formula. 675 00:47:28,480 --> 00:47:31,780 We can put any kind of conversion into that form. 676 00:47:50,990 --> 00:47:58,370 So if we call this thing here delta m, 677 00:47:58,370 --> 00:48:01,340 then I can also move it to the other side and write like this. 678 00:48:04,350 --> 00:48:08,570 And then, what you can say is that if I pick my delta m such 679 00:48:08,570 --> 00:48:13,522 that it's infrared structure, it's asymptotic 680 00:48:13,522 --> 00:48:15,230 structured alpha, is the same as the pole 681 00:48:15,230 --> 00:48:17,510 mass, I can set up something where I 682 00:48:17,510 --> 00:48:52,370 get a renormalon-free mass m R. 683 00:48:52,370 --> 00:48:54,950 So the pole mass has this renormalon 684 00:48:54,950 --> 00:48:56,750 that we've been talking about. 685 00:48:56,750 --> 00:48:59,390 If I set it up so that my subtractions also have 686 00:48:59,390 --> 00:49:02,850 the same kind of structure-- 687 00:49:02,850 --> 00:49:06,380 so for example, with the example that we did, 688 00:49:06,380 --> 00:49:09,200 you would need the nonlogarithmic terms 689 00:49:09,200 --> 00:49:15,930 to be n factorial 2 to the n, beta 0 to the n-- 690 00:49:20,710 --> 00:49:24,280 as long as we have the right terms in the delta m, 691 00:49:24,280 --> 00:49:29,330 we can construct something that's free of that renormalon. 692 00:49:29,330 --> 00:49:32,540 Now, in MS bar, which is what we did, 693 00:49:32,540 --> 00:49:38,650 this R scale is just the MS bar mass itself. 694 00:49:38,650 --> 00:49:43,450 Which is the running quantity, but it's evaluated at m bar. 695 00:49:43,450 --> 00:49:45,280 So the formula that we get would have 696 00:49:45,280 --> 00:49:49,210 m pole is equal to m bar plus m bar times a series in alpha. 697 00:49:49,210 --> 00:49:50,900 That was the conversion. 698 00:49:50,900 --> 00:49:53,830 So the scale R was fixed to be m bar. 699 00:49:57,523 --> 00:49:59,690 In this scheme that we talked about last time, which 700 00:49:59,690 --> 00:50:04,280 is called the 1S scheme, where we talked about defining 701 00:50:04,280 --> 00:50:08,360 a scheme by thinking about two heavy quarks calculating 702 00:50:08,360 --> 00:50:17,520 corrections between calculating perturbatively 703 00:50:17,520 --> 00:50:19,590 the potential between those heavy quarks 704 00:50:19,590 --> 00:50:23,610 and defining a mass that way, that's called the 1S scheme. 705 00:50:23,610 --> 00:50:27,510 In that particular case, you get an extra alpha 706 00:50:27,510 --> 00:50:31,200 that we talked about. 707 00:50:31,200 --> 00:50:34,290 And the corresponding R would actually 708 00:50:34,290 --> 00:50:39,690 be the m 1S mass times an alpha, which 709 00:50:39,690 --> 00:50:50,590 is effectively the inverse Bohr radius for this potential. 710 00:50:50,590 --> 00:50:52,340 But we can even be more general than that. 711 00:50:52,340 --> 00:50:54,800 We can make this R into just a free parameter. 712 00:50:54,800 --> 00:51:03,803 We don't have to fix it in the way that these two schemes do. 713 00:51:03,803 --> 00:51:05,470 We can think of it as a floating cutoff. 714 00:51:15,760 --> 00:51:20,410 And for any R that we pick, the ambiguity delta m 715 00:51:20,410 --> 00:51:22,820 will be R independent. 716 00:51:22,820 --> 00:51:25,420 Just like the m bar was cancelling out when we picked 717 00:51:25,420 --> 00:51:27,760 the MS bar scheme, when you pick the 1S scheme, 718 00:51:27,760 --> 00:51:28,900 that R cancels out. 719 00:51:28,900 --> 00:51:34,870 And generically, the ambiguity, which is just lambda QCD, 720 00:51:34,870 --> 00:51:41,130 is independent of R. 721 00:51:41,130 --> 00:51:43,003 So R is really setting a scale. 722 00:51:43,003 --> 00:51:45,420 So really what you're doing is you're cancelling something 723 00:51:45,420 --> 00:51:48,720 that's happening if you like it in the deep infrared. 724 00:51:48,720 --> 00:51:51,840 But whenever you cancel something, 725 00:51:51,840 --> 00:51:54,420 you have to sort of decide how much of the leftover 726 00:51:54,420 --> 00:51:55,260 do you take. 727 00:51:55,260 --> 00:51:57,780 You're canceling something at 0, but then you're 728 00:51:57,780 --> 00:52:00,420 taking something from 0 to something. 729 00:52:00,420 --> 00:52:04,830 And that "to something" is exactly what this R is. 730 00:52:04,830 --> 00:52:07,810 Just like in dim reg-- 731 00:52:07,810 --> 00:52:10,320 my spelling is atrocious. 732 00:52:16,560 --> 00:52:19,230 Just like in dim reg, when you cancel 733 00:52:19,230 --> 00:52:21,000 the UV pole that's at infinity. 734 00:52:21,000 --> 00:52:23,050 You have 1 over epsilon UV, but the mu comes in, 735 00:52:23,050 --> 00:52:28,030 and the mu is setting kind of a soft cutoff on your integrals. 736 00:52:28,030 --> 00:52:31,680 And you think of that as where you've moved down to. 737 00:52:31,680 --> 00:52:34,410 In the same way, this R is kind of saying, 738 00:52:34,410 --> 00:52:36,600 you're canceling something at 0, and we 739 00:52:36,600 --> 00:52:40,500 have to take along with it some fluctuations up to some scale. 740 00:52:40,500 --> 00:52:42,805 And R us providing that choice. 741 00:52:57,650 --> 00:53:03,290 So we take some infrared fluctuations together 742 00:53:03,290 --> 00:53:04,240 with the pole mass-- 743 00:53:07,310 --> 00:53:10,820 these are the infrared fluctuations that were causing 744 00:53:10,820 --> 00:53:15,140 the [? stability. ?] They are [? dressing ?] the pole mass, 745 00:53:15,140 --> 00:53:18,480 if you like, which is like a point particle mass. 746 00:53:18,480 --> 00:53:22,910 They're making it into a kind of [? dressed ?] mass, 747 00:53:22,910 --> 00:53:25,760 and they're yielding a well-defined mass parameter 748 00:53:25,760 --> 00:53:27,080 that then depends on a scale. 749 00:53:37,110 --> 00:53:39,950 So here's the origin. 750 00:53:39,950 --> 00:53:43,048 I think that you include all the fluctuations 751 00:53:43,048 --> 00:53:44,840 in some sphere around the origin node to R. 752 00:53:44,840 --> 00:53:46,257 That's the right physical picture. 753 00:54:09,700 --> 00:54:11,470 So the right physical picture here 754 00:54:11,470 --> 00:54:14,170 is that when we're using the MS bar scheme, 755 00:54:14,170 --> 00:54:16,000 we're ignoring powers. 756 00:54:16,000 --> 00:54:19,060 And there was actually an improper separation 757 00:54:19,060 --> 00:54:22,030 between certain power law terms. 758 00:54:22,030 --> 00:54:24,070 Those power law terms that you didn't really 759 00:54:24,070 --> 00:54:26,150 see very well in MS bar can be seen 760 00:54:26,150 --> 00:54:28,720 when you look at the asymptotics of the perturbation series. 761 00:54:28,720 --> 00:54:30,773 They come back to haunt you. 762 00:54:30,773 --> 00:54:33,190 You can look at the asymptotics of the perturbation series 763 00:54:33,190 --> 00:54:37,130 using these renormalon Borel-style techniques. 764 00:54:37,130 --> 00:54:38,890 And you can figure out how to define 765 00:54:38,890 --> 00:54:41,950 masses that are sensitive to these problems, 766 00:54:41,950 --> 00:54:43,435 like the MS bar mass. 767 00:54:43,435 --> 00:54:45,610 The MS bar mass has too large an r scale 768 00:54:45,610 --> 00:54:47,740 to be used for b physics. 769 00:54:47,740 --> 00:54:50,613 The 1S mass is something that has a small enough scale 770 00:54:50,613 --> 00:54:52,030 that you can use it for b physics, 771 00:54:52,030 --> 00:54:56,050 although it would have potential problems for top quark physics. 772 00:54:56,050 --> 00:54:58,210 But you can actually make the scale 773 00:54:58,210 --> 00:55:00,250 where you're absorbing the fluctuations up 774 00:55:00,250 --> 00:55:02,860 to a free parameter, and then you're 775 00:55:02,860 --> 00:55:07,000 good to go with any physical problem, 776 00:55:07,000 --> 00:55:11,785 because you can choose it to be an appropriate scale, much 777 00:55:11,785 --> 00:55:13,660 like we choose mu to be an appropriate scale. 778 00:55:19,330 --> 00:55:22,090 So let me give one very definite example 779 00:55:22,090 --> 00:55:28,295 of how you could define a scheme with this R. 780 00:55:28,295 --> 00:55:30,670 So we'll build something that's where we don't have to do 781 00:55:30,670 --> 00:55:34,078 any additional calculations. 782 00:55:34,078 --> 00:55:35,620 What we're going to do is we're going 783 00:55:35,620 --> 00:55:40,450 to use the MS bar mass, the MS bar pole scheme 784 00:55:40,450 --> 00:55:47,760 conversion, and from that we define the a's that we need. 785 00:55:50,980 --> 00:55:58,240 We can take mu equal to R so that the log of mu over R is 0, 786 00:55:58,240 --> 00:56:01,560 and then we don't need any of the other a's. 787 00:56:10,050 --> 00:56:14,160 So then we have a formula that's a definite thing that 788 00:56:14,160 --> 00:56:14,910 looks as follows-- 789 00:56:22,780 --> 00:56:26,280 so we're tweaking the MS bar result, if you like, 790 00:56:26,280 --> 00:56:31,650 so that instead of having a fixed scale, 791 00:56:31,650 --> 00:56:32,880 it has a floating scale. 792 00:56:38,008 --> 00:56:39,300 So these are the MS bar values. 793 00:56:42,900 --> 00:56:45,570 And we don't have the MS bar front, we have an R in front. 794 00:56:52,500 --> 00:56:55,320 And we have R's in these couplings. 795 00:56:55,320 --> 00:56:59,700 And that's a well-defined scheme called the MSR scheme. 796 00:56:59,700 --> 00:57:01,320 And in this scheme, you have a cut off 797 00:57:01,320 --> 00:57:03,960 R that you can just pick as a parameter and adjust it. 798 00:57:03,960 --> 00:57:05,960 It's just a parameter like mu. 799 00:57:05,960 --> 00:57:07,710 It's part of the definition of the scheme. 800 00:57:10,290 --> 00:57:12,570 And so this kind of change here would 801 00:57:12,570 --> 00:57:16,020 be good for doing physics. 802 00:57:16,020 --> 00:57:19,578 This is a good scheme for doing physics 803 00:57:19,578 --> 00:57:21,810 where you want to absorb things up to that cutoff. 804 00:57:21,810 --> 00:57:28,680 So you can sort of-- 805 00:57:31,320 --> 00:57:31,820 well-- 806 00:57:41,840 --> 00:57:44,140 So if you're thinking about HQET, 807 00:57:44,140 --> 00:57:47,920 you can pick the R to be say a GV freely, 808 00:57:47,920 --> 00:57:49,660 and it wouldn't care about whether you're 809 00:57:49,660 --> 00:57:52,735 talking about a top quark or a bottom quark or a charm quark. 810 00:57:52,735 --> 00:57:53,860 It's just a free parameter. 811 00:57:53,860 --> 00:57:57,430 You can pick it to be something fixed, 812 00:57:57,430 --> 00:57:59,800 and it wouldn't depend on what kind of mass 813 00:57:59,800 --> 00:58:02,860 you're starting over here with. 814 00:58:02,860 --> 00:58:05,620 So you're just kind of decoupling this cutoff 815 00:58:05,620 --> 00:58:06,790 from the mass itself. 816 00:58:09,810 --> 00:58:10,310 All right. 817 00:58:18,650 --> 00:58:19,540 Any questions? 818 00:58:23,370 --> 00:58:25,740 AUDIENCE: So when you wrote the series in k that you not 819 00:58:25,740 --> 00:58:28,000 got rid of with this choice, that 820 00:58:28,000 --> 00:58:30,788 was just you being careful about including-- 821 00:58:30,788 --> 00:58:31,830 IAIN STEWART: Everything. 822 00:58:31,830 --> 00:58:32,690 AUDIENCE: [INAUDIBLE]. 823 00:58:32,690 --> 00:58:33,482 IAIN STEWART: Yeah. 824 00:58:33,482 --> 00:58:35,490 I mean, I was just being careful about including 825 00:58:35,490 --> 00:58:37,490 all the kind of things that in principle I 826 00:58:37,490 --> 00:58:41,160 could see in any arbitrary choice of scheme change. 827 00:58:41,160 --> 00:58:42,660 And in any arbitrary choice, there's 828 00:58:42,660 --> 00:58:44,370 both mu, which we defined-- 829 00:58:44,370 --> 00:58:46,320 that was related to the alpha-- 830 00:58:46,320 --> 00:58:48,030 and there was this R showing up. 831 00:58:48,030 --> 00:58:51,300 And we got logs of mu over R. And in MS bar, 832 00:58:51,300 --> 00:58:55,680 the R was in bar, but you still had mu showing up as well. 833 00:58:55,680 --> 00:59:01,170 And mu is related to the running of the MS bar mass. 834 00:59:01,170 --> 00:59:04,020 But in general, the fact that you have both mu and R 835 00:59:04,020 --> 00:59:05,370 is kind of-- 836 00:59:05,370 --> 00:59:06,660 they were separate. 837 00:59:06,660 --> 00:59:09,000 Because you define mu for the log cutoffs. 838 00:59:09,000 --> 00:59:11,880 And you defined R here for the power log cutoffs-- 839 00:59:11,880 --> 00:59:13,515 the power log divergences. 840 00:59:13,515 --> 00:59:15,390 At the end of the day, you might as well just 841 00:59:15,390 --> 00:59:16,900 take those cutoffs to be the same. 842 00:59:16,900 --> 00:59:18,483 And that's what I just did-- mu equals 843 00:59:18,483 --> 00:59:22,608 R. That gets rid of the logs. 844 00:59:22,608 --> 00:59:24,850 AUDIENCE: Do people worry about u equals 1? 845 00:59:24,850 --> 00:59:26,850 IAIN STEWART: Yeah, you can worry about u equals 846 00:59:26,850 --> 00:59:30,000 1. u equals 1/2 is the first thing you should worry about. 847 00:59:30,000 --> 00:59:32,310 If you want to go to u equals 1, then you 848 00:59:32,310 --> 00:59:34,830 need problems that have kind of more information 849 00:59:34,830 --> 00:59:36,032 in perturbation theory. 850 00:59:36,032 --> 00:59:37,740 So if you just have two loop information, 851 00:59:37,740 --> 00:59:40,800 u equals 1/2 is usually perfectly fine. 852 00:59:40,800 --> 00:59:43,140 If you have four loop information, 853 00:59:43,140 --> 00:59:46,710 you probably should start worrying about u equals 1. 854 00:59:46,710 --> 00:59:47,870 That's just kind of rough. 855 00:59:52,160 --> 00:59:53,615 So people don't often talk about u 856 00:59:53,615 --> 00:59:55,700 equals 1, though sometimes they do. 857 01:00:00,450 --> 01:00:00,950 OK. 858 01:00:00,950 --> 01:00:03,740 So I need a little aside here in order 859 01:00:03,740 --> 01:00:08,970 to proceed with what I want to talk about next. 860 01:00:08,970 --> 01:00:12,020 And that is I just want to define for you how 861 01:00:12,020 --> 01:00:14,690 you would define lambda QCD at higher orders of perturbation 862 01:00:14,690 --> 01:00:16,130 theory. 863 01:00:16,130 --> 01:00:20,420 So the formula we used a minute ago was leading log, 864 01:00:20,420 --> 01:00:24,422 but we can define lambda QCD even if we go to higher orders. 865 01:00:24,422 --> 01:00:26,630 And I just want to do a little bit of algebra, set up 866 01:00:26,630 --> 01:00:29,720 some of the notation in order to define this thing at higher 867 01:00:29,720 --> 01:00:31,650 orders. 868 01:00:31,650 --> 01:00:35,263 So MS bar beta function-- 869 01:00:42,840 --> 01:00:48,960 and we're going to just work to all orders formally. 870 01:00:48,960 --> 01:00:52,853 And we'll see that we can construct a solution where it's 871 01:00:52,853 --> 01:00:55,020 kind of obvious how you would work to whatever order 872 01:00:55,020 --> 01:00:55,950 you want to work. 873 01:01:00,730 --> 01:01:02,580 So let's just parameterize the series, 874 01:01:02,580 --> 01:01:04,920 even if we don't know all the terms. 875 01:01:04,920 --> 01:01:06,510 We only up to beta 4. 876 01:01:06,510 --> 01:01:09,060 But let's parameterize the series as a bunch of betas. 877 01:01:22,720 --> 01:01:24,610 I'm going to call the cutoff R, because we've 878 01:01:24,610 --> 01:01:25,610 started calling it that. 879 01:01:25,610 --> 01:01:27,730 But I could call it mu. 880 01:01:34,240 --> 01:01:39,430 I just took mu equal to R. So I can rearrange 881 01:01:39,430 --> 01:01:42,370 that equation in this way-- 882 01:01:42,370 --> 01:01:46,540 I can write it as dR over R is equal to d alpha over beta. 883 01:01:46,540 --> 01:01:49,720 And then I can integrate on both sides between two values. 884 01:01:49,720 --> 01:01:52,147 And on this integral, it's just giving me a log. 885 01:01:52,147 --> 01:01:54,355 This integral, it's giving me an integral over alpha. 886 01:02:15,440 --> 01:02:17,770 So what I'm going to want to do here 887 01:02:17,770 --> 01:02:20,480 is make a change of variables. 888 01:02:20,480 --> 01:02:23,880 And we'll see the following change of intervals convenient. 889 01:02:29,170 --> 01:02:31,690 I'm going to make a change of variables in the integration 890 01:02:31,690 --> 01:02:34,540 to this t. 891 01:02:34,540 --> 01:02:37,450 So I have some dummy variable that I'm integrating over. 892 01:02:37,450 --> 01:02:40,450 Let's change the dummy variable t. 893 01:02:40,450 --> 01:02:44,230 And basically that's going to simplify the lowest order 894 01:02:44,230 --> 01:02:47,260 result. 895 01:02:47,260 --> 01:02:52,030 Because the lowest order result d alpha over beta of alpha, 896 01:02:52,030 --> 01:02:54,877 well, the beta goes like alpha squared at lowest order. 897 01:02:54,877 --> 01:02:56,710 So I'm basically making a change of variable 898 01:02:56,710 --> 01:03:01,043 where this d alpha over beta at lowest order will just be DT. 899 01:03:01,043 --> 01:03:01,960 That's what I'm doing. 900 01:03:08,865 --> 01:03:10,910 I have to change variables in the limits as well. 901 01:03:17,180 --> 01:03:42,960 So I call those t1 and t0 and change variables 902 01:03:42,960 --> 01:03:44,593 consistently everywhere. 903 01:03:52,980 --> 01:04:04,870 And if I do that, this is the kind of form of the result. 904 01:04:04,870 --> 01:04:07,020 So this is just a Laurent series in 1 over t. 905 01:04:15,540 --> 01:04:25,330 And the first term, we had d alpha over minus beta 0 906 01:04:25,330 --> 01:04:28,180 over 2 pi alpha squared. 907 01:04:28,180 --> 01:04:31,000 And that was just minus dt. 908 01:04:31,000 --> 01:04:33,760 So the reason that this thing here is 1 909 01:04:33,760 --> 01:04:35,770 is because my change of variable with dt 910 01:04:35,770 --> 01:04:38,320 was exactly taking into account all the factors 911 01:04:38,320 --> 01:04:40,353 to basically just make it into a 1. 912 01:04:40,353 --> 01:04:42,520 And there's a sign change, but I flipped the limits. 913 01:04:46,908 --> 01:04:48,700 So we made the first term trivial, and then 914 01:04:48,700 --> 01:04:50,110 the higher terms of these terms. 915 01:04:50,110 --> 01:04:54,933 And these b hats we could figure out what they are. 916 01:04:54,933 --> 01:04:57,100 And they're just combinations of the beta functions. 917 01:05:09,520 --> 01:05:13,110 So once we expand the 1 over beta, 918 01:05:13,110 --> 01:05:16,610 we get various combinations of things. 919 01:05:16,610 --> 01:05:17,700 I think I won't write b3. 920 01:05:23,420 --> 01:05:26,900 We could also do the integral-- 921 01:05:26,900 --> 01:05:29,930 pretty easy integral to do. 922 01:05:29,930 --> 01:05:32,210 And so if you want to write a solution for what 923 01:05:32,210 --> 01:05:34,290 the integral is, you could say, well, it's 924 01:05:34,290 --> 01:05:36,920 some function that you get from doing 925 01:05:36,920 --> 01:05:39,500 the indefinite integral evaluated 926 01:05:39,500 --> 01:05:42,500 at the upper limit minus that function evaluated at the lower 927 01:05:42,500 --> 01:05:45,110 limit. 928 01:05:45,110 --> 01:05:50,610 And G is just that function. 929 01:05:50,610 --> 01:05:52,880 So doing the integral of 1 gives t. 930 01:05:55,598 --> 01:05:57,140 Doing the integral of the other time, 931 01:05:57,140 --> 01:06:00,410 and being careful about what's positive and what's negative, 932 01:06:00,410 --> 01:06:02,240 I can write as log of minus t. 933 01:06:09,610 --> 01:06:12,707 So t was negative. 934 01:06:12,707 --> 01:06:14,290 And if you look at my definition of t, 935 01:06:14,290 --> 01:06:17,210 it had a minus sign in it. 936 01:06:17,210 --> 01:06:19,060 So this log is a logical positive number. 937 01:06:24,140 --> 01:06:28,550 So we have this G of t and G prime of t is just b hat. 938 01:06:32,370 --> 01:06:33,840 And if we write a formula-- 939 01:06:33,840 --> 01:06:36,007 which is not what we're actually interested in here, 940 01:06:36,007 --> 01:06:38,390 but we might as well write it anyway-- 941 01:06:38,390 --> 01:06:41,230 it's this way. 942 01:06:41,230 --> 01:06:44,825 Just taking that formula and writing it this way, 943 01:06:44,825 --> 01:06:46,700 then if you think about what this formula is, 944 01:06:46,700 --> 01:06:50,000 it's an all orders relation between the coupling 945 01:06:50,000 --> 01:06:51,830 at one scale and the coupling at another. 946 01:06:55,712 --> 01:06:57,170 It's a formal relation, because you 947 01:06:57,170 --> 01:07:00,200 need to know the coefficients. 948 01:07:00,200 --> 01:07:02,180 But if someone tells you the coefficients, 949 01:07:02,180 --> 01:07:03,680 you can plug them into this formula. 950 01:07:11,860 --> 01:07:14,860 And this is the extension of the usual running formula 951 01:07:14,860 --> 01:07:17,560 that you would write down for relating alpha at one 952 01:07:17,560 --> 01:07:18,910 scale to alpha at another scale. 953 01:07:18,910 --> 01:07:21,535 This is a generalization that includes higher order 954 01:07:21,535 --> 01:07:22,660 terms in the beta function. 955 01:07:30,535 --> 01:07:31,910 So the other thing that we can do 956 01:07:31,910 --> 01:07:35,630 is we can rearrange the formula and take account of the fact 957 01:07:35,630 --> 01:07:39,500 that we have R1's and R0's, and we can make it such 958 01:07:39,500 --> 01:07:42,590 that we have only R1's on one side of the formula 959 01:07:42,590 --> 01:07:46,520 and R0's on the other side of the formula 960 01:07:46,520 --> 01:07:48,140 by splitting the log into two pieces. 961 01:07:51,170 --> 01:07:55,600 And that's what you do to define lambda QCD. 962 01:07:55,600 --> 01:07:58,390 So we can rearrange it as follows-- 963 01:07:58,390 --> 01:07:59,245 taking an exponent. 964 01:08:07,000 --> 01:08:10,560 So now we only have our R0's in this part, R1's in that part. 965 01:08:10,560 --> 01:08:13,750 This thing has to be independent of the choice of the R, 966 01:08:13,750 --> 01:08:16,680 and that's just this constant lambda QCD. 967 01:08:16,680 --> 01:08:18,899 And that's the generalization of this definition 968 01:08:18,899 --> 01:08:20,660 of lambda QCD that we had before, 969 01:08:20,660 --> 01:08:25,873 where it was R0 e to the minus 2 pi over beta 0 alpha. 970 01:08:25,873 --> 01:08:27,540 Now we have the higher terms that I just 971 01:08:27,540 --> 01:08:28,890 decoded in this G function. 972 01:08:36,470 --> 01:08:42,189 So if you didn't remember that formula for the lambda QCD, 973 01:08:42,189 --> 01:08:43,240 now we've derived it. 974 01:08:47,170 --> 01:08:51,170 And we included the higher order terms as well. 975 01:08:51,170 --> 01:08:57,180 So the next term would be this. 976 01:08:57,180 --> 01:09:00,060 [INAUDIBLE] higher order terms. 977 01:09:00,060 --> 01:09:02,970 And usually you call this the leading log expression 978 01:09:02,970 --> 01:09:06,630 for lambda QCD, with the next leading log once you 979 01:09:06,630 --> 01:09:07,850 include that term, et cetera. 980 01:09:11,550 --> 01:09:13,710 But the whole combination is mu independent. 981 01:09:13,710 --> 01:09:16,770 The mu independence of this term is only cancelling to leading 982 01:09:16,770 --> 01:09:18,600 log. 983 01:09:18,600 --> 01:09:22,720 Once you include this term, it's cancelling to next leading log. 984 01:09:22,720 --> 01:09:24,960 The whole thing is mu independent, 985 01:09:24,960 --> 01:09:27,840 getting this from lambda QCD. 986 01:09:27,840 --> 01:09:30,970 So that's end of the aside. 987 01:09:30,970 --> 01:09:33,240 So what I actually want to do is I 988 01:09:33,240 --> 01:09:36,420 want to come back to our definition of this MS bar mass, 989 01:09:36,420 --> 01:09:38,460 and I want to treat it as if we have-- 990 01:09:38,460 --> 01:09:40,950 I want to think about it like an RG. 991 01:09:40,950 --> 01:09:42,180 We have this cutoff. 992 01:09:42,180 --> 01:09:45,090 Let's see if we can write down an renormalization group 993 01:09:45,090 --> 01:09:46,859 equation for that mass. 994 01:09:46,859 --> 01:09:48,218 It has a cutoff-- 995 01:09:48,218 --> 01:09:50,010 there should be some equation that tells us 996 01:09:50,010 --> 01:09:51,180 how to flow in that cutoff. 997 01:09:51,180 --> 01:09:52,763 Let's see what we can learn from that. 998 01:09:55,705 --> 01:09:57,560 So that's end of aside. 999 01:10:09,160 --> 01:10:21,070 So we have a cutoff, and we already 1000 01:10:21,070 --> 01:10:23,140 said that we can treat R as a variable that 1001 01:10:23,140 --> 01:10:24,610 parameterizes the mass scheme. 1002 01:10:37,430 --> 01:10:42,500 And so we can vary R in this MSR scheme, 1003 01:10:42,500 --> 01:10:45,140 just like we varied u in the MS bar scheme. 1004 01:10:55,163 --> 01:10:56,580 So, if you like, what you're doing 1005 01:10:56,580 --> 01:10:58,247 when you're vary mu in the MS bar scheme 1006 01:10:58,247 --> 01:11:00,390 is you're properly taking into account kind of what 1007 01:11:00,390 --> 01:11:02,190 fluctuations you want to put into your mass 1008 01:11:02,190 --> 01:11:04,230 in the ultraviolet. 1009 01:11:04,230 --> 01:11:07,860 And here it's related to this kind of physics that's 1010 01:11:07,860 --> 01:11:12,817 going on in the deep infrared that you're also 1011 01:11:12,817 --> 01:11:13,650 being careful about. 1012 01:11:18,030 --> 01:11:23,880 So we'll have instead of a mu RGE we'll have an R RGE. 1013 01:11:23,880 --> 01:11:25,950 The difference between the mu RGE and the R RGE 1014 01:11:25,950 --> 01:11:29,010 is this time we're talking about powers. 1015 01:11:29,010 --> 01:11:30,765 We'll see what happens. 1016 01:11:35,720 --> 01:11:39,850 So M pole didn't depend on R. So you get 0 for that. 1017 01:11:39,850 --> 01:11:42,960 And if we just look at our scheme change formula, 1018 01:11:42,960 --> 01:11:46,440 then we can figure out what the RGE is. 1019 01:11:56,210 --> 01:12:01,580 This delta M was a function of R. And we can define this as R, 1020 01:12:01,580 --> 01:12:03,923 to get the dimensions right, times some kind 1021 01:12:03,923 --> 01:12:06,215 of anomalous dimension that's just a function of alpha. 1022 01:12:28,850 --> 01:12:32,740 So what this guy would look like is again just a series-- 1023 01:12:48,873 --> 01:12:49,915 some perturbative series. 1024 01:12:52,830 --> 01:13:02,436 RGE is writing out our equation. 1025 01:13:02,436 --> 01:13:06,049 The equation we've come to is this. 1026 01:13:10,540 --> 01:13:14,500 So R d by dR of the mass, because of the fact 1027 01:13:14,500 --> 01:13:16,780 that we're dealing with a power, we 1028 01:13:16,780 --> 01:13:19,130 get an R on the right-hand side of this equation. 1029 01:13:19,130 --> 01:13:20,950 And that's really the only thing that's 1030 01:13:20,950 --> 01:13:24,280 causing a difference from this kind of standard RGE's 1031 01:13:24,280 --> 01:13:25,390 that we've solved so far. 1032 01:13:29,930 --> 01:13:32,330 And the fact that we got this power, which was 1, 1033 01:13:32,330 --> 01:13:34,295 is exactly related to the fact that it was a u 1034 01:13:34,295 --> 01:13:35,750 equals 1/2 renormalon. 1035 01:13:35,750 --> 01:13:38,300 If we looked at the renormalons that are further out, 1036 01:13:38,300 --> 01:13:42,450 like u equals 1 would give R squared. 1037 01:13:42,450 --> 01:13:45,260 We haven't seen really enough to identify that, but that's true. 1038 01:13:47,830 --> 01:13:54,770 So let's solve this guy in the usual way. 1039 01:14:13,725 --> 01:14:14,725 Integrate on both sides. 1040 01:14:26,230 --> 01:14:30,250 From what we just talked about, R is equal to lambda QCD e 1041 01:14:30,250 --> 01:14:33,860 to the minus G of t. 1042 01:14:33,860 --> 01:14:36,550 And we can also therefore write that d 1043 01:14:36,550 --> 01:14:44,280 log R is, from this formula. dt minus G prime of t. 1044 01:14:44,280 --> 01:14:46,780 If we follow that through as a change of variable, that's 1045 01:14:46,780 --> 01:14:48,400 the change of variable we get. 1046 01:14:48,400 --> 01:14:50,570 So let's switch variable from R to dt. 1047 01:14:53,090 --> 01:14:57,080 So d log R is the product of these things, 1048 01:14:57,080 --> 01:15:02,000 and I can write it actually kind of a compact way 1049 01:15:02,000 --> 01:15:14,477 if I write it is lambda QCD dt d by dt of e to the minus G of t. 1050 01:15:14,477 --> 01:15:16,060 So we've got a G prime, and then we've 1051 01:15:16,060 --> 01:15:17,423 got an e to the minus G of t. 1052 01:15:17,423 --> 01:15:19,590 Let's just write that as a total derivative of that. 1053 01:15:44,180 --> 01:15:47,860 So if I write it in terms of t, then that's the solution. 1054 01:15:47,860 --> 01:15:50,530 That's the formal solution in terms of-- 1055 01:15:50,530 --> 01:15:53,590 and I switched variable also in the anomalous dimension. 1056 01:15:53,590 --> 01:15:58,500 And that's the formal solution to the RGE. 1057 01:15:58,500 --> 01:16:01,923 And this is a well-defined integral. 1058 01:16:01,923 --> 01:16:03,340 There's no problems with doing it. 1059 01:16:06,980 --> 01:16:09,050 The place where the integral could start 1060 01:16:09,050 --> 01:16:10,790 causing problems is t equals 0. 1061 01:16:10,790 --> 01:16:12,440 But we're never getting to t equals 0. 1062 01:16:12,440 --> 01:16:14,815 We have cutoffs that are keeping us away from t equals 0. 1063 01:16:25,640 --> 01:16:27,350 So it shouldn't surprise you that it's 1064 01:16:27,350 --> 01:16:30,170 a well-defined integral, because we said that these things are 1065 01:16:30,170 --> 01:16:33,110 supposed to be well defined in masses 1066 01:16:33,110 --> 01:16:35,883 that are not sensitive to these renormalon problems. 1067 01:16:35,883 --> 01:16:37,550 And taking the difference of two of them 1068 01:16:37,550 --> 01:16:40,280 should again give us something that's 1069 01:16:40,280 --> 01:16:42,030 not sensitive to renormalon problems. 1070 01:16:42,030 --> 01:16:47,060 But it's interesting that you see lambda QCD popping out. 1071 01:16:47,060 --> 01:16:49,580 And that's because these two masses and the difference 1072 01:16:49,580 --> 01:16:51,620 of these cutoffs is related to absorbing 1073 01:16:51,620 --> 01:16:54,470 a different amount of these fluctuations which 1074 01:16:54,470 --> 01:16:57,470 were related to lambda QCD. 1075 01:17:32,760 --> 01:17:38,430 So the evolution would just yield a new well-defined mass 1076 01:17:38,430 --> 01:17:44,070 of R1, which absorbs a different amount of IR fluctuations-- 1077 01:17:50,630 --> 01:17:54,050 a shell out to R1 instead of just out to R0 if you like. 1078 01:18:06,760 --> 01:18:09,370 So you're absorbing those fluctuations together with them 1079 01:18:09,370 --> 01:18:12,760 M pole to get something well defined. 1080 01:18:12,760 --> 01:18:15,910 And how much you absorb is related to this cutoff. 1081 01:18:19,630 --> 01:18:22,720 So we could do that integral at whatever order we decide. 1082 01:18:22,720 --> 01:18:24,100 Let's just do it at leading log. 1083 01:18:31,770 --> 01:18:35,870 So at leading log, we take this guy 1084 01:18:35,870 --> 01:18:43,190 and we just take the first term, some constant, alpha over 4 pi. 1085 01:18:43,190 --> 01:18:48,470 And so gamma R of t, switching variables, 1086 01:18:48,470 --> 01:18:50,810 becomes that constant over 2 beta 0 1087 01:18:50,810 --> 01:18:53,870 if we keep all the factors. 1088 01:18:53,870 --> 01:18:55,460 And then a 1 over t-- 1089 01:18:55,460 --> 01:18:56,810 alpha comes 1 over t. 1090 01:19:08,148 --> 01:19:09,940 So if we're at leading log, we should think 1091 01:19:09,940 --> 01:19:11,470 about having the leading log-- 1092 01:19:11,470 --> 01:19:13,240 we'll have the QCD. 1093 01:19:13,240 --> 01:19:15,832 So that's what the 0 means-- 1094 01:19:15,832 --> 01:19:18,540 there's this factor. 1095 01:19:18,540 --> 01:19:24,800 And the integral we'd have to do would be this one, 1096 01:19:24,800 --> 01:19:27,458 e to the minus t over t. 1097 01:19:27,458 --> 01:19:30,000 So you see that t equals 0 is where there could be a problem. 1098 01:19:30,000 --> 01:19:32,070 But away from t equals 0, there's no problem. 1099 01:19:41,220 --> 01:19:45,060 If you do that integral, it gives you an incomplete gamma 1100 01:19:45,060 --> 01:19:46,453 function. 1101 01:19:46,453 --> 01:19:48,120 That's just a way of saying that there's 1102 01:19:48,120 --> 01:19:50,370 a special function that's defined to be that integral. 1103 01:19:55,530 --> 01:19:57,215 Mathematica knows all about it. 1104 01:19:57,215 --> 01:19:58,250 It's very useful. 1105 01:20:04,638 --> 01:20:06,430 And actually, the incomplete gamma function 1106 01:20:06,430 --> 01:20:11,990 is an example of a function that has an asymptotic expansion. 1107 01:20:11,990 --> 01:20:16,540 So if we look at what happens if I expand this incomplete gamma 1108 01:20:16,540 --> 01:20:18,980 function in alpha, t, Remember, is 1 over alpha. 1109 01:20:18,980 --> 01:20:21,190 It can expand about alpha. 1110 01:20:21,190 --> 01:20:26,090 Then it has a series that has factorial growth. 1111 01:20:26,090 --> 01:20:27,370 Let me just write that. 1112 01:20:41,410 --> 01:20:43,870 So expanding about alpha equals 0 is expanding about t 1113 01:20:43,870 --> 01:20:45,108 equals infinity. 1114 01:21:02,220 --> 01:21:09,160 And the gamma function actually has an asymptotic expansion. 1115 01:21:09,160 --> 01:21:15,480 And if you just work it out, plug it into Mathematica-- 1116 01:21:15,480 --> 01:21:17,710 it's a Taylor series-- 1117 01:21:17,710 --> 01:21:24,190 look at the terms, and you exactly see this-- 1118 01:21:24,190 --> 01:21:28,840 2 to the n n factorial, which is a u equals 1/2 renormalon. 1119 01:21:36,582 --> 01:21:38,040 And so what's happening is when you 1120 01:21:38,040 --> 01:21:41,220 take the difference of two of these gammas, 1121 01:21:41,220 --> 01:21:42,968 you're converting this asymptotic series. 1122 01:21:42,968 --> 01:21:45,510 If you take the difference of two of these asymptotic series, 1123 01:21:45,510 --> 01:21:48,070 you're converting it into a convergent series. 1124 01:21:48,070 --> 01:21:50,320 And we'll talk a little bit more about that next time, 1125 01:21:50,320 --> 01:21:51,153 but I'll stop there. 1126 01:21:53,650 --> 01:21:56,830 So we can see, from this renormalization group, 1127 01:21:56,830 --> 01:22:00,130 again the renormalon, and we can see in what way 1128 01:22:00,130 --> 01:22:02,320 it's sort of solved by thinking about making 1129 01:22:02,320 --> 01:22:04,190 it shift from here to here. 1130 01:22:04,190 --> 01:22:06,190 Because it's the difference of these two gammas, 1131 01:22:06,190 --> 01:22:08,720 not one of them individually. 1132 01:22:08,720 --> 01:22:10,630 So we can use normalization group techniques 1133 01:22:10,630 --> 01:22:13,450 to think about these renormalon effects. 1134 01:22:13,450 --> 01:22:15,700 And we'll talk a little bit more about that next time. 1135 01:22:20,570 --> 01:22:23,802 AUDIENCE: So is this a convergent series now? 1136 01:22:23,802 --> 01:22:26,260 IAIN STEWART: The difference of these is convergent series. 1137 01:22:26,260 --> 01:22:28,600 Yeah. 1138 01:22:28,600 --> 01:22:31,502 I'll write that down next time and show you it.