1 00:00:00,000 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,270 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,270 --> 00:00:18,278 at ocw.mit.edu. 8 00:00:21,990 --> 00:00:24,210 IAIN STEWART: So where were we? 9 00:00:24,210 --> 00:00:27,330 So last time, we were talking about MS-bar scheme 10 00:00:27,330 --> 00:00:29,300 and renormalons. 11 00:00:29,300 --> 00:00:31,920 And so we talked about the MS-bar scheme. 12 00:00:31,920 --> 00:00:33,180 We talked about renormalons. 13 00:00:33,180 --> 00:00:35,580 And we said that we could introduce a mass scheme that 14 00:00:35,580 --> 00:00:39,790 has an arbitrary power log cutoff, which we called R 15 00:00:39,790 --> 00:00:44,420 to distinguish it from mu, or lambda or some other cutoff. 16 00:00:44,420 --> 00:00:48,620 And so the idea of this scheme right 17 00:00:48,620 --> 00:00:53,558 here was perturbing a way from MS-bar 18 00:00:53,558 --> 00:00:55,100 to get rid of this renormalon problem 19 00:00:55,100 --> 00:00:59,430 that MS-bar has, but retain all the nice features of MS-bar. 20 00:00:59,430 --> 00:01:02,450 In particular, we don't really want to calculate anything new 21 00:01:02,450 --> 00:01:04,379 if we can avoid it. 22 00:01:04,379 --> 00:01:08,510 And that's what this scheme does because really what it does 23 00:01:08,510 --> 00:01:11,600 is it uses the fact that we know that the MS-bar scheme has 24 00:01:11,600 --> 00:01:13,280 this renormalon. 25 00:01:13,280 --> 00:01:15,740 So this series here has a renormalon. 26 00:01:15,740 --> 00:01:18,530 And it just takes that series over from MS-bar, 27 00:01:18,530 --> 00:01:21,350 puts it in with a cutoff, which is this R, 28 00:01:21,350 --> 00:01:23,795 and defines a new mass scheme using that formula. 29 00:01:27,460 --> 00:01:29,520 So this is like a mass scheme that 30 00:01:29,520 --> 00:01:31,710 has an adjustable cutoff, which we can take to be 31 00:01:31,710 --> 00:01:33,310 at whatever scale we want. 32 00:01:33,310 --> 00:01:34,830 And in particular, at HQET, you'd 33 00:01:34,830 --> 00:01:38,280 want to take that R to be something like a GeV type scale 34 00:01:38,280 --> 00:01:40,980 because that wouldn't spoil your power counting. 35 00:01:40,980 --> 00:01:45,600 So recall, in HQET, that this delta m 36 00:01:45,600 --> 00:01:48,900 should be, by power counting, or order lambda QCD. 37 00:01:48,900 --> 00:01:54,730 So technically, you would take R somewhat greater than lambda 38 00:01:54,730 --> 00:01:57,405 QCD, but of order lambda QCD. 39 00:02:00,070 --> 00:02:00,570 OK. 40 00:02:00,570 --> 00:02:02,310 So now, we have this extra cutoff. 41 00:02:02,310 --> 00:02:05,350 So when we have a cutoff, we have a renormalization group. 42 00:02:05,350 --> 00:02:07,200 And if you think about the renormalization 43 00:02:07,200 --> 00:02:09,789 group for MS-bar mass, it would be something like mu 44 00:02:09,789 --> 00:02:13,710 d by d mu of M of mu is equal to anomalous dimension times M 45 00:02:13,710 --> 00:02:15,325 of mu back again. 46 00:02:15,325 --> 00:02:16,575 This one's a little different. 47 00:02:16,575 --> 00:02:19,140 It's R d by dR of M of R. And then 48 00:02:19,140 --> 00:02:21,060 there's no M on the right-hand side. 49 00:02:21,060 --> 00:02:25,250 It's an additive renormalization to get rid of the renormalon. 50 00:02:25,250 --> 00:02:29,640 And there's a power, also, of R here. 51 00:02:29,640 --> 00:02:32,910 So it's not just summing logs. 52 00:02:32,910 --> 00:02:35,310 It is summing some logs related to the running of alpha, 53 00:02:35,310 --> 00:02:37,058 but it's also got this power. 54 00:02:37,058 --> 00:02:38,850 And when you solve this anomalous dimension 55 00:02:38,850 --> 00:02:41,258 equation, which we talked about last time, 56 00:02:41,258 --> 00:02:42,300 you get an integral like. 57 00:02:42,300 --> 00:02:45,060 In this t variable and at leading log order, 58 00:02:45,060 --> 00:02:48,030 it looks like this, difference of two incomplete gamma 59 00:02:48,030 --> 00:02:49,200 functions. 60 00:02:49,200 --> 00:02:51,000 Because of the mass dimensions, it 61 00:02:51,000 --> 00:02:52,530 has to be made up by mass dimensions 62 00:02:52,530 --> 00:02:53,940 on the right-hand side. 63 00:02:53,940 --> 00:02:55,530 Coupling is dimensionless. 64 00:02:55,530 --> 00:02:58,230 The only thing that has dimensions is lambda QCD. 65 00:02:58,230 --> 00:03:00,660 And that's exactly what pops out of solving this equation. 66 00:03:00,660 --> 00:03:05,890 The R gets converted to a lambda QCD by this formula here. 67 00:03:05,890 --> 00:03:09,270 This g of t was just t at lowest order. 68 00:03:09,270 --> 00:03:11,140 So this is an all-order form here. 69 00:03:11,140 --> 00:03:14,345 And this is the leading log solution there. 70 00:03:14,345 --> 00:03:15,720 So if we look at that leading log 71 00:03:15,720 --> 00:03:18,060 and what is the t0 or the t1, that's 72 00:03:18,060 --> 00:03:21,540 just the same formula here, but with alpha 73 00:03:21,540 --> 00:03:25,560 at R1 and R0, the boundary conditions. 74 00:03:25,560 --> 00:03:27,570 We want to run from R0 to R1. 75 00:03:27,570 --> 00:03:29,860 And this formula tells us how to do that. 76 00:03:29,860 --> 00:03:32,760 And we get a result here that has the coupling at R0 77 00:03:32,760 --> 00:03:35,820 and the coupling at R1. 78 00:03:35,820 --> 00:03:39,510 OK, so if we look at that result, it's interesting. 79 00:03:39,510 --> 00:03:42,360 Because if we just had one of these gamma functions, 80 00:03:42,360 --> 00:03:44,280 these incomplete gamma functions, then we 81 00:03:44,280 --> 00:03:45,950 would have a renormalon. 82 00:03:45,950 --> 00:03:50,050 And that's where I stopped last time. 83 00:03:50,050 --> 00:03:54,760 So just looking at this lambda QCD times gamma 84 00:03:54,760 --> 00:03:58,860 and expanding in a series of about 0, which 85 00:03:58,860 --> 00:04:03,860 is expanding in the coupling at infinity, 86 00:04:03,860 --> 00:04:06,540 you get this series, which has a 2 to the n n factorial. 87 00:04:06,540 --> 00:04:07,860 It's an asymptotic series. 88 00:04:07,860 --> 00:04:10,020 This is a classic example of a function 89 00:04:10,020 --> 00:04:13,570 that has an asymptotic series, this incomplete gamma function. 90 00:04:13,570 --> 00:04:15,860 There's some exponential that combines together 91 00:04:15,860 --> 00:04:19,829 with the lambda QCD to give an R. So I pulled that out front. 92 00:04:19,829 --> 00:04:22,200 And then we have this series, which is exactly u 93 00:04:22,200 --> 00:04:23,850 equals 1/2 renormalon. 94 00:04:23,850 --> 00:04:26,490 And the u equals 1/2 had to do with this power 2 to the n 95 00:04:26,490 --> 00:04:29,190 here, the fact that it's 2 to the n and not 96 00:04:29,190 --> 00:04:32,770 some other number. 97 00:04:32,770 --> 00:04:34,310 But if we take the difference here-- 98 00:04:34,310 --> 00:04:36,190 and this is what I said in words last time, but now 99 00:04:36,190 --> 00:04:37,360 I'll write it down in equations. 100 00:04:37,360 --> 00:04:39,310 If we take the difference there, that actually 101 00:04:39,310 --> 00:04:43,000 doesn't have this renormalon. 102 00:04:43,000 --> 00:04:52,710 So this is the difference of two series, each of which 103 00:04:52,710 --> 00:04:53,710 is asymptotic. 104 00:05:02,932 --> 00:05:04,390 If we want to compare those series, 105 00:05:04,390 --> 00:05:09,565 we should expand them in the same coupling constant. 106 00:05:17,840 --> 00:05:18,720 So let me do that. 107 00:05:41,880 --> 00:05:45,290 So if I take the difference of those two series using this 108 00:05:45,290 --> 00:05:49,260 formula, but now I re-expand all these alphas in terms of alphas 109 00:05:49,260 --> 00:05:51,123 at the R1 scale-- 110 00:05:51,123 --> 00:05:52,540 so when they're at the R0 scale, I 111 00:05:52,540 --> 00:05:55,790 re-expand them in terms of alpha at the R1 scale. 112 00:05:55,790 --> 00:05:57,050 Do some rearranging. 113 00:05:57,050 --> 00:06:00,440 I can write the result in this form as this. 114 00:06:00,440 --> 00:06:03,330 So the key thing is that difference there. 115 00:06:03,330 --> 00:06:05,810 And if you think about what this sum here is, 116 00:06:05,810 --> 00:06:08,630 this is kind of like an exponential, something 117 00:06:08,630 --> 00:06:10,250 to a power, k factorial. 118 00:06:10,250 --> 00:06:12,597 Except it's limited to n. 119 00:06:12,597 --> 00:06:14,930 So if I really went all the way up to n equals infinity, 120 00:06:14,930 --> 00:06:16,880 this would be just giving actually exponential 121 00:06:16,880 --> 00:06:18,275 of R1 over R0. 122 00:06:18,275 --> 00:06:23,150 It would cancel the R0 over R1, which would then cancel the 1. 123 00:06:23,150 --> 00:06:26,420 You can rearrange the series a little bit 124 00:06:26,420 --> 00:06:29,616 to make it more obvious that it's convergent. 125 00:06:47,925 --> 00:06:49,550 Because there is this n factorial here. 126 00:06:49,550 --> 00:06:51,650 And you might worry, well, this thing here 127 00:06:51,650 --> 00:06:55,560 has to fall fast enough to get rid of that n factorial. 128 00:06:55,560 --> 00:06:58,130 So write this thing out, this 1, as an infinite series 129 00:06:58,130 --> 00:07:01,340 of terms from 0 to infinity of the same form as this. 130 00:07:01,340 --> 00:07:05,390 And then what you're left with is the following thing. 131 00:07:05,390 --> 00:07:08,000 All the terms from 0 to n cancel. 132 00:07:08,000 --> 00:07:10,250 And then you're just left with the terms from n plus 1 133 00:07:10,250 --> 00:07:11,396 to infinity. 134 00:07:21,962 --> 00:07:23,670 And then you can see that the n factorial 135 00:07:23,670 --> 00:07:26,040 is being tamed by another factorial growth, which 136 00:07:26,040 --> 00:07:29,710 is always of a greater power, n plus 1. 137 00:07:29,710 --> 00:07:32,670 OK, so this is a number less than 1. 138 00:07:32,670 --> 00:07:36,900 And this thing here, beta 0 alpha over 2 pi times the log, 139 00:07:36,900 --> 00:07:39,368 is also something that's less than 1. 140 00:07:39,368 --> 00:07:41,160 It's exactly the thing that you sum up when 141 00:07:41,160 --> 00:07:42,928 you're running the coupling. 142 00:07:42,928 --> 00:07:44,220 So this is a convergent series. 143 00:07:55,300 --> 00:07:58,600 And basically the physics of it is 144 00:07:58,600 --> 00:08:02,350 we had this mass that didn't have a renormalon, 145 00:08:02,350 --> 00:08:04,257 but it had an arbitrary scale. 146 00:08:04,257 --> 00:08:06,340 And what the renormalization group allows us to do 147 00:08:06,340 --> 00:08:08,243 is move to another scale. 148 00:08:08,243 --> 00:08:09,910 When we move to another scale, it better 149 00:08:09,910 --> 00:08:12,340 be that we don't reintroduce the renormalon. 150 00:08:12,340 --> 00:08:15,510 And the fact that I can write this as a convergent series 151 00:08:15,510 --> 00:08:17,170 shows very explicitly that I'm not 152 00:08:17,170 --> 00:08:20,140 having that problem in the difference between these two. 153 00:08:28,870 --> 00:08:32,330 So it's renormalon free. 154 00:08:32,330 --> 00:08:36,870 And it remains renormalon free when we change the scale. 155 00:08:36,870 --> 00:08:39,150 And we can sum up these logarithms 156 00:08:39,150 --> 00:08:41,780 that have to do with changing the scale. 157 00:08:50,610 --> 00:08:51,970 Why would you care about that? 158 00:08:51,970 --> 00:08:56,080 Well, imagine that you extract this mass from some B physics. 159 00:08:56,080 --> 00:08:59,040 So you take an r. that's of order GeV. 160 00:08:59,040 --> 00:09:01,410 And then you say, well, that's a very precise number. 161 00:09:01,410 --> 00:09:04,835 But how do I use it for, say, a LHC phenomenology? 162 00:09:04,835 --> 00:09:06,960 Well, if I want to connect it to LHC phenomenology, 163 00:09:06,960 --> 00:09:10,020 I probably want to convert to the MS-bar scheme 164 00:09:10,020 --> 00:09:11,430 because LHC is high energy. 165 00:09:11,430 --> 00:09:12,950 Maybe I'm doing Higgs to bb-bar. 166 00:09:12,950 --> 00:09:18,600 Bb-bar-- very energetic, so we want to use the MS-bar scheme. 167 00:09:18,600 --> 00:09:20,903 But I've got this mass at a very low scale. 168 00:09:20,903 --> 00:09:22,320 What you would want to do is you'd 169 00:09:22,320 --> 00:09:24,240 want to use them the renormalization group, 170 00:09:24,240 --> 00:09:29,260 run it up to the mass scale Mb, switch schemes from this scheme 171 00:09:29,260 --> 00:09:31,740 where you extracted the mass to MS-bar. 172 00:09:31,740 --> 00:09:34,170 You'd get a correspondingly precise value of a MS-bar 173 00:09:34,170 --> 00:09:37,170 because the series between this mass and the MS-bar mass 174 00:09:37,170 --> 00:09:39,920 is, again, a renormalon free series. 175 00:09:39,920 --> 00:09:42,150 And then you could take that mass at the Mb scale, 176 00:09:42,150 --> 00:09:44,160 run it up to, say, the Higgs scale, 177 00:09:44,160 --> 00:09:46,122 and use it for phenomenology. 178 00:09:46,122 --> 00:09:47,580 So you always want to be able to go 179 00:09:47,580 --> 00:09:49,165 back and forth between schemes. 180 00:09:49,165 --> 00:09:51,540 And if different schemes have different scales associated 181 00:09:51,540 --> 00:09:55,020 to them, then you need to the renormalization group 182 00:09:55,020 --> 00:09:58,200 in order to put them at the same scale where you 183 00:09:58,200 --> 00:10:00,900 want to do the conversion, OK? 184 00:10:00,900 --> 00:10:03,900 So that's why being able to sum up 185 00:10:03,900 --> 00:10:06,930 these logarithms without reintroducing any renormalon 186 00:10:06,930 --> 00:10:09,758 problems is important. 187 00:10:13,180 --> 00:10:18,370 And in general, if you were to try to do this with MS-bar, 188 00:10:18,370 --> 00:10:19,130 it wouldn't work. 189 00:10:23,780 --> 00:10:25,730 You don't really have a way of treating 190 00:10:25,730 --> 00:10:27,800 the renormalon in the MS-bar scheme when 191 00:10:27,800 --> 00:10:30,800 you're talking about physics below the mass of the particle. 192 00:10:34,040 --> 00:10:37,500 OK, so we can generalize this little discussion here, 193 00:10:37,500 --> 00:10:40,730 which was a fixed order leading log discussion. 194 00:10:40,730 --> 00:10:43,430 And I can write down for you just what would happen 195 00:10:43,430 --> 00:10:46,070 if I were to formally integrate this integral here 196 00:10:46,070 --> 00:10:49,640 without making any expansion in alpha S, OK? 197 00:11:00,830 --> 00:11:04,590 So express gamma R of t as an infinite series. 198 00:11:04,590 --> 00:11:06,500 And it's an infant series in alpha S. 199 00:11:06,500 --> 00:11:09,140 That means it's an infinite series in 1 over t. 200 00:11:09,140 --> 00:11:13,550 We had an expression for this capital G of t last time, which 201 00:11:13,550 --> 00:11:16,310 again was an infinite series. 202 00:11:16,310 --> 00:11:18,200 Just plug in those infinite series. 203 00:11:18,200 --> 00:11:19,490 Do all the integrals. 204 00:11:19,490 --> 00:11:23,420 They all turned out to be incomplete gamma functions, OK? 205 00:11:23,420 --> 00:11:29,280 So we can write down a formula which 206 00:11:29,280 --> 00:11:31,350 is an all-order generalization of that result 207 00:11:31,350 --> 00:11:34,737 just to see what happens at higher orders. 208 00:11:34,737 --> 00:11:36,570 So basically, we would have something like-- 209 00:11:49,495 --> 00:11:52,260 if I look back at what g of t was, 210 00:11:52,260 --> 00:11:54,450 we'd have something like this. 211 00:11:54,450 --> 00:11:56,760 And then all the rest of the terms 212 00:11:56,760 --> 00:12:05,430 we can expand out in an inverse Taylor series in 1 over t. 213 00:12:05,430 --> 00:12:07,193 So just there's two terms in g of t 214 00:12:07,193 --> 00:12:08,610 I want to keep in the exponential. 215 00:12:08,610 --> 00:12:10,860 And then the rest of them I can expand out and combine 216 00:12:10,860 --> 00:12:11,568 with these terms. 217 00:12:11,568 --> 00:12:13,440 And that just gives me some series. 218 00:12:13,440 --> 00:12:17,620 And I can do these integrals. 219 00:12:17,620 --> 00:12:37,390 So writing the integrand out, we can just establish some 220 00:12:37,390 --> 00:12:47,858 notation for it for that infinite series of 1 of t's. 221 00:12:47,858 --> 00:12:49,525 And we'll just call the coefficients Sj. 222 00:12:55,290 --> 00:12:57,075 There is 1 over t. 223 00:13:02,170 --> 00:13:04,050 Yeah. 224 00:13:04,050 --> 00:13:07,198 So this gamma bar starts as a 1 over t. 225 00:13:07,198 --> 00:13:12,438 So really it should be a [INAUDIBLE] 1 over t here. 226 00:13:12,438 --> 00:13:13,980 That's where this 1 over t came from. 227 00:13:16,600 --> 00:13:20,730 So then integrate that, and that gives us 228 00:13:20,730 --> 00:13:25,770 a generalization of this formula here where we 229 00:13:25,770 --> 00:13:31,590 say that M of R1 minus M of R0 where we work 230 00:13:31,590 --> 00:13:36,180 at some order in the resummation, what you would say 231 00:13:36,180 --> 00:13:37,680 is, N to the KLL. 232 00:13:37,680 --> 00:13:39,930 So you have leading log, next leading log, 233 00:13:39,930 --> 00:13:42,120 next to next leading log. 234 00:13:42,120 --> 00:13:44,970 And when you have K of them, you say NKLL. 235 00:13:47,840 --> 00:13:50,520 And the solution of that equation 236 00:13:50,520 --> 00:14:00,600 would be the k-th order lambda QCD and then this series. 237 00:14:07,590 --> 00:14:12,060 And instead of just the 0-th incomplete gamma function, 238 00:14:12,060 --> 00:14:14,430 we get a slightly different one. 239 00:14:14,430 --> 00:14:16,290 But it's got the same kind of structure 240 00:14:16,290 --> 00:14:18,420 where we get the difference of gamma functions. 241 00:14:31,960 --> 00:14:33,550 And these S's are just whatever you 242 00:14:33,550 --> 00:14:35,230 get from the anomalous dimension, 243 00:14:35,230 --> 00:14:37,985 so kind of simple notation. 244 00:14:50,132 --> 00:14:51,630 [INAUDIBLE] 245 00:14:51,630 --> 00:14:54,220 So they're just some combination of numbers. 246 00:14:54,220 --> 00:14:56,110 And the twiddles here is just my shorthand 247 00:14:56,110 --> 00:15:00,790 for accounting for a fact that there's 248 00:15:00,790 --> 00:15:04,500 some beta 0s floating around. 249 00:15:04,500 --> 00:15:05,500 So this is just algebra. 250 00:15:09,540 --> 00:15:11,650 And these b1 hats and b2 hats were 251 00:15:11,650 --> 00:15:15,790 things that showed up in the beta function of QCD, 252 00:15:15,790 --> 00:15:19,250 which we also talked about last time. 253 00:15:19,250 --> 00:15:20,920 So we defined those last time. 254 00:15:20,920 --> 00:15:24,370 The b1 hat and b2 hat were just combinations of the beta 1 255 00:15:24,370 --> 00:15:29,120 and beta 2 and beta 0. 256 00:15:29,120 --> 00:15:31,600 So these are some numbers that you can calculate 257 00:15:31,600 --> 00:15:33,338 given the anomalous dimensions. 258 00:15:33,338 --> 00:15:35,630 Given those numbers, you can plug it into this formula. 259 00:15:35,630 --> 00:15:38,860 And then you have a generalization 260 00:15:38,860 --> 00:15:40,070 of the previous result. 261 00:15:40,070 --> 00:15:42,445 So whatever order you know the anomalous dimension to you 262 00:15:42,445 --> 00:15:43,750 can use that solution. 263 00:15:43,750 --> 00:15:46,375 And we'll use it a little later when I talk about some numbers. 264 00:15:51,180 --> 00:15:53,750 All right, so any questions so far? 265 00:15:59,380 --> 00:16:01,380 There's a similar thing for MS-bar mass. 266 00:16:01,380 --> 00:16:03,395 If you were summing logs of the MS-bar mass, 267 00:16:03,395 --> 00:16:04,770 there's an all-orders formula you 268 00:16:04,770 --> 00:16:07,263 could write down and use once you 269 00:16:07,263 --> 00:16:09,130 know the anomalous dimensions. 270 00:16:09,130 --> 00:16:13,340 You could find it in the particle data book I think. 271 00:16:13,340 --> 00:16:18,810 All right, so just coming back to this comment about the fact 272 00:16:18,810 --> 00:16:21,450 that the anomalous dimension is renormalon free-- 273 00:16:26,430 --> 00:16:29,205 so the fact that the solution here was renormalon free, 274 00:16:29,205 --> 00:16:31,080 that the difference here was renormalon free, 275 00:16:31,080 --> 00:16:32,600 was because there was no renormalon 276 00:16:32,600 --> 00:16:35,100 on the right-hand side here and that the anomalous dimension 277 00:16:35,100 --> 00:16:37,668 was renormalon free. 278 00:16:37,668 --> 00:16:39,210 And it was constructed from something 279 00:16:39,210 --> 00:16:44,940 that had a renormalon, which is this delta M, 280 00:16:44,940 --> 00:16:48,330 but it was a derivative of that delta M. 281 00:16:48,330 --> 00:16:53,440 And so the derivative kills the renormalon. 282 00:16:53,440 --> 00:17:00,430 So this thing is free of the delta M of order lambda QCD 283 00:17:00,430 --> 00:17:04,655 renormalon because that renormalon is a constant. 284 00:17:04,655 --> 00:17:06,030 And when you take the derivative, 285 00:17:06,030 --> 00:17:07,170 you kill the constant. 286 00:17:07,170 --> 00:17:09,869 That's how you can think of it. 287 00:17:09,869 --> 00:17:13,500 If you start writing out the gamma functions, 288 00:17:13,500 --> 00:17:16,714 then you see that it looks like you're finding the renormalon. 289 00:17:16,714 --> 00:17:18,089 But then there's some differences 290 00:17:18,089 --> 00:17:21,030 that show up and cancel it off. 291 00:17:21,030 --> 00:17:24,810 So the way that that works, it's kind 292 00:17:24,810 --> 00:17:26,310 of like this where you sort of think 293 00:17:26,310 --> 00:17:27,630 that you have the renormalon, but then you 294 00:17:27,630 --> 00:17:29,760 realize there's another term and it cancels it off. 295 00:17:48,830 --> 00:17:55,760 So if we look back at our formula for delta M, 296 00:17:55,760 --> 00:17:58,660 this had an infinite series of a's. 297 00:17:58,660 --> 00:18:01,310 This was like a sum over sum a n's, and some alpha 298 00:18:01,310 --> 00:18:05,300 to the n, some other factors. 299 00:18:05,300 --> 00:18:07,370 And these gammas are just related to the a's. 300 00:18:12,260 --> 00:18:13,940 But they're not just related to the a's. 301 00:18:13,940 --> 00:18:17,330 They're also factors of the beta function that come in. 302 00:18:17,330 --> 00:18:20,390 So just to give you an idea-- 303 00:18:25,625 --> 00:18:28,890 so once you know the scheme conversion to the scheme 304 00:18:28,890 --> 00:18:33,670 and you know the anomalous dimension 305 00:18:33,670 --> 00:18:35,930 and if you look at this series and you think about 306 00:18:35,930 --> 00:18:37,910 whether the series for the anomalous dimension 307 00:18:37,910 --> 00:18:48,570 is convergent or asymptotic, you can sort of 308 00:18:48,570 --> 00:18:51,702 identify the pattern of how these coefficients look. 309 00:18:51,702 --> 00:18:53,160 And basically, if you, for example, 310 00:18:53,160 --> 00:18:56,460 look at the bubble sum, this a n plus 1 311 00:18:56,460 --> 00:18:59,250 would be something that has an n factorial 312 00:18:59,250 --> 00:19:03,720 2 beta 0 to the n plus 1-- 313 00:19:03,720 --> 00:19:07,060 sorry, to the n growth. 314 00:19:07,060 --> 00:19:14,640 And then this thing here is n minus 1 factorial to beta 0 315 00:19:14,640 --> 00:19:17,040 to the n minus 1. 316 00:19:17,040 --> 00:19:21,690 And then there's this extra 2n beta 0 that multiplies it. 317 00:19:21,690 --> 00:19:24,390 And you can see that there is an exact cancellation. 318 00:19:27,220 --> 00:19:29,220 One way of probing the renormalon 319 00:19:29,220 --> 00:19:30,780 is just to look at the bubble sum. 320 00:19:30,780 --> 00:19:33,900 And the bubble sum gave us a particular value of these a's. 321 00:19:33,900 --> 00:19:37,115 And you can see that the bubble sum renormalon is cancelling 322 00:19:37,115 --> 00:19:38,240 out between the terms here. 323 00:19:47,170 --> 00:19:50,550 So that's just an expression of what I said over here. 324 00:19:50,550 --> 00:19:53,400 That's the kind of more definite expression of the fact 325 00:19:53,400 --> 00:19:55,657 that this thing is actually free of the renormalon. 326 00:19:59,717 --> 00:20:01,550 So I'm going to put this aside for a minute. 327 00:20:01,550 --> 00:20:03,470 And we'll come back in to use this formula a little bit 328 00:20:03,470 --> 00:20:04,580 later in today's lecture. 329 00:20:08,698 --> 00:20:11,240 So there's sort of two ways that you can use this technology. 330 00:20:11,240 --> 00:20:14,660 You can use it as a means of doing what I said, 331 00:20:14,660 --> 00:20:16,450 of connecting masses at different scales 332 00:20:16,450 --> 00:20:17,450 and doing phenomenology. 333 00:20:17,450 --> 00:20:20,160 And we'll come back to that a little bit later. 334 00:20:20,160 --> 00:20:23,163 But you could also use it as a probe of the renormalon. 335 00:20:23,163 --> 00:20:25,580 So one thing that you might complain about with the bubble 336 00:20:25,580 --> 00:20:28,757 sum is that it's just some arbitrarily chosen way 337 00:20:28,757 --> 00:20:29,840 of probing the renormalon. 338 00:20:29,840 --> 00:20:32,150 And maybe there's some problem where you don't 339 00:20:32,150 --> 00:20:33,620 have light fermions around. 340 00:20:33,620 --> 00:20:35,500 Maybe you only have gluons. 341 00:20:35,500 --> 00:20:38,160 And there could still be a renormalon in those problems. 342 00:20:38,160 --> 00:20:40,730 So how would you deal with that if you 343 00:20:40,730 --> 00:20:43,280 didn't have the light quarks available to make up 344 00:20:43,280 --> 00:20:44,000 this bubble sum? 345 00:20:44,000 --> 00:20:45,950 Or maybe there's some renormalon that 346 00:20:45,950 --> 00:20:49,010 just so happens you don't see it with the quarks. 347 00:20:49,010 --> 00:20:50,270 There's no guarantee of that. 348 00:20:50,270 --> 00:20:52,220 So you'd like to have some other mechanism 349 00:20:52,220 --> 00:20:53,900 for looking for renormalons. 350 00:20:53,900 --> 00:20:56,570 And you can actually use the renormalization group 351 00:20:56,570 --> 00:20:58,657 to do that, which is kind of interesting. 352 00:20:58,657 --> 00:20:59,990 So I'll show you how that works. 353 00:21:10,935 --> 00:21:12,560 A nice thing about this is that it also 354 00:21:12,560 --> 00:21:15,310 makes it clear how the renormalon relates 355 00:21:15,310 --> 00:21:16,427 to the Landau pole. 356 00:21:16,427 --> 00:21:17,510 So we'll see that as well. 357 00:21:25,420 --> 00:21:28,912 So take our solution for the RGE, 358 00:21:28,912 --> 00:21:30,870 which we could write in terms of this integral. 359 00:21:41,768 --> 00:21:43,810 And let's consider what this integral actually is 360 00:21:43,810 --> 00:21:47,060 doing in the complex t plane. 361 00:21:47,060 --> 00:21:58,210 So the t's are negative, and so they look like this. 362 00:21:58,210 --> 00:22:01,540 The Landau pole where the coupling blows up 363 00:22:01,540 --> 00:22:03,920 is where the t goes to 0. 364 00:22:03,920 --> 00:22:05,645 So the Landau pole is at the origin. 365 00:22:08,822 --> 00:22:10,780 So this is what the complex t plane looks like. 366 00:22:13,482 --> 00:22:15,440 And what we're doing when we do the integral is 367 00:22:15,440 --> 00:22:19,040 we're just doing an integral over that little range 368 00:22:19,040 --> 00:22:20,690 here, right? 369 00:22:20,690 --> 00:22:22,460 So we're far away from the Landau pole. 370 00:22:22,460 --> 00:22:24,418 And that's why everything was nicely convergent 371 00:22:24,418 --> 00:22:28,210 and you could connect one mass to another. 372 00:22:28,210 --> 00:22:31,220 But you could ask, what happens if I take this t0 373 00:22:31,220 --> 00:22:34,240 and I move it somewhere else in this picture? 374 00:22:34,240 --> 00:22:36,720 And that's what we're going to do. 375 00:22:36,720 --> 00:22:39,980 So let's consider the limit where R0 goes to 0. 376 00:22:44,090 --> 00:22:47,240 If R goes to 0 and you look back at the formula 377 00:22:47,240 --> 00:22:50,150 that we had that converts between the MS-bar mass 378 00:22:50,150 --> 00:22:56,375 and this M of R, you'll see that M of R0 goes to M pole. 379 00:22:59,750 --> 00:23:02,420 Because we had a power of R times a series of alpha 380 00:23:02,420 --> 00:23:04,010 of R's, which are just logs of R. 381 00:23:04,010 --> 00:23:06,380 The power always wins, and so the mass 382 00:23:06,380 --> 00:23:11,990 goes to the pole mass and that limit that R0 goes to 0. 383 00:23:11,990 --> 00:23:20,420 And if you look at what T0 is, T0 384 00:23:20,420 --> 00:23:22,730 is related to the coupling, which 385 00:23:22,730 --> 00:23:25,850 I could write in terms of a log of lambda QCD. 386 00:23:25,850 --> 00:23:30,390 And T0 is basically in this limit going to plus infinity. 387 00:23:30,390 --> 00:23:33,200 So what happens is, if I want to get back to the pole mass 388 00:23:33,200 --> 00:23:37,010 from one of these masses, I basically have to take this guy 389 00:23:37,010 --> 00:23:39,830 and move out to infinity, which means 390 00:23:39,830 --> 00:23:42,800 I have to make a choice about going around that Landau pole. 391 00:24:01,900 --> 00:24:04,036 So basically, you're taking your RGE 392 00:24:04,036 --> 00:24:06,530 and you're pushing it into a region 393 00:24:06,530 --> 00:24:12,680 where you're no longer completely perturbative 394 00:24:12,680 --> 00:24:16,640 and that you're forced to continue by Landau pole. 395 00:24:21,140 --> 00:24:23,698 Now, you can actually avoid the Landau pole. 396 00:24:28,550 --> 00:24:30,770 It's true that the series becomes 397 00:24:30,770 --> 00:24:33,230 non-perturbative in this region, but you could go way up 398 00:24:33,230 --> 00:24:36,350 here and come back over there. 399 00:24:36,350 --> 00:24:39,800 So you can actually formally stay 400 00:24:39,800 --> 00:24:41,720 in a region in the complex plane where 401 00:24:41,720 --> 00:24:45,615 you're perturbation theory is still applicable. 402 00:24:45,615 --> 00:24:47,490 But you still have to decide which way you're 403 00:24:47,490 --> 00:24:50,320 going to go around it. 404 00:24:50,320 --> 00:24:52,490 And so that introduces an ambiguity. 405 00:25:04,370 --> 00:25:19,627 So doing that, say we go above or something. 406 00:25:27,420 --> 00:25:34,520 And if you plug in this result, the lowest order leading log 407 00:25:34,520 --> 00:25:48,600 result, you get this integral. 408 00:25:48,600 --> 00:25:50,580 And this integral, you know, you have 409 00:25:50,580 --> 00:25:55,603 to go around the t equals 0 pole because t1 is negative. 410 00:25:55,603 --> 00:25:58,020 So it turns out that you can actually take this integral-- 411 00:25:58,020 --> 00:26:00,180 and I'm going to leave this for your problem set-- 412 00:26:08,690 --> 00:26:21,115 and write it in the form of a Borel integral 413 00:26:21,115 --> 00:26:25,360 with an exponential that's exactly the Borel transform 414 00:26:25,360 --> 00:26:26,750 variable. 415 00:26:26,750 --> 00:26:30,410 So I just change the variable from t to u. 416 00:26:30,410 --> 00:26:32,840 So I get this integral. 417 00:26:32,840 --> 00:26:38,330 And if you do that, what you find is that f of u 418 00:26:38,330 --> 00:26:43,800 is proportional to 1 over u minus 1/2. 419 00:26:43,800 --> 00:26:45,800 AUDIENCE: What's the lower limit of integration? 420 00:26:45,800 --> 00:26:47,136 IAIN STEWART: 0. 421 00:26:47,136 --> 00:26:47,636 Yeah. 422 00:26:51,080 --> 00:26:55,590 OK, so you can actually see that the RGE and what was the Landau 423 00:26:55,590 --> 00:26:58,710 pole here is effectively, in this integral here, 424 00:26:58,710 --> 00:27:01,050 becoming this Borel pole at u equals 1/2. 425 00:27:11,700 --> 00:27:14,970 And that makes a little more clear the fact 426 00:27:14,970 --> 00:27:18,570 that these poles are related to the non-perturbative physics 427 00:27:18,570 --> 00:27:22,260 that's in the coupling and the infrared physics. 428 00:27:24,820 --> 00:27:26,220 So formally, this actually allows 429 00:27:26,220 --> 00:27:31,950 you to set up a method of trying to find the renormalon. 430 00:27:31,950 --> 00:27:36,900 You can ask, given some series, if I sort of construct 431 00:27:36,900 --> 00:27:40,393 this thing, what if I just had this attitude 432 00:27:40,393 --> 00:27:42,810 that we want to pick that series to cancel the renormalon? 433 00:27:42,810 --> 00:27:44,227 Now, take the attitude, what if we 434 00:27:44,227 --> 00:27:47,150 didn't know that that series had a renormalon? 435 00:27:47,150 --> 00:27:49,860 I could still throw it in, make a change of variable 436 00:27:49,860 --> 00:27:51,420 to some mass. 437 00:27:51,420 --> 00:27:53,160 If it's a convergent series, then I'm 438 00:27:53,160 --> 00:27:55,285 just defining some new mass. 439 00:27:55,285 --> 00:27:56,910 And I could go through this technology. 440 00:27:56,910 --> 00:28:03,430 And it should turn out that, if the series was OK, so to speak, 441 00:28:03,430 --> 00:28:06,510 then you wouldn't find this renormalon. 442 00:28:06,510 --> 00:28:07,920 You wouldn't find a pole there. 443 00:28:11,100 --> 00:28:12,540 I won't go through this in detail 444 00:28:12,540 --> 00:28:17,520 because it is a bit tricky to derive. 445 00:28:17,520 --> 00:28:21,570 But you can derive what's called a sum rule for the renormalon, 446 00:28:21,570 --> 00:28:25,560 which is another way of probing it by basically 447 00:28:25,560 --> 00:28:28,270 taking our all-order solution. 448 00:28:28,270 --> 00:28:32,400 So I took our leading log solution here, right? 449 00:28:32,400 --> 00:28:35,850 If I did the same thing with the all-order solution, 450 00:28:35,850 --> 00:28:39,360 I could also take the Borel transform of that. 451 00:28:39,360 --> 00:28:42,675 And that, if I do that, gives me this thing called the sum rule. 452 00:29:01,570 --> 00:29:03,870 So it's just manipulating an infinite series. 453 00:29:03,870 --> 00:29:08,530 And Mathematica can sum some of them up for you. 454 00:29:08,530 --> 00:29:14,430 So it's just tedious. 455 00:29:20,070 --> 00:29:29,890 And what you find is a formula for the residue of the Borel 456 00:29:29,890 --> 00:29:31,610 pole. 457 00:29:31,610 --> 00:29:33,850 So let me call that residue P 1/2. 458 00:29:43,220 --> 00:29:45,110 And the answer is actually pretty simple. 459 00:29:57,780 --> 00:29:59,450 So these S's were these coefficients 460 00:29:59,450 --> 00:30:01,040 that were showing up in our solution 461 00:30:01,040 --> 00:30:03,620 for the all-orders RGE. 462 00:30:03,620 --> 00:30:06,410 P 1/2 is a series which will tell me 463 00:30:06,410 --> 00:30:08,810 the residue of the pole. 464 00:30:08,810 --> 00:30:13,980 So that means that, if P 1/2 is not equal to 0, 465 00:30:13,980 --> 00:30:15,650 then you have a u equals 1/2 pole. 466 00:30:24,520 --> 00:30:32,900 And if 1/2 is equal to 0, then you have no u equals 1/2 pole. 467 00:30:32,900 --> 00:30:36,140 Now, in practice, you don't have an all-orders result. 468 00:30:36,140 --> 00:30:38,497 You only know a finite number of these S's. 469 00:30:38,497 --> 00:30:40,580 So what you actually do is you look at this series 470 00:30:40,580 --> 00:30:44,000 and try to figure out where it's converging to, OK? 471 00:30:44,000 --> 00:30:46,160 But actually, even with the knowledge 472 00:30:46,160 --> 00:30:50,120 that we have of the series, for example, for the b quark mass, 473 00:30:50,120 --> 00:30:53,390 you can very quickly see that it has a u equals 1/2 pole. 474 00:30:53,390 --> 00:30:55,072 And you can even do some error analysis 475 00:30:55,072 --> 00:30:56,030 by varying some things. 476 00:30:56,030 --> 00:30:59,810 And you get some kind of ID of the theoretical uncertainty 477 00:30:59,810 --> 00:31:02,000 from higher order terms in the series. 478 00:31:02,000 --> 00:31:06,232 If you just formally, say, take a series that's convergent, 479 00:31:06,232 --> 00:31:07,940 if I hand you a series that's convergent, 480 00:31:07,940 --> 00:31:10,760 you can calculate all these SK's. 481 00:31:10,760 --> 00:31:14,410 And then you can very easily see that actually there's no u 482 00:31:14,410 --> 00:31:15,080 equals 1/2 pole. 483 00:31:15,080 --> 00:31:17,740 What happens is that all the terms are cancelling in this 484 00:31:17,740 --> 00:31:20,953 series, and you just get 0. 485 00:31:20,953 --> 00:31:23,120 So it's kind of like the cancellations I was talking 486 00:31:23,120 --> 00:31:26,480 about where I was cancelling a poll in the anomalous 487 00:31:26,480 --> 00:31:29,000 dimension, but here it's happening in this thing if I 488 00:31:29,000 --> 00:31:30,080 have a convergent series. 489 00:31:30,080 --> 00:31:33,050 So this is a way of probing for renormalons, which 490 00:31:33,050 --> 00:31:34,790 doesn't rely on the bubble sum. 491 00:31:34,790 --> 00:31:37,160 And it basically uses the perturbative information 492 00:31:37,160 --> 00:31:41,900 that you have available, whether it's non-abelian or abelian 493 00:31:41,900 --> 00:31:42,560 information. 494 00:31:42,560 --> 00:31:43,380 AUDIENCE: Iain? 495 00:31:43,380 --> 00:31:45,100 IAIN STEWART: Yeah? 496 00:31:45,100 --> 00:31:47,930 AUDIENCE: So when you did, well, this calculation 497 00:31:47,930 --> 00:31:51,740 here, that capital F, is it the same result as we found a week 498 00:31:51,740 --> 00:31:54,840 and a half ago or something that has the renormalon at u 499 00:31:54,840 --> 00:31:56,545 equals 1 and all that stuff as well? 500 00:31:56,545 --> 00:31:58,788 IAIN STEWART: No, it's just the u equals 1/2 pole. 501 00:31:58,788 --> 00:31:59,830 AUDIENCE: So [INAUDIBLE]. 502 00:31:59,830 --> 00:32:00,390 IAIN STEWART: Yeah. 503 00:32:00,390 --> 00:32:00,560 Yeah. 504 00:32:00,560 --> 00:32:02,340 AUDIENCE: So what happened to the [INAUDIBLE]?? 505 00:32:02,340 --> 00:32:03,173 IAIN STEWART: Right. 506 00:32:03,173 --> 00:32:05,540 So what we've done here-- 507 00:32:05,540 --> 00:32:07,607 and actually, I'm just going to come to this, 508 00:32:07,607 --> 00:32:08,440 but I'll foreshadow. 509 00:32:08,440 --> 00:32:10,690 So what we've done here is we set up 510 00:32:10,690 --> 00:32:13,870 a construction to remove the u equals 1/2 pole. 511 00:32:13,870 --> 00:32:17,320 And we did not remove any higher poles. 512 00:32:17,320 --> 00:32:19,460 We said the u equals 1/2 is the most important. 513 00:32:19,460 --> 00:32:22,390 Let's get rid of that one from the MS-bar mass. 514 00:32:22,390 --> 00:32:24,710 There would be, in principle, higher ones. 515 00:32:24,710 --> 00:32:27,500 And higher ones, basically, we had 516 00:32:27,500 --> 00:32:33,340 something that was proportional to R. That's for u equals 1/2. 517 00:32:33,340 --> 00:32:34,840 If you wanted to remove higher ones, 518 00:32:34,840 --> 00:32:37,660 you'd need terms proportional to R squared, for example, for u 519 00:32:37,660 --> 00:32:40,580 equals 1, et cetera. 520 00:32:40,580 --> 00:32:43,707 So we could have added more terms to do more, 521 00:32:43,707 --> 00:32:44,290 but we didn't. 522 00:32:44,290 --> 00:32:46,930 We just sort of removed the problem at u equals 1/2. 523 00:32:46,930 --> 00:32:48,910 And that's why, when we go through this, 524 00:32:48,910 --> 00:32:51,376 you only see a u equals 1/2 pole. 525 00:32:51,376 --> 00:32:54,328 AUDIENCE: So if you wanted to do [INAUDIBLE] analysis 526 00:32:54,328 --> 00:32:56,620 or something, you'd have to change the result of this-- 527 00:32:56,620 --> 00:32:56,870 IAIN STEWART: Yeah. 528 00:32:56,870 --> 00:32:58,375 So the way you should think about it 529 00:32:58,375 --> 00:33:00,250 is that basically what you're doing is you're 530 00:33:00,250 --> 00:33:04,300 setting up a scheme change that perturbatively takes you out 531 00:33:04,300 --> 00:33:06,580 of MS-bar towards something else. 532 00:33:06,580 --> 00:33:10,260 You remove the u equals 1/2, and you get rid of that problem. 533 00:33:10,260 --> 00:33:12,760 If you thought you had enough perturbative accuracy that you 534 00:33:12,760 --> 00:33:14,620 were seeing problems related to u equals 1, 535 00:33:14,620 --> 00:33:16,730 then you could make further scheme change 536 00:33:16,730 --> 00:33:19,490 with another series that's proportional to R squared 537 00:33:19,490 --> 00:33:21,982 to get rid of that one. 538 00:33:21,982 --> 00:33:23,860 AUDIENCE: Wait, so the Landau pole 539 00:33:23,860 --> 00:33:25,570 would tell you about all the renormalons? 540 00:33:25,570 --> 00:33:27,280 IAIN STEWART: It would tell you about these ones, too. yeah, 541 00:33:27,280 --> 00:33:28,430 absolutely. 542 00:33:28,430 --> 00:33:33,610 So you can actually generalize this for R to the P. 543 00:33:33,610 --> 00:33:35,680 And you can derive a sum rule for R to the P. 544 00:33:35,680 --> 00:33:37,990 And it's a slightly more complicated formula here. 545 00:33:37,990 --> 00:33:40,825 And then you'd have the P over 2 pole. 546 00:33:45,170 --> 00:33:57,810 All right, OK, so the sum rule is a probe, alternate probe, 547 00:33:57,810 --> 00:33:59,468 for the renormalon. 548 00:34:08,432 --> 00:34:13,219 The nf bubbles are the classic one that people know about. 549 00:34:13,219 --> 00:34:16,159 And this provides an alternate. 550 00:34:16,159 --> 00:34:17,659 The nice thing about this is it also 551 00:34:17,659 --> 00:34:20,480 provides a series that you can look at the convergence, where 552 00:34:20,480 --> 00:34:23,600 the nf bubbles just get a result. See, nf bubbles are 553 00:34:23,600 --> 00:34:26,449 good for finding out whether or not 554 00:34:26,449 --> 00:34:28,250 this thing has a renormalon. 555 00:34:28,250 --> 00:34:31,909 If you get some non-0 result, then you expect that it does. 556 00:34:31,909 --> 00:34:35,300 This way you could actually calculate the residue 557 00:34:35,300 --> 00:34:38,900 if you wanted to know that value for some reason. 558 00:34:38,900 --> 00:34:39,400 OK. 559 00:34:43,837 --> 00:34:46,123 Now, that actually might be useful, for example, 560 00:34:46,123 --> 00:34:48,290 if you wanted to think about higher renormalon poles 561 00:34:48,290 --> 00:34:50,060 because you might want to say, well, 562 00:34:50,060 --> 00:34:52,690 let me get rid of the first one with whatever residue 563 00:34:52,690 --> 00:34:54,800 I can best approximate for and see if there's 564 00:34:54,800 --> 00:34:56,827 another one underneath. 565 00:34:56,827 --> 00:34:58,910 There's not really enough perturbative information 566 00:34:58,910 --> 00:35:01,590 that we have about QCD to be able to do that type of thing. 567 00:35:01,590 --> 00:35:04,185 There's a few cases where u equals 1/2 is absent, 568 00:35:04,185 --> 00:35:05,810 and then you can do that kind of thing. 569 00:35:05,810 --> 00:35:07,010 I mean, there's actually many cases 570 00:35:07,010 --> 00:35:08,300 where u equals 1/2 is absent, and you 571 00:35:08,300 --> 00:35:09,230 can look for u equals 1. 572 00:35:09,230 --> 00:35:10,897 But looking for a sub-leading renormalon 573 00:35:10,897 --> 00:35:15,080 is something that nobody's found one. 574 00:35:15,080 --> 00:35:19,370 People sometimes speculate about them 575 00:35:19,370 --> 00:35:23,997 because maybe the residue of the first one happens to be small. 576 00:35:23,997 --> 00:35:25,580 This would allow you to test for that. 577 00:35:30,340 --> 00:35:32,320 In the tau decays people talk about that. 578 00:35:35,530 --> 00:35:39,700 OK, so let's come back and actually show you 579 00:35:39,700 --> 00:35:42,340 how you can use this technology for some phenomenological 580 00:35:42,340 --> 00:35:43,570 stuff. 581 00:35:43,570 --> 00:35:46,180 And I want to do that by generalizing our discussion 582 00:35:46,180 --> 00:35:50,590 slightly away from masses towards just a general operator 583 00:35:50,590 --> 00:35:53,180 product expansion. 584 00:35:53,180 --> 00:35:55,240 So everything that we've said here for masses 585 00:35:55,240 --> 00:36:02,020 can be applied more generally to an operator product expansion. 586 00:36:02,020 --> 00:36:05,153 So let me show you what I mean. 587 00:36:05,153 --> 00:36:07,570 So let's first consider what I mean by an operator product 588 00:36:07,570 --> 00:36:09,980 expansion at MS-bar. 589 00:36:09,980 --> 00:36:12,847 So you can think of this like maybe I'm integrating out 590 00:36:12,847 --> 00:36:13,555 a heavy particle. 591 00:36:16,860 --> 00:36:19,690 And I'm writing a formula down, and it 592 00:36:19,690 --> 00:36:21,100 has some Wilson coefficients. 593 00:36:21,100 --> 00:36:24,040 It has some operators. 594 00:36:24,040 --> 00:36:26,590 And it's a Taylor series in this parameter 595 00:36:26,590 --> 00:36:31,035 q, which is the hard scale, high energy scale. 596 00:36:31,035 --> 00:36:32,410 And I'm basically just expanding, 597 00:36:32,410 --> 00:36:37,700 if you like, in lambda QCD over Q. And the operators, 598 00:36:37,700 --> 00:36:40,750 they're set by lambda QCD. 599 00:36:40,750 --> 00:36:43,960 So I'm going to take this guy to be dimensionless. 600 00:36:43,960 --> 00:36:47,970 This is some dimensionless observable. 601 00:36:47,970 --> 00:36:49,560 You can always make it dimensionless 602 00:36:49,560 --> 00:36:54,330 by a suitably choosing, multiplying by appropriate math 603 00:36:54,330 --> 00:36:55,810 scales. 604 00:36:55,810 --> 00:36:58,150 So then this here will be a dimensionless Wilson 605 00:36:58,150 --> 00:37:00,180 coefficient. 606 00:37:00,180 --> 00:37:03,460 And it's a Wilson coefficient in MS-bar. 607 00:37:03,460 --> 00:37:05,900 That's what the bar means. 608 00:37:05,900 --> 00:37:09,230 And so this is an MS-bar operator, MS-bar matrix element 609 00:37:09,230 --> 00:37:10,464 of some operator. 610 00:37:13,070 --> 00:37:14,562 And it's also dimensionless. 611 00:37:17,730 --> 00:37:21,365 So this guy will be dimensionless, too. 612 00:37:21,365 --> 00:37:23,490 And finally, this guy will have dimension equals 1. 613 00:37:28,730 --> 00:37:30,830 Now, at MS-bar, what's beautiful about MS-bar 614 00:37:30,830 --> 00:37:35,150 is that the series for these coefficients are very simple. 615 00:37:35,150 --> 00:37:37,940 That's why people like to compute in MS-bar. 616 00:37:49,570 --> 00:37:56,030 So if you look at what the series for C bar looks like, 617 00:37:56,030 --> 00:37:58,670 the form of it is as follows. 618 00:38:04,400 --> 00:38:09,170 It's got dependence on mu over Q and alpha s of mu. 619 00:38:13,850 --> 00:38:17,330 And these guys here are not an arbitrary function 620 00:38:17,330 --> 00:38:20,450 of mu over Q, but they're really just some logs. 621 00:38:20,450 --> 00:38:22,430 So this looks like a series of logs. 622 00:38:25,520 --> 00:38:29,830 So in MS-bar, you're only seeing these logarithms. 623 00:38:29,830 --> 00:38:43,400 So that's good from a computational point of view, 624 00:38:43,400 --> 00:38:45,080 but we also know that generically, 625 00:38:45,080 --> 00:38:48,500 in this MS-bar approach, you might get sensitivity 626 00:38:48,500 --> 00:38:50,090 to renormalons. 627 00:38:56,330 --> 00:38:58,370 Also, another thing we like about MS-bar 628 00:38:58,370 --> 00:39:01,370 is that it satisfies all the symmetries we want to keep. 629 00:39:01,370 --> 00:39:02,480 It doesn't mess them up. 630 00:39:02,480 --> 00:39:05,292 So it's Lorentz and gauge invariant, 631 00:39:05,292 --> 00:39:06,500 which is always a good thing. 632 00:39:13,280 --> 00:39:16,190 And not only is it simple for multi-loop computations, 633 00:39:16,190 --> 00:39:19,310 but multi-loop computations are not even done in other schemes 634 00:39:19,310 --> 00:39:21,118 because they're just too difficult. 635 00:39:21,118 --> 00:39:23,285 So it really makes multi-loop computations possible. 636 00:39:28,970 --> 00:39:31,460 And the but, there's only really this one but, 637 00:39:31,460 --> 00:39:36,060 that it has this renormalon issue. 638 00:39:41,120 --> 00:39:43,400 So we talked about renormalon for a mass. 639 00:39:43,400 --> 00:39:46,910 And I just showed you what it was, and then we explored it. 640 00:39:46,910 --> 00:39:50,960 Here, let me tell you what the renormalon would be. 641 00:39:50,960 --> 00:39:54,170 So generically, what the renormalon would be 642 00:39:54,170 --> 00:39:54,920 is the following. 643 00:39:54,920 --> 00:39:58,250 You would have an ambiguity in the coefficient C0 644 00:39:58,250 --> 00:40:06,700 bar that's of the size lambda QCD over whatever the scale 645 00:40:06,700 --> 00:40:07,240 Q is. 646 00:40:11,540 --> 00:40:17,950 And you would have the same ambiguity in this theta 1 bar. 647 00:40:17,950 --> 00:40:20,470 So this theta 1 bar, which is an MS-bar matrix element 648 00:40:20,470 --> 00:40:24,820 has an ambiguity, which is of order lambda QCD. 649 00:40:24,820 --> 00:40:26,440 This guy has an ambiguity. 650 00:40:26,440 --> 00:40:28,570 And those ambiguities cancel. 651 00:40:28,570 --> 00:40:31,180 So you're dividing up the physics again in MS-bar 652 00:40:31,180 --> 00:40:33,850 into short distance physics and long distance physics. 653 00:40:33,850 --> 00:40:36,970 And the interesting thing is that there's 654 00:40:36,970 --> 00:40:40,180 a piece of long distance physics trapped in the C0 655 00:40:40,180 --> 00:40:42,340 that, if you really wanted, should 656 00:40:42,340 --> 00:40:44,340 be physically in the theta 1 bar and vice versa. 657 00:40:47,450 --> 00:40:49,660 So really what's the problem with MS-bar 658 00:40:49,660 --> 00:40:51,160 is that there's a piece of something 659 00:40:51,160 --> 00:40:53,470 that you want in theta 1 bar, but ends up just 660 00:40:53,470 --> 00:40:55,710 being in the C0 bar. 661 00:40:55,710 --> 00:40:58,400 And that's what I mean by this that there is something here, 662 00:40:58,400 --> 00:41:00,280 which is infrared sensitive. 663 00:41:00,280 --> 00:41:03,730 Correspondingly, there is kind of an ultraviolet sensitivity 664 00:41:03,730 --> 00:41:04,660 in this theta 1 bar. 665 00:41:04,660 --> 00:41:07,013 And you'd like to get that rearranged 666 00:41:07,013 --> 00:41:07,930 in an appropriate way. 667 00:41:07,930 --> 00:41:10,030 And that's like this kind of rearrangement 668 00:41:10,030 --> 00:41:12,910 we were making by changing schemes for masses. 669 00:41:12,910 --> 00:41:15,670 Except here it's going to be changing schemes 670 00:41:15,670 --> 00:41:17,348 for Wilson coefficients. 671 00:41:20,220 --> 00:41:26,480 So this thing here is what we would call a u 672 00:41:26,480 --> 00:41:28,253 equals 1 renormalon. 673 00:41:32,330 --> 00:41:34,860 So what happens with u equals 1 is 674 00:41:34,860 --> 00:41:37,470 that you make a connection between the leading order term 675 00:41:37,470 --> 00:41:40,370 and the power suppressed term. 676 00:41:40,370 --> 00:41:43,030 That's what happens with the delta C0 bar. 677 00:41:47,100 --> 00:41:49,900 So in order to flesh out a little bit better what I just 678 00:41:49,900 --> 00:41:52,240 said in words, let's pretend that we 679 00:41:52,240 --> 00:41:53,530 understand what the theory is. 680 00:41:53,530 --> 00:41:55,280 And let's just pretend it's some integral. 681 00:41:57,660 --> 00:42:00,160 And I'll just show you what is going on here. 682 00:42:15,300 --> 00:42:20,503 Let's imagine that our theory is an integral. 683 00:42:20,503 --> 00:42:21,920 Instead of a quantum field theory, 684 00:42:21,920 --> 00:42:23,253 it's a one-dimensional integral. 685 00:42:27,510 --> 00:42:30,680 And it has two scales, one which I'll 686 00:42:30,680 --> 00:42:33,800 call lambda QCD, which is some low scale, 687 00:42:33,800 --> 00:42:39,170 and one which I'll call Q, which is some high scale. 688 00:42:39,170 --> 00:42:42,500 And I'm using a notation where the integral could diverge 689 00:42:42,500 --> 00:42:46,190 and I regulate it with a dimensional regularization type 690 00:42:46,190 --> 00:42:46,800 parameter. 691 00:42:46,800 --> 00:42:48,383 So I'm doing a one dimensional undergo 692 00:42:48,383 --> 00:42:50,415 but continuing it into epsilon dimensions 693 00:42:50,415 --> 00:42:52,450 to regulate any divergences. 694 00:42:55,090 --> 00:42:59,540 So what MS-bar does with this integral is it just says, 695 00:42:59,540 --> 00:43:02,920 well, I'm not going to change the limits of the integral. 696 00:43:02,920 --> 00:43:05,110 I'm just going to either expand out this f 697 00:43:05,110 --> 00:43:08,440 or expand out this denominator. 698 00:43:08,440 --> 00:43:10,000 I'll do two different Taylor series, 699 00:43:10,000 --> 00:43:12,250 and then I'll put them together. 700 00:43:12,250 --> 00:43:17,330 And that's basically how I'm identifying the terms 701 00:43:17,330 --> 00:43:18,140 in my OPE. 702 00:43:22,330 --> 00:43:45,190 So MS-bar-- so I sort of associate 703 00:43:45,190 --> 00:43:49,420 there being a high energy piece where k is of order Q. 704 00:43:49,420 --> 00:43:53,253 And then there's some low energy piece. 705 00:43:53,253 --> 00:43:54,295 I don't change the limit. 706 00:43:58,490 --> 00:44:02,230 So in the low energy piece, I keep the full function f. 707 00:44:02,230 --> 00:44:05,830 And then I expand out the other thing. 708 00:44:10,230 --> 00:44:13,360 And if I drop the dots in this formula, 709 00:44:13,360 --> 00:44:16,720 I can identify these integrals as toy models 710 00:44:16,720 --> 00:44:20,860 for the things that are appearing in my OPE. 711 00:44:20,860 --> 00:44:22,270 So let me write that OPE again. 712 00:44:29,602 --> 00:44:33,100 So here these things are 1, this example. 713 00:44:33,100 --> 00:44:35,860 This integral here, which is over high energy scales 714 00:44:35,860 --> 00:44:38,140 k [INAUDIBLE] by Q is this guy. 715 00:44:40,980 --> 00:44:43,510 And this integral here, where k is of order lambda QCD, 716 00:44:43,510 --> 00:44:45,190 effectively, there's a 1 over Q, which 717 00:44:45,190 --> 00:44:49,630 is this 1 over Q. That's giving me the operator, theta 1 bar. 718 00:44:52,700 --> 00:44:54,850 So Wilson coefficient is the high energy piece 719 00:44:54,850 --> 00:44:57,250 of the original thing. 720 00:44:57,250 --> 00:44:59,080 Matrix element here is the low energy piece 721 00:44:59,080 --> 00:45:04,420 which is suppressed by 1 over Q, but they both 722 00:45:04,420 --> 00:45:07,200 came from the same physics in the beginning. 723 00:45:07,200 --> 00:45:08,950 And I just was expanding it because that's 724 00:45:08,950 --> 00:45:10,117 how I wanted to organize it. 725 00:45:12,258 --> 00:45:14,800 Now, the thing about MS-bar is that you integrate all the way 726 00:45:14,800 --> 00:45:15,700 down to 0 there. 727 00:45:15,700 --> 00:45:19,000 And you integrate all the way up to infinity here. 728 00:45:19,000 --> 00:45:21,160 And that's where the renormalon problems come from. 729 00:45:31,330 --> 00:45:35,040 So this little procedure here separates the long and short 730 00:45:35,040 --> 00:45:37,920 distance physics for the logs. 731 00:45:43,040 --> 00:45:46,100 It does that correctly. 732 00:45:46,100 --> 00:45:50,570 But for powers, we effectively rely on the fact 733 00:45:50,570 --> 00:45:51,535 that it's at scale. 734 00:45:51,535 --> 00:45:52,670 This integral goes to 0. 735 00:45:58,940 --> 00:46:02,240 That's what's forced upon us by the definition 736 00:46:02,240 --> 00:46:05,440 of MS-bar and dimensional regularization. 737 00:46:08,582 --> 00:46:11,710 And what goes wrong is basically that physically we're still 738 00:46:11,710 --> 00:46:14,770 integrating over regions of those integrals which are not 739 00:46:14,770 --> 00:46:18,190 associated to the physics that should be in those parameters. 740 00:47:01,140 --> 00:47:04,903 So, now, this is a toy example, but the same thing 741 00:47:04,903 --> 00:47:06,570 is happening in the quantum field theory 742 00:47:06,570 --> 00:47:09,070 when you think about what you're doing when you write down 743 00:47:09,070 --> 00:47:11,470 Feynman diagram and you construct a Wilson coefficient. 744 00:47:11,470 --> 00:47:14,053 It's effectively the same thing in dimensional regularization. 745 00:47:14,053 --> 00:47:16,620 You're integrating over all values of momentum. 746 00:47:16,620 --> 00:47:18,310 And this little kind of toy analogy 747 00:47:18,310 --> 00:47:22,300 is actually apt for what's going on in a true operator product 748 00:47:22,300 --> 00:47:24,730 expansion. 749 00:47:24,730 --> 00:47:30,650 What would happen if you did a Wilsonian picture in our toy 750 00:47:30,650 --> 00:47:32,870 model? 751 00:47:32,870 --> 00:47:35,120 Well, then you would cut off the integrals explicitly. 752 00:47:35,120 --> 00:47:40,660 You'd say, let's introduce some scale lambda f. 753 00:47:40,660 --> 00:47:43,970 I could still think of it expanding in the same way. 754 00:47:48,050 --> 00:47:50,882 I don't need the dimensional regularization anymore. 755 00:47:50,882 --> 00:47:53,410 So I'll set that, get rid of that. 756 00:47:56,873 --> 00:47:59,540 I think of breaking the integral into two pieces, the low energy 757 00:47:59,540 --> 00:48:01,040 and high energy pieces. 758 00:48:01,040 --> 00:48:03,740 And in each of those pieces, I expand the integral 759 00:48:03,740 --> 00:48:04,640 in different ways-- 760 00:48:08,620 --> 00:48:13,160 so same type of expansion, I'm just regulating it differently. 761 00:48:13,160 --> 00:48:15,510 Now, I'm guaranteed that this is high energy 762 00:48:15,510 --> 00:48:18,120 and this is low energy. 763 00:48:18,120 --> 00:48:20,250 And you can write down sort of what this is 764 00:48:20,250 --> 00:48:21,680 a proxy for as a Wilsonian OPE. 765 00:48:33,880 --> 00:48:37,560 Mu gets replaced by the scale lambda f. 766 00:48:37,560 --> 00:48:39,360 Everything gets a W for Wilsonian. 767 00:48:47,350 --> 00:48:50,680 Again, this is one with our toy model. 768 00:48:50,680 --> 00:48:53,200 This guy is this guy now. 769 00:48:53,200 --> 00:48:56,270 This guy is this guy now. 770 00:48:56,270 --> 00:48:58,270 So it's a very similar kind of association, 771 00:48:58,270 --> 00:48:59,770 but just the way that we're treating 772 00:48:59,770 --> 00:49:01,740 the integral is different. 773 00:49:01,740 --> 00:49:04,090 In this way of doing things, you don't have any problem 774 00:49:04,090 --> 00:49:06,020 with the renormalon. 775 00:49:06,020 --> 00:49:07,780 You just have an explicit cutoff, 776 00:49:07,780 --> 00:49:10,900 which says there's no low energy part of this, 777 00:49:10,900 --> 00:49:14,160 and there's no high energy part of that. 778 00:49:14,160 --> 00:49:18,290 I'm not integrating all the way down to 0 779 00:49:18,290 --> 00:49:20,517 in the first integral, so I don't have any problems 780 00:49:20,517 --> 00:49:21,350 with the renormalon. 781 00:49:21,350 --> 00:49:24,020 The renormalon came from infrared physics that 782 00:49:24,020 --> 00:49:26,990 was happening near k equals 0. 783 00:49:26,990 --> 00:49:30,750 And in this one, there's no sort of problem 784 00:49:30,750 --> 00:49:34,510 which would come from infinity. 785 00:49:34,510 --> 00:49:39,000 So in some sense, this is what we'd like to do. 786 00:49:46,110 --> 00:49:51,540 And the but here is that it causes difficulties 787 00:49:51,540 --> 00:49:52,380 with symmetries. 788 00:49:59,800 --> 00:50:04,780 In particular, it generically breaks gauge invariance 789 00:50:04,780 --> 00:50:07,007 and Lorentz invariance. 790 00:50:12,254 --> 00:50:15,330 These get broken. 791 00:50:15,330 --> 00:50:19,050 And the calculations that you would like to do 792 00:50:19,050 --> 00:50:27,570 are just too difficult. I don't know of anybody that's done-- 793 00:50:27,570 --> 00:50:30,030 maybe somebody's done two loops in the Wilsonian picture, 794 00:50:30,030 --> 00:50:32,580 but nobody's ever done three. 795 00:50:32,580 --> 00:50:35,820 So in some sense, you have some nice things here. 796 00:50:35,820 --> 00:50:37,268 You have some nice things here. 797 00:50:37,268 --> 00:50:38,810 But they're not the same nice things. 798 00:50:38,810 --> 00:50:41,735 And you'd like to have something that does the best of both. 799 00:50:41,735 --> 00:50:43,110 And that's effectively what we're 800 00:50:43,110 --> 00:50:45,395 doing when we construct these R schemes. 801 00:50:45,395 --> 00:50:46,770 What we're doing is we're saying, 802 00:50:46,770 --> 00:50:48,570 well, let's take MS-bar as a starting point 803 00:50:48,570 --> 00:50:51,060 and try to get rid of the problem that it has 804 00:50:51,060 --> 00:50:53,320 by perturbing towards a Wilsonian picture. 805 00:51:00,160 --> 00:51:02,940 So what would an R scheme OPEC look like? 806 00:51:07,200 --> 00:51:10,905 Start with MS-bar and make a scheme change. 807 00:51:15,280 --> 00:51:28,660 And make a scheme change here both to the operators 808 00:51:28,660 --> 00:51:32,093 and make the same scheme change to the coefficient. 809 00:51:32,093 --> 00:51:33,760 So I'm just really moving things around. 810 00:51:45,053 --> 00:51:46,470 So let me just formally write down 811 00:51:46,470 --> 00:51:48,220 what such a scheme change would look like. 812 00:52:04,920 --> 00:52:07,220 I set up some coefficients dn. 813 00:52:11,710 --> 00:52:16,112 And I do a subtraction on the operators and an addition 814 00:52:16,112 --> 00:52:18,070 on the coefficients in the way I've written it. 815 00:52:22,250 --> 00:52:24,550 So that's just a rearrangement of the physics. 816 00:52:24,550 --> 00:52:31,980 And we've set it up so that the result of that looks like this. 817 00:52:42,310 --> 00:52:50,000 I took, in this example, C1 equal to 1 simplicity. 818 00:52:50,000 --> 00:52:53,080 A little more complicated if I want to keep it, 819 00:52:53,080 --> 00:52:56,150 but we could keep it if we needed to. 820 00:52:56,150 --> 00:52:58,930 So basically, I've made a rearrangement to some 821 00:52:58,930 --> 00:53:02,300 equivalent OPE by just changing the thing, 822 00:53:02,300 --> 00:53:05,740 the definitions here, but I have now the freedom to pick these 823 00:53:05,740 --> 00:53:06,410 d's. 824 00:53:06,410 --> 00:53:08,035 And what I can do is I can pick the d's 825 00:53:08,035 --> 00:53:11,350 to have the same renormalon as the MS-bar scheme. 826 00:53:11,350 --> 00:53:13,960 And, therefore, I can iteratively, again, 827 00:53:13,960 --> 00:53:17,400 remove the first problematic term of this 828 00:53:17,400 --> 00:53:22,300 to get a better defined theta 1 in C0 829 00:53:22,300 --> 00:53:25,520 just like we were doing for the masses. 830 00:53:25,520 --> 00:53:28,165 So you can remove the u equals 1, for example. 831 00:53:33,310 --> 00:53:43,450 You can remove the u equals 1 renormalon by choosing the d. 832 00:53:43,450 --> 00:53:45,440 And that's exactly what the Wilsonian 833 00:53:45,440 --> 00:53:47,530 was trying to do for you. 834 00:53:47,530 --> 00:53:50,910 And that's what it was doing for you with putting this cutoff. 835 00:53:50,910 --> 00:53:52,690 MS-bar was having problems at 0. 836 00:53:52,690 --> 00:53:56,340 And we're sort of perturbatively removing those problems, 837 00:53:56,340 --> 00:53:59,393 perturbatively going towards the Wilsonian picture. 838 00:53:59,393 --> 00:54:01,060 And the nice thing about this is that we 839 00:54:01,060 --> 00:54:03,850 can maintain the symmetries, still gauge invariant, still 840 00:54:03,850 --> 00:54:05,157 Lorentz invariant. 841 00:54:19,580 --> 00:54:22,130 And we don't get the full sort of power of the Wilsonian, 842 00:54:22,130 --> 00:54:25,560 but we get closer to it. 843 00:54:25,560 --> 00:54:27,470 And we can, in some sense, get as close 844 00:54:27,470 --> 00:54:31,850 as we want, again, by sort of perturbing this from order 845 00:54:31,850 --> 00:54:34,020 by order. 846 00:54:34,020 --> 00:54:37,160 So what we introduce is power law dependence in R. 847 00:54:37,160 --> 00:54:39,987 And that power law dependence, it's 848 00:54:39,987 --> 00:54:41,570 the analog of the power law dependence 849 00:54:41,570 --> 00:54:43,640 that would be in the Wilsonian scheme. 850 00:54:43,640 --> 00:54:46,720 So when you have integrals like this, you would have powers. 851 00:54:46,720 --> 00:54:48,470 There's nothing that stops you from having 852 00:54:48,470 --> 00:54:51,200 a complicated function of Q over lambda. 853 00:54:51,200 --> 00:54:54,770 And what we're saying is that the dominant kind of power law 854 00:54:54,770 --> 00:54:57,320 sensitive terms here and here are captured 855 00:54:57,320 --> 00:54:58,640 by making this scheme change. 856 00:55:23,690 --> 00:55:25,790 So we decide that we like MS-bar because we 857 00:55:25,790 --> 00:55:31,330 have calculations that other people have done hopefully. 858 00:55:31,330 --> 00:55:34,020 If they're three-loop order, you certainly 859 00:55:34,020 --> 00:55:36,443 hope that someone else spent two years of their life on it 860 00:55:36,443 --> 00:55:37,110 rather than you. 861 00:55:44,300 --> 00:55:52,940 And you can take and perturb your results from MS-bar 862 00:55:52,940 --> 00:55:55,130 towards a gauge invariant, Lorentz invariant 863 00:55:55,130 --> 00:56:00,450 version of the Wilsonian, of the Wilsonian picture. 864 00:56:04,130 --> 00:56:05,740 So let me give you one example of that 865 00:56:05,740 --> 00:56:11,000 to show you in practice, putting in some numbers. 866 00:56:11,000 --> 00:56:12,880 Oh, I'm not quite there yet. 867 00:56:15,550 --> 00:56:19,840 OK, so let me give you a well-defined scheme which 868 00:56:19,840 --> 00:56:23,560 is analogous to the scheme that we just talked about, 869 00:56:23,560 --> 00:56:25,130 which was MSR scheme for the mass. 870 00:56:25,130 --> 00:56:28,330 There's also an MSR scheme for the OPE. 871 00:56:28,330 --> 00:56:31,120 And again, the attitude is let's not calculate anything new 872 00:56:31,120 --> 00:56:33,160 that we don't have to. 873 00:56:33,160 --> 00:56:37,000 So we'll reuse the coefficients of MS-bar. 874 00:56:37,000 --> 00:56:40,990 Those coefficients had the renormalon. 875 00:56:40,990 --> 00:56:43,960 So if those coefficients were, we called them 876 00:56:43,960 --> 00:56:48,160 a few minutes ago, bn mu over Q, we just 877 00:56:48,160 --> 00:56:49,660 reuse these at a different scale. 878 00:56:53,980 --> 00:56:57,760 So what we do is we say dn of mu over R 879 00:56:57,760 --> 00:57:02,740 is defined to be the bn of my over R. 880 00:57:02,740 --> 00:57:04,810 So these guys were the MS-bar coefficients. 881 00:57:04,810 --> 00:57:06,280 These guys are whatever we decide 882 00:57:06,280 --> 00:57:07,930 to put into the series over there. 883 00:57:07,930 --> 00:57:09,560 Just take them to be equal. 884 00:57:09,560 --> 00:57:12,130 That's the MSR scheme. 885 00:57:12,130 --> 00:57:16,570 And then if you look at what the coefficient is, if we just 886 00:57:16,570 --> 00:57:19,680 write out explicitly what the coefficient is 887 00:57:19,680 --> 00:57:23,530 by plugging in the series, you can again 888 00:57:23,530 --> 00:57:26,720 see how it's providing a cutoff. 889 00:57:41,120 --> 00:57:42,950 So this was our original MS-bar series, 890 00:57:42,950 --> 00:57:46,460 bn of mu over Q. We introduce a subtraction, which 891 00:57:46,460 --> 00:57:51,560 is suppressed by R over Q times the same series bn. 892 00:57:51,560 --> 00:57:54,500 And this R is providing kind of like a power law 893 00:57:54,500 --> 00:57:58,310 cutoff on a problem that this guy had, 894 00:57:58,310 --> 00:58:00,660 exactly the power law cutoff. 895 00:58:00,660 --> 00:58:03,080 So again, if you look at the renormalon in this thing, 896 00:58:03,080 --> 00:58:06,530 this combined thing, the renormalon is independent of R 897 00:58:06,530 --> 00:58:18,060 and Q. So you would just plug in the bubble series result. 898 00:58:18,060 --> 00:58:21,820 You'd find, actually, that the R over Q here would cancel out. 899 00:58:21,820 --> 00:58:24,630 And so that these would just exactly cancel each other. 900 00:58:33,180 --> 00:58:36,197 The u equals 1 [INAUDIBLE] 1. 901 00:58:36,197 --> 00:58:38,280 And again, if I wanted to probe other renormalons, 902 00:58:38,280 --> 00:58:40,615 I'd have to consider other powers besides R over Q. 903 00:58:40,615 --> 00:58:42,810 But those actually, in this OPE, would be 904 00:58:42,810 --> 00:58:45,000 connected to other operators. 905 00:58:45,000 --> 00:58:47,467 So there is other sub-leading renormalons here, 906 00:58:47,467 --> 00:58:49,800 which could be connected to 1 over Q squared, et cetera, 907 00:58:49,800 --> 00:58:51,530 et cetera. 908 00:58:51,530 --> 00:58:54,390 And in MS-bar, that generically will happen. 909 00:58:54,390 --> 00:58:56,580 But usually, it suffices just to worry 910 00:58:56,580 --> 00:58:59,220 about the first one since many people 911 00:58:59,220 --> 00:59:01,800 in the community that do these multi-loop calculations don't 912 00:59:01,800 --> 00:59:03,092 even worry about the first one. 913 00:59:06,090 --> 00:59:12,180 Anyway, OK, so you can again, after you've 914 00:59:12,180 --> 00:59:15,600 got this definition, you could actually formally write 915 00:59:15,600 --> 00:59:20,760 this as C0 bar [INAUDIBLE] Q, mu if you want to think about what 916 00:59:20,760 --> 00:59:26,550 it is, minus R over Q, same Wilson coefficient, 917 00:59:26,550 --> 00:59:30,378 replacing all the Q's by R. So that's 918 00:59:30,378 --> 00:59:31,920 another way of saying what the scheme 919 00:59:31,920 --> 00:59:36,620 change is from MS-bar results to this result. 920 00:59:36,620 --> 00:59:42,345 And this R is acting like an IR cutoff 921 00:59:42,345 --> 00:59:43,930 to get rid of the bad problem. 922 00:59:49,860 --> 00:59:53,000 And again, we have a renormalization group. 923 00:59:53,000 --> 00:59:59,515 So R d by dR of this Wilson coefficient, 924 00:59:59,515 --> 01:00:01,890 it's convenient to think about that renormalization group 925 01:00:01,890 --> 01:00:07,920 with the log cutoff mu set equal to the power law cutoff R. 926 01:00:07,920 --> 01:00:09,660 And it's very much like our masses. 927 01:00:09,660 --> 01:00:14,280 There's some function with an R in front. 928 01:00:14,280 --> 01:00:18,940 And we could formally write down solutions to it. 929 01:00:18,940 --> 01:00:21,420 And we can explicitly plug in results 930 01:00:21,420 --> 01:00:22,920 given how many terms in the series 931 01:00:22,920 --> 01:00:25,480 we know in the MS-bar series. 932 01:00:28,330 --> 01:00:31,400 So all the anomalous dimensions of this Wilson coefficient 933 01:00:31,400 --> 01:00:33,160 would be determined by the MS-bar series, 934 01:00:33,160 --> 01:00:35,420 the original one, because the scheme 935 01:00:35,420 --> 01:00:38,600 that redefining is set up so that it uses those [INAUDIBLE] 936 01:00:38,600 --> 01:00:39,782 coefficients. 937 01:00:43,020 --> 01:00:46,730 So what would the RGE look like? 938 01:01:03,665 --> 01:01:05,290 It's the same kind of structure as what 939 01:01:05,290 --> 01:01:14,008 we have for masses, some gamma functions. 940 01:01:27,470 --> 01:01:30,730 And you can think of this as some C0. 941 01:01:30,730 --> 01:01:33,620 If you want kind of to have a more standard notation, 942 01:01:33,620 --> 01:01:37,700 you would say that this is some U function. 943 01:01:37,700 --> 01:01:40,030 And this is a bit of abuse of notation, actually, here 944 01:01:40,030 --> 01:01:46,363 because it's additive, not multiplicative. 945 01:01:46,363 --> 01:01:48,280 But if you allow me to put this guy into here, 946 01:01:48,280 --> 01:01:50,660 then I could always write it this way. 947 01:01:50,660 --> 01:01:52,880 And I'll just do that for convenience. 948 01:01:52,880 --> 01:01:55,510 So I want to show you one example of how this plays out 949 01:01:55,510 --> 01:01:57,020 in practice. 950 01:01:57,020 --> 01:02:00,200 And it's related to our discussion of HQET. 951 01:02:00,200 --> 01:02:02,430 So one thing that we said about HQET 952 01:02:02,430 --> 01:02:04,555 is that you could calculate the following quantity. 953 01:02:16,195 --> 01:02:18,658 So HQET makes some predictions about the B star 954 01:02:18,658 --> 01:02:19,950 and the B. They were connected. 955 01:02:19,950 --> 01:02:21,810 They were in a symmetry multiplet. 956 01:02:21,810 --> 01:02:25,370 And when you took the difference of the B star and B 957 01:02:25,370 --> 01:02:27,120 and you took this combination, it actually 958 01:02:27,120 --> 01:02:28,245 is purely perturbative. 959 01:02:28,245 --> 01:02:30,120 Because there's a non-perturbative parameter. 960 01:02:30,120 --> 01:02:32,440 But if you treat the charm quark as heavy, 961 01:02:32,440 --> 01:02:34,380 it's the same for charm and bottom. 962 01:02:34,380 --> 01:02:35,220 And it cancels out. 963 01:02:37,900 --> 01:02:42,130 So you can actually write down an MS-bar operator product 964 01:02:42,130 --> 01:02:48,790 expansion, which is technically defined in HQET. 965 01:02:48,790 --> 01:02:51,100 And what it looks like is the first thing 966 01:02:51,100 --> 01:02:53,930 is just a ratio of Wilson coefficients. 967 01:02:53,930 --> 01:02:55,957 So we talked about this. 968 01:02:55,957 --> 01:02:58,040 And we plugged in this sort of leading log result. 969 01:02:58,040 --> 01:03:00,207 And we saw that it was working actually pretty well. 970 01:03:03,950 --> 01:03:06,170 And if you were to kind of look at higher order 971 01:03:06,170 --> 01:03:08,510 terms in that OPE, there would be some power corrections 972 01:03:08,510 --> 01:03:09,010 to that. 973 01:03:11,682 --> 01:03:13,140 They've got some traditional names. 974 01:03:13,140 --> 01:03:16,530 The names are not too important. 975 01:03:16,530 --> 01:03:20,517 But this is like the analog of the 1 over Q term. 976 01:03:20,517 --> 01:03:21,600 This is the 1 over Q term. 977 01:03:21,600 --> 01:03:24,210 This is the leading order term, which was purely perturbative. 978 01:03:24,210 --> 01:03:25,502 There's no matrix element here. 979 01:03:25,502 --> 01:03:26,640 It's just 1. 980 01:03:26,640 --> 01:03:28,680 This is a higher dimension operator. 981 01:03:28,680 --> 01:03:34,260 This guy scales like lambda QCD cubed over lambda QCD squared. 982 01:03:34,260 --> 01:03:35,930 This is just lambda QCD. 983 01:03:38,610 --> 01:03:41,307 So that's like the lambda QCD over Q term. 984 01:03:41,307 --> 01:03:43,140 And this is the formula you would write down 985 01:03:43,140 --> 01:03:45,182 if you wanted to make a prediction for this thing 986 01:03:45,182 --> 01:03:49,380 at higher precision in MS-bar. 987 01:03:49,380 --> 01:03:51,930 So what happens if I think about this perturbative result 988 01:03:51,930 --> 01:03:53,860 at higher orders? 989 01:03:53,860 --> 01:03:56,000 So people have actually calculated that coefficient 990 01:03:56,000 --> 01:03:57,508 to three-loop order. 991 01:03:57,508 --> 01:03:58,800 And here's where it looks like. 992 01:04:02,190 --> 01:04:06,865 So at order alpha, you get minus 0.113. 993 01:04:06,865 --> 01:04:11,930 At order alpha squared, you get minus 0.078. 994 01:04:11,930 --> 01:04:19,174 And then you get another minus 0.0755 at alpha cubed. 995 01:04:19,174 --> 01:04:22,683 It doesn't look so pretty, all right? 996 01:04:22,683 --> 01:04:24,600 So each order that you calculate is giving you 997 01:04:24,600 --> 01:04:29,040 a correction about the same size as the previous order. 998 01:04:29,040 --> 01:04:31,230 Well, you might say, of course, there's logs. 999 01:04:31,230 --> 01:04:33,600 So I should resum the logs. 1000 01:04:33,600 --> 01:04:38,790 So let's resum the logs, so leading log. 1001 01:04:42,350 --> 01:04:45,360 And then since I have higher order anomalous dimensions, 1002 01:04:45,360 --> 01:04:49,440 let me put in next leading log. 1003 01:04:49,440 --> 01:04:50,940 And actually, I have one more order. 1004 01:04:50,940 --> 01:05:00,300 So let me put that in, too, [? 908 ?] next leading log. 1005 01:05:00,300 --> 01:05:03,240 Uh-oh, all right? 1006 01:05:03,240 --> 01:05:05,820 So this is the one that we actually talked about earlier. 1007 01:05:05,820 --> 01:05:09,080 And that one is actually close to the data. 1008 01:05:09,080 --> 01:05:12,300 The leading log result was sort of close to what the data was. 1009 01:05:14,820 --> 01:05:17,422 But when you think about whether that's actually right 1010 01:05:17,422 --> 01:05:19,380 and you calculate the higher order corrections, 1011 01:05:19,380 --> 01:05:23,160 you find that you're moving away from the data, right? 1012 01:05:23,160 --> 01:05:24,700 And again, it's not convergent. 1013 01:05:24,700 --> 01:05:27,100 Summing the logs has not helped you. 1014 01:05:27,100 --> 01:05:29,100 And that's because what we're talking about here 1015 01:05:29,100 --> 01:05:30,525 has nothing to do with logs. 1016 01:05:30,525 --> 01:05:31,710 It has to do with powers. 1017 01:05:31,710 --> 01:05:36,360 It's the renormalon that's causing this problem, OK? 1018 01:05:36,360 --> 01:05:40,050 So we could throw up our hands and say, oh, it doesn't work. 1019 01:05:40,050 --> 01:05:42,780 Or we could try to do something better. 1020 01:05:42,780 --> 01:05:45,750 So let me show you what happens if we do some of this stuff 1021 01:05:45,750 --> 01:05:48,190 that I've been telling you about. 1022 01:05:48,190 --> 01:05:52,462 So this guy has a u equals 1 renormalon. 1023 01:05:52,462 --> 01:05:55,200 You can check that very easily by doing a bubble sum. 1024 01:06:00,220 --> 01:06:01,530 Switch to the MSR scheme. 1025 01:06:07,720 --> 01:06:10,250 So we reuse our perturbative information in MS-bar, 1026 01:06:10,250 --> 01:06:13,240 but just organize in a more intelligent fashion. 1027 01:06:18,630 --> 01:06:34,493 So we define a coefficient as above, 1028 01:06:34,493 --> 01:06:35,910 saying that this problem must have 1029 01:06:35,910 --> 01:06:38,260 to do with that renormalon. 1030 01:06:38,260 --> 01:06:47,790 And then if we write down what the OPE looks like, 1031 01:06:47,790 --> 01:06:51,680 we have, again, an OPE. 1032 01:06:51,680 --> 01:06:54,270 And we've changed matrix examined definitions as well 1033 01:06:54,270 --> 01:06:57,060 as Wilson coefficients definitions. 1034 01:06:57,060 --> 01:06:59,070 So I'm putting everything in this scale R0. 1035 01:07:06,450 --> 01:07:09,090 So that's just rewriting the OPE in this new scheme. 1036 01:07:14,210 --> 01:07:17,670 When we look at this scheme and we think about what's going on, 1037 01:07:17,670 --> 01:07:20,710 we want to choose the R0 to be of order lambda 1038 01:07:20,710 --> 01:07:23,395 QCD or a bit bigger. 1039 01:07:23,395 --> 01:07:24,770 And the reason we want to do that 1040 01:07:24,770 --> 01:07:26,030 is to preserve power counting. 1041 01:07:30,480 --> 01:07:34,370 So power counting is important when we're making expansions. 1042 01:07:34,370 --> 01:07:37,530 In some sense, it's key. 1043 01:07:37,530 --> 01:07:43,520 And if we thought that the original MS-bar matrix 1044 01:07:43,520 --> 01:07:47,040 element was dominated by the scale lambda QCD, 1045 01:07:47,040 --> 01:07:52,670 we better not screw that up when we make a change of variable. 1046 01:07:52,670 --> 01:07:59,145 And so the kind of change of variable you're making 1047 01:07:59,145 --> 01:08:01,020 is you're taking the original matrix element. 1048 01:08:01,020 --> 01:08:05,520 You're subtracting a series with a proportionality constant R0. 1049 01:08:05,520 --> 01:08:07,170 This guy was lambda QCD squared. 1050 01:08:07,170 --> 01:08:11,950 And you want this to be not too far different than lambda QCD, 1051 01:08:11,950 --> 01:08:14,370 so that this whole thing here is lambda QCD cubed just 1052 01:08:14,370 --> 01:08:18,910 like this, so that this thing is still 1053 01:08:18,910 --> 01:08:21,064 of order lambda QCD cubed. 1054 01:08:21,064 --> 01:08:22,439 So you don't want to do something 1055 01:08:22,439 --> 01:08:25,859 like choose the R0 to be Mb or something because then you'd 1056 01:08:25,859 --> 01:08:29,010 be making a huge change in the value of this matrix element. 1057 01:08:29,010 --> 01:08:31,790 You really want to take the R0 cutoff to be of order, 1058 01:08:31,790 --> 01:08:33,397 say, g of r or something. 1059 01:08:37,069 --> 01:08:41,109 So you can use the RGE to sum up logs from that scale R0 1060 01:08:41,109 --> 01:08:42,090 up to the scale of MQ. 1061 01:08:46,560 --> 01:08:52,149 So again, we can sum up logs and have a logarithmically improved 1062 01:08:52,149 --> 01:08:59,109 result. But actually the most important thing here 1063 01:08:59,109 --> 01:09:01,960 is getting rid of the renormalon. 1064 01:09:01,960 --> 01:09:04,270 I just wanted to kind of show you 1065 01:09:04,270 --> 01:09:09,920 what the result would look like with both the RGE improvement, 1066 01:09:09,920 --> 01:09:15,069 just sort of using the full set of tools that we have. 1067 01:09:15,069 --> 01:09:16,430 So here's the scales. 1068 01:09:16,430 --> 01:09:20,000 You have three physical scales, b quark mass, charm mass, 1069 01:09:20,000 --> 01:09:21,319 lambda QCD. 1070 01:09:21,319 --> 01:09:22,227 You have two cutoffs. 1071 01:09:22,227 --> 01:09:23,810 You have our R0, which I just told you 1072 01:09:23,810 --> 01:09:26,390 you should pick to be close to lambda QCD. 1073 01:09:26,390 --> 01:09:29,800 And then you can think that there's these other scales, Mb 1074 01:09:29,800 --> 01:09:31,760 and M charm. 1075 01:09:31,760 --> 01:09:34,170 If this is some low scale and that's some high scale, 1076 01:09:34,170 --> 01:09:35,359 I'd like to sum logs. 1077 01:09:35,359 --> 01:09:39,350 And I can do that by running up from R0 to R1, 1078 01:09:39,350 --> 01:09:44,779 which in this case I'll just pick between Mb and M charm. 1079 01:09:44,779 --> 01:09:50,779 So if you did that, you'd get some result 1080 01:09:50,779 --> 01:09:58,230 that looks like this, just putting 1081 01:09:58,230 --> 01:10:00,400 in some notation for the resummation here. 1082 01:10:08,940 --> 01:10:10,565 And then this guy is at the low scale. 1083 01:10:20,477 --> 01:10:21,560 So now, everybody's happy. 1084 01:10:21,560 --> 01:10:23,120 The Wilson coefficients here can be expanded 1085 01:10:23,120 --> 01:10:24,180 in perturbation theory. 1086 01:10:24,180 --> 01:10:26,060 We don't think there's any large logs. 1087 01:10:26,060 --> 01:10:28,550 This U sums up any large logs between the low scales 1088 01:10:28,550 --> 01:10:29,960 and the high scales. 1089 01:10:29,960 --> 01:10:33,560 And this guy's living at a scale that he likes to live at. 1090 01:10:33,560 --> 01:10:37,350 And there's no renormalon, no u equals 1 renormalon. 1091 01:10:37,350 --> 01:10:41,570 So what does it look like if I write down this guy? 1092 01:10:41,570 --> 01:10:44,570 So first of all, if I look at this thing order by order, 1093 01:10:44,570 --> 01:10:47,718 this is how the convergence looks. 1094 01:10:47,718 --> 01:10:49,760 You get a downward shift by about the same amount 1095 01:10:49,760 --> 01:10:53,550 as before, but the next shifts are small. 1096 01:10:53,550 --> 01:10:56,240 So the series actually looks like it's 1097 01:10:56,240 --> 01:10:58,510 converging to something. 1098 01:10:58,510 --> 01:11:01,565 So that's the first three orders if you use this approach. 1099 01:11:04,100 --> 01:11:08,410 And if you use the complete power of this at the highest 1100 01:11:08,410 --> 01:11:16,270 order, you can write down that you get this 0.860 from 1101 01:11:16,270 --> 01:11:19,975 the perturbative part, which is not too far from the 0.88 1102 01:11:19,975 --> 01:11:23,240 or the 0.8517, which I said was close to the data. 1103 01:11:23,240 --> 01:11:26,260 So this actually is pretty good. 1104 01:11:26,260 --> 01:11:28,750 You can make some error estimate for how big this thing is. 1105 01:11:36,230 --> 01:11:37,730 And you can make some error estimate 1106 01:11:37,730 --> 01:11:40,930 for the perturbative uncertainty from higher orders 1107 01:11:40,930 --> 01:11:43,590 by varying scales. 1108 01:11:43,590 --> 01:11:48,330 And you get a prediction that looks like that. 1109 01:11:48,330 --> 01:11:50,390 So when you want to estimate uncertainties, 1110 01:11:50,390 --> 01:11:52,940 what you typically do with your cutoffs is you vary them. 1111 01:11:52,940 --> 01:11:54,440 You say, well, it has to be up here, 1112 01:11:54,440 --> 01:11:56,000 but I don't have to take it exactly there. 1113 01:11:56,000 --> 01:11:58,170 I could take it up here or take it a little lower, 1114 01:11:58,170 --> 01:11:59,090 move this guy around. 1115 01:11:59,090 --> 01:12:01,340 That gives you some idea of higher order uncertainties 1116 01:12:01,340 --> 01:12:03,380 because the dependence on these scales 1117 01:12:03,380 --> 01:12:06,190 cancels out order by order in perturbation theory. 1118 01:12:06,190 --> 01:12:08,990 And so varying it is a way of getting 1119 01:12:08,990 --> 01:12:10,680 an idea of higher order uncertainties 1120 01:12:10,680 --> 01:12:11,835 from perturbation theory. 1121 01:12:11,835 --> 01:12:13,460 And that's what I've done, for example, 1122 01:12:13,460 --> 01:12:14,793 for this perturbative term here. 1123 01:12:14,793 --> 01:12:19,490 This is varying R1 up and down by a factor of 2. 1124 01:12:19,490 --> 01:12:23,600 And just see kind of what residual uncertainty you have. 1125 01:12:23,600 --> 01:12:25,027 This one's more interesting. 1126 01:12:25,027 --> 01:12:26,360 So that one's like mu variation. 1127 01:12:26,360 --> 01:12:28,400 This one's more interesting. 1128 01:12:28,400 --> 01:12:31,100 And that's because we have this other cutoff, R0. 1129 01:12:31,100 --> 01:12:35,210 But what R0 does is it connects a leading order term 1130 01:12:35,210 --> 01:12:37,690 to a sub-leading power term. 1131 01:12:37,690 --> 01:12:40,460 The R0 dependence cancels between something 1132 01:12:40,460 --> 01:12:42,860 that's order 1 in the power counting, something 1133 01:12:42,860 --> 01:12:45,520 that's higher order in the power counting. 1134 01:12:45,520 --> 01:12:47,630 But when I vary R0, it still cancels 1135 01:12:47,630 --> 01:12:48,740 between these two pieces. 1136 01:12:48,740 --> 01:12:52,370 Because remember, if you look at the top of the board, 1137 01:12:52,370 --> 01:12:56,350 I've included higher power terms in my Wilson coefficient. 1138 01:12:56,350 --> 01:12:59,000 And so varying the R0, you can actually 1139 01:12:59,000 --> 01:13:00,860 get an idea of how big this thing is 1140 01:13:00,860 --> 01:13:04,200 kind of in a naive dimensional analysis type way. 1141 01:13:04,200 --> 01:13:07,215 And not gives this number here. 1142 01:13:07,215 --> 01:13:09,500 So R0 variation actually allows you 1143 01:13:09,500 --> 01:13:12,650 to test for the size of the power corrections 1144 01:13:12,650 --> 01:13:14,900 if you don't know them. 1145 01:13:14,900 --> 01:13:17,720 And in this case, it turns out that the power correction is 1146 01:13:17,720 --> 01:13:19,500 kind of small in this scheme. 1147 01:13:19,500 --> 01:13:21,410 But it didn't have to be that small. 1148 01:13:21,410 --> 01:13:24,810 It could have been 165, 265. 1149 01:13:24,810 --> 01:13:27,770 It happens to be 0.065, OK? 1150 01:13:27,770 --> 01:13:30,230 So everything looks much nicer after we just 1151 01:13:30,230 --> 01:13:31,607 get rid of the renormalon. 1152 01:13:31,607 --> 01:13:33,440 Someone else did the three-loop calculation. 1153 01:13:33,440 --> 01:13:39,080 You made the prediction, just true in this case. 1154 01:13:41,880 --> 01:13:43,270 OK, so questions? 1155 01:13:46,340 --> 01:13:50,930 So the moral of the story is that, if you really 1156 01:13:50,930 --> 01:13:54,225 want to use calculations that are out there, 1157 01:13:54,225 --> 01:13:55,725 you have to think about the physics. 1158 01:13:55,725 --> 01:13:57,805 You can't just calculate, sorry. 1159 01:14:05,640 --> 01:14:08,490 So these renormalons, when you have kind of order 1160 01:14:08,490 --> 01:14:10,270 alpha squared or alpha cubed information, 1161 01:14:10,270 --> 01:14:12,187 then you have to worry about things like this. 1162 01:14:12,187 --> 01:14:14,820 And if you didn't, what looked like was working beautifully 1163 01:14:14,820 --> 01:14:17,490 at leading log order might get spoiled. 1164 01:14:17,490 --> 01:14:18,990 You're leading log might be the best 1165 01:14:18,990 --> 01:14:21,625 prediction unless you think about this that you can make. 1166 01:14:21,625 --> 01:14:23,250 Because your just higher orders, you're 1167 01:14:23,250 --> 01:14:25,960 getting sensitive to this renormalon problem. 1168 01:14:25,960 --> 01:14:30,480 And what that means is generically this is changing 1169 01:14:30,480 --> 01:14:32,225 and this is changing. 1170 01:14:32,225 --> 01:14:33,600 The power correction is changing. 1171 01:14:33,600 --> 01:14:35,510 And so at each order that this changes, 1172 01:14:35,510 --> 01:14:37,800 I could compensate by changing this. 1173 01:14:37,800 --> 01:14:39,970 But it would have to be a pretty big change, right? 1174 01:14:39,970 --> 01:14:41,490 I changed this guy by this amount. 1175 01:14:41,490 --> 01:14:43,073 And then I have to compensate by that. 1176 01:14:43,073 --> 01:14:45,120 Whoops, now, it even changes by a larger amount. 1177 01:14:45,120 --> 01:14:48,780 Well, compensate by changing this by order 1 amount. 1178 01:14:48,780 --> 01:14:52,740 That's exactly this fact, that both things have this problem. 1179 01:14:52,740 --> 01:14:55,973 Order by order, I can cancel out the problem, 1180 01:14:55,973 --> 01:14:58,140 but then I can never really assign a number to this. 1181 01:14:58,140 --> 01:14:59,567 It's an order dependent number. 1182 01:14:59,567 --> 01:15:01,650 Each order in perturbation theory that I use here, 1183 01:15:01,650 --> 01:15:02,940 this number changes. 1184 01:15:02,940 --> 01:15:04,440 And that's not really what you think 1185 01:15:04,440 --> 01:15:06,212 of for a physical concept. 1186 01:15:06,212 --> 01:15:07,920 This matrix element should have a meaning 1187 01:15:07,920 --> 01:15:10,320 that it shouldn't be changing by order 1 1188 01:15:10,320 --> 01:15:13,170 depending on the perturbative order that you're working. 1189 01:15:13,170 --> 01:15:14,880 In this way of thinking, this thing 1190 01:15:14,880 --> 01:15:17,780 is stable, much more stable anyway. 1191 01:15:17,780 --> 01:15:20,970 And you can assign a number to it. 1192 01:15:20,970 --> 01:15:24,210 That's kind of the moral. 1193 01:15:24,210 --> 01:15:25,097 All right. 1194 01:15:25,097 --> 01:15:25,930 AUDIENCE: Hey, Iain? 1195 01:15:25,930 --> 01:15:26,722 IAIN STEWART: Yeah. 1196 01:15:26,722 --> 01:15:28,978 AUDIENCE: That [INAUDIBLE] looks [INAUDIBLE].. 1197 01:15:31,790 --> 01:15:34,542 IAIN STEWART: So the thing is that the convergence, 1198 01:15:34,542 --> 01:15:35,750 it's not about the precision. 1199 01:15:35,750 --> 01:15:38,390 It's about the convergence, right? 1200 01:15:38,390 --> 01:15:41,450 So you would be fine in some sense. 1201 01:15:41,450 --> 01:15:45,860 It would not be technically a problem if this was 0, right? 1202 01:15:45,860 --> 01:15:49,400 If this guy was not much of a shift-- 1203 01:15:49,400 --> 01:15:50,870 and this was a 15 shift. 1204 01:15:50,870 --> 01:15:52,310 This is a 7 shift. 1205 01:15:52,310 --> 01:15:56,085 If this guy was small again, you'd be fine, right? 1206 01:15:56,085 --> 01:15:57,460 Because then you'd say, OK, well, 1207 01:15:57,460 --> 01:16:02,310 I extract this thing in the same order that I include that. 1208 01:16:02,310 --> 01:16:04,520 And maybe this thing gives a plus 0.07, 1209 01:16:04,520 --> 01:16:05,960 and this guy is a minus 0.07. 1210 01:16:05,960 --> 01:16:07,252 And then I'm close to the data. 1211 01:16:07,252 --> 01:16:08,570 And you're happy. 1212 01:16:08,570 --> 01:16:10,460 The issue is that you go to a higher order, 1213 01:16:10,460 --> 01:16:12,950 and you get even bigger shift. 1214 01:16:12,950 --> 01:16:14,540 That's the real problem, that you 1215 01:16:14,540 --> 01:16:16,250 don't, in the perturbative part alone, 1216 01:16:16,250 --> 01:16:18,980 have any sign of convergence. 1217 01:16:18,980 --> 01:16:20,570 That's the issue. 1218 01:16:20,570 --> 01:16:24,050 So the fact that this number is sort of close to that number 1219 01:16:24,050 --> 01:16:25,965 is, in some sense, OK because you 1220 01:16:25,965 --> 01:16:28,340 think that this thing should generically be of that size. 1221 01:16:28,340 --> 01:16:30,975 That's what dimensional analysis would say. 1222 01:16:30,975 --> 01:16:32,600 In dimensional analysis, you might even 1223 01:16:32,600 --> 01:16:35,310 say it's a little bigger. 1224 01:16:35,310 --> 01:16:37,760 So the real issue is that you want your predictions 1225 01:16:37,760 --> 01:16:40,252 to be stable. 1226 01:16:40,252 --> 01:16:40,752 Yeah. 1227 01:16:40,752 --> 01:16:44,760 AUDIENCE: But I thought that precision should be higher 1228 01:16:44,760 --> 01:16:45,962 than the [INAUDIBLE]. 1229 01:16:45,962 --> 01:16:46,670 IAIN STEWART: No. 1230 01:16:46,670 --> 01:16:50,630 This, actually, here is connected to that term. 1231 01:16:50,630 --> 01:16:53,060 I'm saying, even if I leave this term out, 1232 01:16:53,060 --> 01:16:55,970 I actually get something that agrees with the data. 1233 01:16:55,970 --> 01:16:58,760 And that's because it turns out that the power correction here 1234 01:16:58,760 --> 01:17:00,080 is of this size. 1235 01:17:00,080 --> 01:17:02,840 And I can get an idea of how big it is by varying 1236 01:17:02,840 --> 01:17:04,520 the R0 in this prediction. 1237 01:17:04,520 --> 01:17:08,280 And that changes this number by that amount. 1238 01:17:08,280 --> 01:17:11,090 And since this thing is also changing in exactly a way that 1239 01:17:11,090 --> 01:17:12,620 compensates that R0 dependence, I'm 1240 01:17:12,620 --> 01:17:14,810 getting an estimate for this thing. 1241 01:17:14,810 --> 01:17:15,960 That's what I did there. 1242 01:17:15,960 --> 01:17:18,650 So if I were to put this thing in, I'd have a new parameter. 1243 01:17:18,650 --> 01:17:21,460 And I could exactly fit the data, right? 1244 01:17:21,460 --> 01:17:26,440 But what I wrote was a little different. 1245 01:17:26,440 --> 01:17:28,225 Yeah. 1246 01:17:28,225 --> 01:17:29,600 I kind of went over that quickly. 1247 01:17:32,160 --> 01:17:36,230 All right, so that's actually it for HQET. 1248 01:17:40,780 --> 01:17:44,040 So before we get to SCET, which is coming up 1249 01:17:44,040 --> 01:17:47,710 after spring break, we're going to talk 1250 01:17:47,710 --> 01:17:53,695 about one more topic, which I'll just basically introduce. 1251 01:17:53,695 --> 01:17:56,470 And then we'll stop and continue on Thursday. 1252 01:18:03,178 --> 01:18:05,720 So there's going to be one more example of an effective field 1253 01:18:05,720 --> 01:18:07,830 theory. 1254 01:18:07,830 --> 01:18:10,550 And this is going to be an example of an effective field 1255 01:18:10,550 --> 01:18:13,400 theory with what looks like a problem. 1256 01:18:13,400 --> 01:18:15,460 It's got a fine tuning. 1257 01:18:15,460 --> 01:18:17,877 So usually, if the whole notion of effective field theory 1258 01:18:17,877 --> 01:18:20,210 is against the idea that there should be a fine tuning-- 1259 01:18:20,210 --> 01:18:21,860 because you're making dimensional analysis 1260 01:18:21,860 --> 01:18:22,693 estimates of things. 1261 01:18:22,693 --> 01:18:25,310 If there's a fine tuning, that means your dimensional analysis 1262 01:18:25,310 --> 01:18:26,810 failed. 1263 01:18:26,810 --> 01:18:31,010 In this example, we'll see that you can make one fine tuning. 1264 01:18:31,010 --> 01:18:33,560 And you can understand what's going on with that fine tuning 1265 01:18:33,560 --> 01:18:36,693 and actually propagate it to change your power counting, 1266 01:18:36,693 --> 01:18:38,110 to change your power counting such 1267 01:18:38,110 --> 01:18:41,165 that it takes into account the existence of that fine tuning. 1268 01:18:41,165 --> 01:18:44,060 It builds the whole effective theory around the idea 1269 01:18:44,060 --> 01:18:47,480 that there was this fine tuning from original, perhaps, 1270 01:18:47,480 --> 01:18:49,490 dimension counting point of view. 1271 01:18:49,490 --> 01:18:51,680 But we can adopt a different power counting 1272 01:18:51,680 --> 01:18:54,590 that actually organizes the physics in exactly 1273 01:18:54,590 --> 01:18:56,017 the right way that we want to. 1274 01:18:56,017 --> 01:18:58,350 And in fact, it's going to be such an easy example where 1275 01:18:58,350 --> 01:19:02,390 we can just basically calculate all the Feynman diagrams very 1276 01:19:02,390 --> 01:19:05,030 simply, one line. 1277 01:19:05,030 --> 01:19:06,530 So that's what we'll do. 1278 01:19:11,330 --> 01:19:16,120 It'll be a very simple example of an effective theory field. 1279 01:19:16,120 --> 01:19:18,560 And we'll prove some things about quantum mechanics that 1280 01:19:18,560 --> 01:19:24,080 would be very difficult if we weren't using effective field 1281 01:19:24,080 --> 01:19:25,790 theory along the way. 1282 01:19:30,423 --> 01:19:32,090 So we'll investigate an effective theory 1283 01:19:32,090 --> 01:19:37,640 that has a naively irrelevant operator that must 1284 01:19:37,640 --> 01:19:40,600 be promoted to being relevant. 1285 01:19:48,150 --> 01:19:50,800 So by dimensional analysis, it would be irrelevant. 1286 01:19:50,800 --> 01:19:52,800 And one way of thinking about it is that it just 1287 01:19:52,800 --> 01:19:55,140 has such a large anomalous dimension that it actually 1288 01:19:55,140 --> 01:19:57,870 ends up being totally relevant. 1289 01:19:57,870 --> 01:20:02,110 And that is actually not a bad way of thinking about it. 1290 01:20:02,110 --> 01:20:05,400 So the example we'll talk about is 1291 01:20:05,400 --> 01:20:13,382 something called two-nucleon non-relativistic 1292 01:20:13,382 --> 01:20:14,340 effective field theory. 1293 01:20:20,130 --> 01:20:22,655 So you have two nucleons, a neutron and a proton, 1294 01:20:22,655 --> 01:20:23,155 for example. 1295 01:20:26,080 --> 01:20:29,030 It's a bottom-up effective theory. 1296 01:20:29,030 --> 01:20:31,550 We're not going to be thinking about calculating nucleons 1297 01:20:31,550 --> 01:20:33,210 in QCD. 1298 01:20:33,210 --> 01:20:35,212 We're going to work at momenta that 1299 01:20:35,212 --> 01:20:37,420 are so small that we actually integrate out the pion. 1300 01:20:43,671 --> 01:20:45,970 So all exchange particles are integrated out. 1301 01:21:01,190 --> 01:21:03,500 So there's nothing to exchange and really just have 1302 01:21:03,500 --> 01:21:06,090 contact type interactions. 1303 01:21:06,090 --> 01:21:08,450 So something that you might think of as like A pion 1304 01:21:08,450 --> 01:21:12,890 exchange between two nucleons gets 1305 01:21:12,890 --> 01:21:16,650 represented by some local operators between the nucleons. 1306 01:21:22,240 --> 01:21:23,925 All right, so let me stop there. 1307 01:21:23,925 --> 01:21:25,300 And we'll come back and talk more 1308 01:21:25,300 --> 01:21:27,460 about this theory next time. 1309 01:21:27,460 --> 01:21:31,810 And we'll see, first, that these operators of this type 1310 01:21:31,810 --> 01:21:34,180 actually can organize some facts about quantum 1311 01:21:34,180 --> 01:21:36,280 mechanics in a very nice way. 1312 01:21:36,280 --> 01:21:38,920 And then we'll see how to think about fine tuning 1313 01:21:38,920 --> 01:21:43,180 from this contact interaction theory 1314 01:21:43,180 --> 01:21:45,588 and also how to think about how we 1315 01:21:45,588 --> 01:21:47,380 want to organize the power counting, what's 1316 01:21:47,380 --> 01:21:49,530 MS-bar say, et cetera. 1317 01:21:53,820 --> 01:21:56,040 It's kind of a fun theory.