1 00:00:00,000 --> 00:00:02,520 The following content is provided under a Creative 2 00:00:02,520 --> 00:00:03,970 Commons license. 3 00:00:03,970 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,660 continue to offer high quality educational resources for free. 5 00:00:10,660 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,190 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,190 --> 00:00:18,370 at ocw.mit.edu. 8 00:00:25,225 --> 00:00:26,100 PROFESSOR: All right. 9 00:00:26,100 --> 00:00:31,110 So, so far we've recently been talking about examples in SET2, 10 00:00:31,110 --> 00:00:34,030 and we're going to continue to do so today. 11 00:00:34,030 --> 00:00:36,090 So the example that we did last time 12 00:00:36,090 --> 00:00:38,070 was the plan photon form factor. 13 00:00:38,070 --> 00:00:40,140 That did not have any soft degrees of freedom. 14 00:00:40,140 --> 00:00:43,060 It just had colinear and higher degrees of freedom. 15 00:00:43,060 --> 00:00:45,570 So it was a particularly simple example of something 16 00:00:45,570 --> 00:00:47,790 we could think of in SET2. 17 00:00:47,790 --> 00:00:50,370 We'll start with a slightly more complicated example, 18 00:00:50,370 --> 00:00:54,480 this decay, B to D pi, where we have both colinear 19 00:00:54,480 --> 00:00:55,860 and soft degrees of freedom. 20 00:00:55,860 --> 00:00:57,690 This was an example that we mentioned 21 00:00:57,690 --> 00:00:59,940 at the very beginning of our discussion of SET, 22 00:00:59,940 --> 00:01:03,462 and now we're going to see how factorization looks for it. 23 00:01:03,462 --> 00:01:04,920 And then we'll talk about something 24 00:01:04,920 --> 00:01:07,920 called the rapidity renormalization group, 25 00:01:07,920 --> 00:01:10,860 which has to do with situations in SET2 26 00:01:10,860 --> 00:01:13,020 where the separation of degrees of freedom 27 00:01:13,020 --> 00:01:15,570 is a little more complicated than in the previous examples. 28 00:01:15,570 --> 00:01:18,660 And we'll see that there can be a new type of divergence that 29 00:01:18,660 --> 00:01:19,650 shows up. 30 00:01:19,650 --> 00:01:22,050 And that new type of divergence leads to a new type 31 00:01:22,050 --> 00:01:25,800 of renormalization group. 32 00:01:25,800 --> 00:01:32,340 So B to D pie. 33 00:01:32,340 --> 00:01:39,270 So there's going to be, in some sense, three hard scales 34 00:01:39,270 --> 00:01:41,790 of this problem. 35 00:01:41,790 --> 00:01:44,275 The mass of the B quark and the mass of the charm quark 36 00:01:44,275 --> 00:01:46,650 are going to be taken to be much greater than lambda QCD, 37 00:01:46,650 --> 00:01:48,960 so we'll have an HQET type description of the B 38 00:01:48,960 --> 00:01:50,670 quark and the charm quark. 39 00:01:50,670 --> 00:01:53,423 And also the energy of the pion, which 40 00:01:53,423 --> 00:01:55,590 is in some sense, related to the difference of the B 41 00:01:55,590 --> 00:01:58,080 quark and the charm quark mass, will also 42 00:01:58,080 --> 00:02:02,050 take that to be much greater than the lambda QCD. 43 00:02:02,050 --> 00:02:11,190 So just by kinematics this thing is proportional to MB minus MC 44 00:02:11,190 --> 00:02:12,353 roughly. 45 00:02:12,353 --> 00:02:13,770 You could say, it's the difference 46 00:02:13,770 --> 00:02:17,500 of the squares of the hadron masses. 47 00:02:17,500 --> 00:02:18,000 OK. 48 00:02:18,000 --> 00:02:21,480 So let's first-- we know how to treat this decay if we were 49 00:02:21,480 --> 00:02:23,490 integrating out the W. This is a weak decay, 50 00:02:23,490 --> 00:02:25,540 so B is changing to charm. 51 00:02:25,540 --> 00:02:28,650 Integrate of the W boson, run down to the scale MB, 52 00:02:28,650 --> 00:02:30,720 which is the larger scale here, that's 53 00:02:30,720 --> 00:02:32,040 the electroweak Hamiltonian. 54 00:02:34,590 --> 00:02:39,690 So that's what we'll call the QCD operators, which 55 00:02:39,690 --> 00:02:42,920 are the relevant description at the scale of order MB. 56 00:02:54,030 --> 00:02:54,990 Some pre-factor. 57 00:03:02,430 --> 00:03:05,610 And I'll write the operators in the following way-- 58 00:03:05,610 --> 00:03:07,440 a slightly different basis than we 59 00:03:07,440 --> 00:03:13,480 used previously, or a long time ago 60 00:03:13,480 --> 00:03:19,910 when we were talking about this particular case. 61 00:03:19,910 --> 00:03:22,495 So just a different color basis, singlet and octet. 62 00:03:39,530 --> 00:03:41,780 OK, so that's our description, where 63 00:03:41,780 --> 00:03:48,340 p left is projecting us onto, the left-handed components. 64 00:03:48,340 --> 00:03:51,160 So what we want to do is we want to factorize the amplitude. 65 00:03:51,160 --> 00:03:52,840 This is an exclusive process where 66 00:03:52,840 --> 00:03:57,520 we make a transition between specific states. 67 00:03:57,520 --> 00:04:06,580 So we'd like to separate scales in the D pi, 68 00:04:06,580 --> 00:04:12,380 and then we have O0 or O8 in this matrix element. 69 00:04:12,380 --> 00:04:16,779 So we have two matrix elements, one with O0 and one with O8. 70 00:04:16,779 --> 00:04:19,370 And so, what could it possibly look like? 71 00:04:19,370 --> 00:04:21,790 Well, we already talked about the degrees of freedom here. 72 00:04:21,790 --> 00:04:25,330 The D is going to be soft and the B is going to be soft. 73 00:04:25,330 --> 00:04:30,350 So this is soft, this is soft. 74 00:04:30,350 --> 00:04:31,570 This is going to be colinear. 75 00:04:31,570 --> 00:04:33,280 And so, if it's going to factorize, 76 00:04:33,280 --> 00:04:34,905 and the soft degrees of freedom are not 77 00:04:34,905 --> 00:04:36,550 going to talk to the colinear, the kind 78 00:04:36,550 --> 00:04:39,890 of thing that you would expect to show 79 00:04:39,890 --> 00:04:44,740 at leading order in the lambda expansion 80 00:04:44,740 --> 00:04:48,100 is that you have the following kind of process. 81 00:04:55,240 --> 00:04:56,430 Let me reclaim this space. 82 00:05:11,550 --> 00:05:13,280 So here's a heavy quark. 83 00:05:13,280 --> 00:05:16,880 Here's one of these operators. 84 00:05:16,880 --> 00:05:19,760 Here's the valence quarks in the pion. 85 00:05:19,760 --> 00:05:36,930 Another heavy quark-- this was B charm, U and D. 86 00:05:36,930 --> 00:05:38,610 And there's an anti-quark, and we 87 00:05:38,610 --> 00:05:40,620 have to address this by gluons. 88 00:05:40,620 --> 00:05:43,410 And if it's going to factorize, then the way 89 00:05:43,410 --> 00:05:46,990 that we should dress it by gluons is as follows. 90 00:05:46,990 --> 00:05:50,850 We would have soft gluons here, and they 91 00:05:50,850 --> 00:05:53,730 could interact, if you like, between things 92 00:05:53,730 --> 00:05:57,990 in the B and the D, because the B and the D are both soft. 93 00:05:57,990 --> 00:06:02,370 We can also have back and polarization diagrams. 94 00:06:02,370 --> 00:06:09,180 And that is going to factorize from things in the pion which 95 00:06:09,180 --> 00:06:10,240 are going to be colinear. 96 00:06:10,240 --> 00:06:14,790 So we have our colinear gluons and colinear quarks 97 00:06:14,790 --> 00:06:19,007 inside here, and maybe there's some Wilson lines too. 98 00:06:19,007 --> 00:06:21,090 So we would expect some kind of picture like that. 99 00:06:21,090 --> 00:06:24,570 And that's actually going to be what we do find. 100 00:06:24,570 --> 00:06:27,300 But exactly what happens at this vertex, what kind 101 00:06:27,300 --> 00:06:31,530 of convolutions there are, that we have to work out. 102 00:06:31,530 --> 00:06:32,250 All right. 103 00:06:32,250 --> 00:06:35,700 So what factorization in this context means 104 00:06:35,700 --> 00:06:40,650 is that there's no gluons that are directly 105 00:06:40,650 --> 00:06:43,500 exchanged between the B to D part and the pion part. 106 00:06:52,100 --> 00:06:55,490 So that it effectively factorizes into a matrix 107 00:06:55,490 --> 00:06:57,260 element that's like a B to D transition 108 00:06:57,260 --> 00:07:00,770 and a vacuum to pion transition. 109 00:07:00,770 --> 00:07:03,500 So you can even guess what kind of objects 110 00:07:03,500 --> 00:07:05,180 this would depend on. 111 00:07:05,180 --> 00:07:07,550 If you have something like this green thing, 112 00:07:07,550 --> 00:07:10,670 that's a B to D form factor. 113 00:07:10,670 --> 00:07:17,780 So we expect a B to D form factor. 114 00:07:22,520 --> 00:07:24,770 And for the pion, if you have something like this, 115 00:07:24,770 --> 00:07:26,812 well, we already talked about something like this 116 00:07:26,812 --> 00:07:29,900 when we were talking about gamma star, gamma to pi zero. 117 00:07:29,900 --> 00:07:33,500 So for the pion, we expect the pi zero-- 118 00:07:33,500 --> 00:07:42,800 the light cone distribution, which 119 00:07:42,800 --> 00:07:47,210 is the sort of leading order operator for the pion. 120 00:07:47,210 --> 00:07:50,330 So we'd expect a 5pi of x. 121 00:07:50,330 --> 00:07:53,210 And we'll see that we do indeed find that. 122 00:08:03,160 --> 00:08:07,010 So the B and D have P squared of order lambda QCD 123 00:08:07,010 --> 00:08:08,600 squared for their constituents. 124 00:08:08,600 --> 00:08:13,790 The pion is colinear, and its constituents 125 00:08:13,790 --> 00:08:16,670 have P squared of order lambda QCD2, but they're boosted. 126 00:08:19,330 --> 00:08:20,950 We can again use SET-- 127 00:08:20,950 --> 00:08:25,650 this is SET2, but we can use SET1 as an intermediate step, 128 00:08:25,650 --> 00:08:27,190 just like we did-- 129 00:08:27,190 --> 00:08:28,450 we talked about last time. 130 00:08:42,770 --> 00:08:45,130 So let's do that again. 131 00:08:45,130 --> 00:08:49,390 So match-- step one was to match QCD onto the SCET1. 132 00:08:52,670 --> 00:08:53,990 So there was some hard scale. 133 00:08:53,990 --> 00:08:55,365 And the harder scale in this case 134 00:08:55,365 --> 00:08:57,990 could be any one of these three. 135 00:08:57,990 --> 00:09:05,933 So collectively I just denote them by Q. 136 00:09:05,933 --> 00:09:07,850 And so what's going to happen in that matching 137 00:09:07,850 --> 00:09:10,040 is we take these operators O0 and O8, 138 00:09:10,040 --> 00:09:12,200 and we have to match onto SET operators. 139 00:09:20,720 --> 00:09:22,490 So let me call the SET operators Q0. 140 00:09:26,510 --> 00:09:35,950 I just there's-- because of the fact that there's a heavy quark 141 00:09:35,950 --> 00:09:40,960 in the way that that works, there's two possible spin 142 00:09:40,960 --> 00:09:43,730 structures here. 143 00:09:43,730 --> 00:09:46,630 But we'll see that actually only one of them 144 00:09:46,630 --> 00:09:49,900 has the quantum numbers in the end. 145 00:09:49,900 --> 00:09:56,650 So we have heavy quark fields, charm, and bottom, 146 00:09:56,650 --> 00:09:59,120 that we can imagine that type of operator. 147 00:09:59,120 --> 00:10:02,380 And then the colinear part, we can dress it with the Wilson 148 00:10:02,380 --> 00:10:04,240 line, as always. 149 00:10:04,240 --> 00:10:05,800 And let me put the flavor upstairs. 150 00:10:15,282 --> 00:10:17,490 And let me put in the most general Wilson coefficient 151 00:10:17,490 --> 00:10:20,980 that I can think of for this process. 152 00:10:20,980 --> 00:10:24,900 This could also depend on V dot V prime. 153 00:10:24,900 --> 00:10:26,530 I didn't denote that. 154 00:10:26,530 --> 00:10:27,540 But in general, it will. 155 00:10:27,540 --> 00:10:31,530 And it could depend on any of the scales Q. 156 00:10:31,530 --> 00:10:35,400 So it can depend on the large momentum of the colinear 157 00:10:35,400 --> 00:10:36,478 fields, as always. 158 00:10:36,478 --> 00:10:38,520 And there's one combination that's not restricted 159 00:10:38,520 --> 00:10:41,075 by momentum conservation. 160 00:10:41,075 --> 00:10:42,450 So there's one combination that's 161 00:10:42,450 --> 00:10:44,880 not related to these scales, and that's 162 00:10:44,880 --> 00:10:49,673 the, in my notation, the P bar plus operator. 163 00:10:49,673 --> 00:10:51,090 And then there's likewise, there's 164 00:10:51,090 --> 00:10:52,350 another thing with the TA. 165 00:10:52,350 --> 00:11:00,780 So same thing, TA, everything the same, TA. 166 00:11:00,780 --> 00:11:06,420 And then this has got a different coefficient 167 00:11:06,420 --> 00:11:09,400 like that. 168 00:11:09,400 --> 00:11:11,895 And so, what are the Dirac structures just to be explicit? 169 00:11:14,640 --> 00:11:22,620 The heavy one-- so the light one is going to be M bar 170 00:11:22,620 --> 00:11:28,680 slash over 4 in my notation, the 1 minus gamma 5. 171 00:11:28,680 --> 00:11:32,440 And for the heavy one, you could have either 1 or gamma 5. 172 00:11:32,440 --> 00:11:37,020 And that's because you originally have a left handed-- 173 00:11:37,020 --> 00:11:39,420 originally here you have a left handed guy 174 00:11:39,420 --> 00:11:42,930 between the charm and the B. But remember that a mass term 175 00:11:42,930 --> 00:11:47,070 connects left and right. 176 00:11:47,070 --> 00:11:49,800 So after you integrate out the mass of these quarks, 177 00:11:49,800 --> 00:11:51,330 you don't know whether-- 178 00:11:51,330 --> 00:11:54,278 you don't know the chirality anymore in this. 179 00:11:54,278 --> 00:11:56,820 So that's why you can have both the possibilities 1 and gamma 180 00:11:56,820 --> 00:11:57,320 5. 181 00:12:01,330 --> 00:12:03,820 So if you put any other Dirac structure-- so here 182 00:12:03,820 --> 00:12:04,900 you could use chirality. 183 00:12:04,900 --> 00:12:07,767 And so, you know that these guys should be left handed still, 184 00:12:07,767 --> 00:12:09,850 and that's why that this should be this structure. 185 00:12:09,850 --> 00:12:11,320 You know it should be an M bar slash, 186 00:12:11,320 --> 00:12:13,320 because any other structure that you would stick 187 00:12:13,320 --> 00:12:16,510 in here between them would give you something 188 00:12:16,510 --> 00:12:18,640 that's power suppressed. 189 00:12:18,640 --> 00:12:25,160 Because you know that N slash, when CN 0, and also CN 190 00:12:25,160 --> 00:12:30,744 bar gamma [? per mu ?] with any kind of P left or whatever, 191 00:12:30,744 --> 00:12:32,530 is also 0. 192 00:12:32,530 --> 00:12:38,450 So you'd have to have something more complicated. 193 00:12:38,450 --> 00:12:40,310 All right, so there's those operators. 194 00:12:40,310 --> 00:12:43,280 And when you do this matching, it is non-trivial in the sense 195 00:12:43,280 --> 00:12:45,590 that these two operators-- it's not diagonal. 196 00:12:45,590 --> 00:12:48,560 It's not like O0 goes to Q0, and O8 goes to Q8. 197 00:12:48,560 --> 00:12:51,830 As soon as you start adding loop corrections these two mix, 198 00:12:51,830 --> 00:12:54,320 and then they give you some contributions 199 00:12:54,320 --> 00:12:57,390 to these coefficients C0 and C8. 200 00:12:57,390 --> 00:12:57,950 OK? 201 00:12:57,950 --> 00:13:01,423 So what you mean by octet operator in the electroweak 202 00:13:01,423 --> 00:13:02,840 Hamiltonian is different than what 203 00:13:02,840 --> 00:13:10,360 you mean by octet operator in the SET1 factorized result. 204 00:13:10,360 --> 00:13:14,320 But that's just a complication that you deal with when 205 00:13:14,320 --> 00:13:15,700 you are doing the matching. 206 00:13:15,700 --> 00:13:18,100 This guy can be proportional to the Wilson coefficient 207 00:13:18,100 --> 00:13:25,420 C0F and C08, and that's not really a big deal. 208 00:13:25,420 --> 00:13:28,440 Any questions so far? 209 00:13:28,440 --> 00:13:29,790 STUDENT: [INAUDIBLE] 210 00:13:29,790 --> 00:13:31,000 PROFESSOR: Yeah 211 00:13:31,000 --> 00:13:33,920 STUDENT: So is the point of matching to get one of them 212 00:13:33,920 --> 00:13:35,930 as being [INAUDIBLE] to get the softs? 213 00:13:35,930 --> 00:13:36,650 PROFESSOR: Yeah. 214 00:13:36,650 --> 00:13:37,400 STUDENT: So you're going to distribute them-- 215 00:13:37,400 --> 00:13:37,710 PROFESSOR: That's right. 216 00:13:37,710 --> 00:13:38,690 I'm going to do that right now. 217 00:13:38,690 --> 00:13:39,330 STUDENT: OK, so that's-- 218 00:13:39,330 --> 00:13:39,997 PROFESSOR: Yeah. 219 00:13:39,997 --> 00:13:49,490 So then step two, field redefinition in the SCET1. 220 00:13:55,560 --> 00:13:59,660 And let me not put superscript zeros, 221 00:13:59,660 --> 00:14:02,450 let me just make it as a replacement. 222 00:14:02,450 --> 00:14:08,600 So then we get RQ0 again. 223 00:14:08,600 --> 00:14:11,960 And OK. 224 00:14:11,960 --> 00:14:15,470 For this guy, it's exactly the same, 225 00:14:15,470 --> 00:14:21,200 because in the Q01 comma 5, all the Y's cancel. 226 00:14:25,630 --> 00:14:29,260 In the octet guy, it's not quite that way, because there is-- 227 00:14:29,260 --> 00:14:32,680 because we do have-- 228 00:14:32,680 --> 00:14:35,530 in that case, we do have the Wilson and lines 229 00:14:35,530 --> 00:14:37,030 getting trapped by the TA. 230 00:14:37,030 --> 00:14:38,470 So let me write out that case. 231 00:14:42,691 --> 00:15:21,210 So we have-- so that's what we would get after the field 232 00:15:21,210 --> 00:15:24,780 redefinition for two operators. 233 00:15:24,780 --> 00:15:25,290 OK. 234 00:15:25,290 --> 00:15:29,640 So the next thing to do would be to instead of calling them 235 00:15:29,640 --> 00:15:31,140 Y's call them S's. 236 00:15:31,140 --> 00:15:34,740 So, but there's one more thing I can do too. 237 00:15:34,740 --> 00:15:37,430 So these are, remember, these are soft fields 238 00:15:37,430 --> 00:15:38,680 and these are colinear fields. 239 00:15:38,680 --> 00:15:40,470 So this isn't factorized, because we 240 00:15:40,470 --> 00:15:44,430 have contractions between these Y's in the fields over here. 241 00:15:44,430 --> 00:15:47,010 Gluons can attach to heavy quarks. 242 00:15:47,010 --> 00:15:50,400 So in order to factorize we want to move those Y's from there 243 00:15:50,400 --> 00:15:52,450 over to here. 244 00:15:52,450 --> 00:15:53,690 And we can do that. 245 00:15:53,690 --> 00:15:55,440 So here's how that works. 246 00:15:55,440 --> 00:15:59,530 This is a formula that I could have told you earlier. 247 00:15:59,530 --> 00:16:02,820 So if you have a Y that lies around a TA, 248 00:16:02,820 --> 00:16:06,030 that's just actually the adjoint Wilson line. 249 00:16:14,445 --> 00:16:19,450 So this is a formula that relates fundamental Wilson 250 00:16:19,450 --> 00:16:21,210 lines and an adjoint Wilson line. 251 00:16:26,820 --> 00:16:31,200 So in the adjoint Wilson line, you'd build it out of matrices. 252 00:16:31,200 --> 00:16:32,670 If you like, they're like this. 253 00:16:39,500 --> 00:16:42,616 So the matrix indices would be B and C, 254 00:16:42,616 --> 00:16:46,220 and instead of having fundamental indices for the TA 255 00:16:46,220 --> 00:16:51,053 alpha beta, you have an FABC, and this is the kind of thing 256 00:16:51,053 --> 00:16:52,220 that you would exponentiate. 257 00:16:55,970 --> 00:16:57,810 But other than that it's the same thing. 258 00:16:57,810 --> 00:17:01,372 And there's just a color identity relating them. 259 00:17:01,372 --> 00:17:03,080 So because of this identity, you can also 260 00:17:03,080 --> 00:17:10,220 write down another identity, which is Y dagger TAY. 261 00:17:10,220 --> 00:17:18,475 This guy is-- so Y dagger TAY is just the other YAB-- 262 00:17:18,475 --> 00:17:23,790 this guy's an orthogonal matrix, TB. 263 00:17:23,790 --> 00:17:29,700 So if you reverse the indices then you get the opposite way. 264 00:17:29,700 --> 00:17:32,700 And also, this guy is-- 265 00:17:32,700 --> 00:17:35,860 remember this is a matrix just in the AB space. 266 00:17:35,860 --> 00:17:38,150 So if I use this formula in here, 267 00:17:38,150 --> 00:17:40,650 that allows me to take these Wilson lines here and move them 268 00:17:40,650 --> 00:17:42,310 over here. 269 00:17:42,310 --> 00:17:42,810 Right? 270 00:17:42,810 --> 00:17:45,870 Because I can take them, write them as a Y, 271 00:17:45,870 --> 00:17:47,483 and then the Y is just something that 272 00:17:47,483 --> 00:17:49,650 doesn't care about-- it just moves over because it's 273 00:17:49,650 --> 00:17:52,000 contracted with that index A. 274 00:17:52,000 --> 00:17:54,570 I guess I've got some problems with 275 00:17:54,570 --> 00:17:57,303 capitals and small letters. 276 00:17:57,303 --> 00:17:58,470 Let's make them all capital. 277 00:18:03,955 --> 00:18:05,580 So then not going to move it over here. 278 00:18:05,580 --> 00:18:08,850 And then I can convert it back to a Y dagger Y if I want to. 279 00:18:08,850 --> 00:18:09,350 OK. 280 00:18:09,350 --> 00:18:14,200 So we can take this guy and this thing 281 00:18:14,200 --> 00:18:26,310 and write it over there as HV Y TAY dagger HV prime. 282 00:18:29,130 --> 00:18:31,050 I was careful about that. 283 00:18:31,050 --> 00:18:32,220 This was the prime. 284 00:18:36,790 --> 00:18:39,370 And then all the soft gluons-- 285 00:18:39,370 --> 00:18:43,150 all the ultra soft ones are over here in this matrix element 286 00:18:43,150 --> 00:18:46,150 and all the colinear fields are over there. 287 00:18:46,150 --> 00:18:49,730 So if you like, you could say, we get this, 288 00:18:49,730 --> 00:18:52,300 and then we get our colinear matrix element that has TA, 289 00:18:52,300 --> 00:19:00,700 but it has now no ultra softs. 290 00:19:00,700 --> 00:19:02,770 And so we have a product of colinear and ultra 291 00:19:02,770 --> 00:19:07,610 soft things tied together by one index A. OK, 292 00:19:07,610 --> 00:19:12,330 so that's just a little color rearrangement. 293 00:19:12,330 --> 00:19:14,880 It's useful, because now they're really factorized. 294 00:19:14,880 --> 00:19:16,560 And now when you take matrix elements, 295 00:19:16,560 --> 00:19:18,090 the matrix elements will factorize. 296 00:19:41,330 --> 00:19:44,460 Oh, sorry, before we take matrix elements, 297 00:19:44,460 --> 00:19:46,090 let's switch to SCET 2. 298 00:19:54,080 --> 00:19:56,560 So this is, again, an example where it's trivial. 299 00:19:56,560 --> 00:19:58,582 Because what we have is, we have one type 300 00:19:58,582 --> 00:20:00,040 of operator that we're considering, 301 00:20:00,040 --> 00:20:02,140 this weak transition, and we don't 302 00:20:02,140 --> 00:20:04,240 have a time limited product of any type 303 00:20:04,240 --> 00:20:06,490 of two operators in SET2. 304 00:20:06,490 --> 00:20:09,290 We just have a single operator that has both types of fields, 305 00:20:09,290 --> 00:20:13,105 and then we have the Lagrangians So this is, again, simple. 306 00:20:18,640 --> 00:20:28,210 So we have one mixed operator plus L0 colinear and L0 soft. 307 00:20:28,210 --> 00:20:29,920 And those things are already decoupled, 308 00:20:29,920 --> 00:20:30,970 and so this is simple. 309 00:20:34,560 --> 00:20:39,320 And so, we simply replace Y's by S, renaming 310 00:20:39,320 --> 00:20:40,940 it soft instead of ultra soft. 311 00:20:40,940 --> 00:20:42,470 Really nothing is changing. 312 00:20:42,470 --> 00:20:44,990 And these colinears, we just put them down 313 00:20:44,990 --> 00:20:49,430 onto SET2 colinears from SET1 colinears. 314 00:20:49,430 --> 00:21:00,610 OK, so to make it look like I've done something, 315 00:21:00,610 --> 00:21:01,830 I'll write it out again. 316 00:21:01,830 --> 00:21:06,950 But there's really nothing happening 317 00:21:06,950 --> 00:21:10,440 except that now the fields are in SET2, with the correct SET2 318 00:21:10,440 --> 00:21:10,940 scaling. 319 00:21:17,940 --> 00:21:19,820 So there's no Wilson coefficient that's 320 00:21:19,820 --> 00:21:21,230 generated by this stuff-- 321 00:21:21,230 --> 00:21:23,540 there's no additional Wilson coefficient 322 00:21:23,540 --> 00:21:25,790 because of that fact. 323 00:21:25,790 --> 00:21:29,010 So these are SCET2 now. 324 00:21:29,010 --> 00:21:30,500 And similarly for the octet. 325 00:21:34,690 --> 00:21:36,340 So here we would really have the-- 326 00:21:50,680 --> 00:21:51,180 OK. 327 00:21:53,930 --> 00:21:55,827 So now we can take matrix elements. 328 00:21:55,827 --> 00:21:56,660 STUDENT: [INAUDIBLE] 329 00:21:56,660 --> 00:21:56,870 PROFESSOR: Yeah? 330 00:21:56,870 --> 00:21:58,580 STUDENT: What do the coefficients 331 00:21:58,580 --> 00:22:01,530 look like when they're not just 1? 332 00:22:01,530 --> 00:22:04,160 PROFESSOR: So they would be functions of say, 333 00:22:04,160 --> 00:22:06,530 plus times minus momenta. 334 00:22:06,530 --> 00:22:08,460 So we could have-- if it wasn't 1, 335 00:22:08,460 --> 00:22:11,417 what would happen is effectively-- so yeah, 336 00:22:11,417 --> 00:22:13,250 we talked a little bit about this last time, 337 00:22:13,250 --> 00:22:14,960 but let me remind you. 338 00:22:14,960 --> 00:22:17,510 If it wasn't 1, that would happen in a situation 339 00:22:17,510 --> 00:22:20,660 where you had something like this. 340 00:22:20,660 --> 00:22:27,820 Some off shell field, O, say like this. 341 00:22:27,820 --> 00:22:30,890 So this is a T product of two things 342 00:22:30,890 --> 00:22:34,220 rather than just one thing that mix off the colinear. 343 00:22:34,220 --> 00:22:37,910 And this field here was off shell in a way 344 00:22:37,910 --> 00:22:42,560 that basically, this field here is an off shell field that 345 00:22:42,560 --> 00:22:45,170 would be a product of the plus and minus momentum 346 00:22:45,170 --> 00:22:45,817 of these guys. 347 00:22:45,817 --> 00:22:47,900 And so, you could get something that's effectively 348 00:22:47,900 --> 00:22:50,600 living at this hard colinear scale 349 00:22:50,600 --> 00:22:53,270 below the hard scale in the problem, 350 00:22:53,270 --> 00:22:57,200 from sort of T products of soft and colinear operators. 351 00:22:57,200 --> 00:23:00,680 This is getting a little sketchy, but-- 352 00:23:00,680 --> 00:23:03,470 since here we only have one operator, that couldn't happen, 353 00:23:03,470 --> 00:23:05,950 because you just start attaching softs. 354 00:23:05,950 --> 00:23:08,163 And colinear is been-- it's already factored. 355 00:23:08,163 --> 00:23:09,830 So there's no way that you could sort of 356 00:23:09,830 --> 00:23:12,620 get this intermediate off shell guy. 357 00:23:12,620 --> 00:23:13,460 STUDENT: [INAUDIBLE] 358 00:23:13,460 --> 00:23:15,460 PROFESSOR: So this guy would be colinear, right? 359 00:23:15,460 --> 00:23:18,920 And the way to think about this is like this. 360 00:23:18,920 --> 00:23:21,270 This is just a diagram that exists. 361 00:23:21,270 --> 00:23:22,940 But then you go over to the SET2, 362 00:23:22,940 --> 00:23:26,750 and then you have your change of colinear to the right colinear. 363 00:23:26,750 --> 00:23:28,260 But this guy doesn't change. 364 00:23:28,260 --> 00:23:31,252 He's still hard colinear. 365 00:23:31,252 --> 00:23:33,920 STUDENT: OK, so it's a matching when the material actually 366 00:23:33,920 --> 00:23:35,563 has a different scale than matching-- 367 00:23:35,563 --> 00:23:36,230 PROFESSOR: Yeah. 368 00:23:36,230 --> 00:23:38,180 This scale would show actually up. 369 00:23:38,180 --> 00:23:40,830 And so if it doesn't show up-- 370 00:23:40,830 --> 00:23:42,395 so this would be kind of a situation, 371 00:23:42,395 --> 00:23:48,210 an going from step one. 372 00:23:48,210 --> 00:23:50,358 And actually, we could have done some examples 373 00:23:50,358 --> 00:23:51,150 where that happens. 374 00:23:53,760 --> 00:23:57,120 But I'm choosing to do this rapidity renormalization 375 00:23:57,120 --> 00:23:58,590 group instead. 376 00:23:58,590 --> 00:24:00,888 Basically this happens if you look at matrix elements, 377 00:24:00,888 --> 00:24:02,430 and you could look at matrix elements 378 00:24:02,430 --> 00:24:04,890 where you have some subleading interactions. 379 00:24:04,890 --> 00:24:07,470 And there are examples in exclusive decays 380 00:24:07,470 --> 00:24:09,480 where you could have this happen. 381 00:24:09,480 --> 00:24:14,100 One example is if you look at B0 to D0, pi 0, 382 00:24:14,100 --> 00:24:18,203 just having all neutral charges, then actually this will happen. 383 00:24:18,203 --> 00:24:19,870 It'll be-- and it'll be more complicated 384 00:24:19,870 --> 00:24:20,610 than what I'm telling you. 385 00:24:20,610 --> 00:24:23,085 But you can derive a factorization theorem for this. 386 00:24:23,085 --> 00:24:26,100 It's power suppressed relative to the one we're talking about, 387 00:24:26,100 --> 00:24:29,610 because the one we're talking about always has a charge pion, 388 00:24:29,610 --> 00:24:31,410 and it turns out that that happens 389 00:24:31,410 --> 00:24:33,690 at leading order, whereas the neutral pion 390 00:24:33,690 --> 00:24:36,630 process with the neutral B and neutral D 391 00:24:36,630 --> 00:24:39,362 is something that's power suppressed. 392 00:24:49,484 --> 00:24:52,020 Hope I'm remembering that right. 393 00:24:52,020 --> 00:24:53,500 Yeah. 394 00:24:53,500 --> 00:24:55,450 I am. 395 00:24:55,450 --> 00:24:55,950 All right. 396 00:24:55,950 --> 00:25:00,990 So number four, this is an aside. 397 00:25:00,990 --> 00:25:09,025 Number four, take matrix elements, 398 00:25:09,025 --> 00:25:11,400 and here we find actually that one of the matrix elements 399 00:25:11,400 --> 00:25:13,560 is just 0, the one with the octet. 400 00:25:16,200 --> 00:25:18,090 So let me write the nonzero ones first. 401 00:25:42,860 --> 00:25:45,256 I wasn't too careful about the two different-- 402 00:25:53,830 --> 00:25:55,530 about this. 403 00:25:55,530 --> 00:25:57,170 So there's some guy that's just giving 404 00:25:57,170 --> 00:26:00,350 a convolution between the Wilson coefficient and 5 pi. 405 00:26:00,350 --> 00:26:12,620 And then this guy, which is some normalization 406 00:26:12,620 --> 00:26:14,840 factor times a form factor. 407 00:26:18,800 --> 00:26:22,520 These things are all mu dependent in general. 408 00:26:22,520 --> 00:26:26,010 And this thing here is the Isgur-Wise function, 409 00:26:26,010 --> 00:26:27,440 which is the HQT form factor. 410 00:26:32,320 --> 00:26:37,210 And W0 is kind of the kinematic variable 411 00:26:37,210 --> 00:26:38,720 that that form factor can depend on, 412 00:26:38,720 --> 00:26:41,020 which is V dot V prime, the labels on the fields, 413 00:26:41,020 --> 00:26:43,570 and that encodes the momentum transfer. 414 00:26:43,570 --> 00:26:45,520 Which here is just related to the kinematics. 415 00:26:45,520 --> 00:26:52,878 So W0 is some function of MB and MC, 416 00:26:52,878 --> 00:26:54,670 which I'm not going to bother writing down. 417 00:27:03,454 --> 00:27:06,640 STUDENT: [INAUDIBLE] 418 00:27:06,640 --> 00:27:07,890 PROFESSOR: Just some constant. 419 00:27:07,890 --> 00:27:11,460 I mean, it could have some kinematic factors that makes up 420 00:27:11,460 --> 00:27:13,590 the dimension, like some-- 421 00:27:13,590 --> 00:27:15,075 depends on my gamma. 422 00:27:15,075 --> 00:27:16,320 Yeah, it's just some number. 423 00:27:16,320 --> 00:27:19,230 STUDENT: [INAUDIBLE] 424 00:27:31,870 --> 00:27:34,540 PROFESSOR: OK so these are the singlet operators. 425 00:27:38,020 --> 00:27:41,940 So in the singlet case, we have initial state and final state. 426 00:27:41,940 --> 00:27:44,130 In all cases we have initial state and final states, 427 00:27:44,130 --> 00:27:46,050 which are color singlets. 428 00:27:46,050 --> 00:27:48,210 And these operators are color singlets. 429 00:27:48,210 --> 00:27:51,780 In the case of the octet, we also have color singlet states. 430 00:27:51,780 --> 00:27:54,720 And we factorize such that we do have 431 00:27:54,720 --> 00:27:58,090 a matrix element for example. 432 00:27:58,090 --> 00:28:01,920 So let me just write one of them, and no. 433 00:28:10,700 --> 00:28:13,790 And this is 0, because there's nothing 434 00:28:13,790 --> 00:28:17,000 that could carry the index A in this matrix element. 435 00:28:20,750 --> 00:28:22,120 So the octet matrix element's 0. 436 00:28:22,120 --> 00:28:23,620 It's important that we factorized it 437 00:28:23,620 --> 00:28:24,745 for that to be true, right? 438 00:28:24,745 --> 00:28:29,320 If we had D pi, then we'd have a color singlet operator here, 439 00:28:29,320 --> 00:28:31,750 B. So we wouldn't have been able to make this statement 440 00:28:31,750 --> 00:28:34,273 in the original operator in the electroweak Hamiltonian. 441 00:28:34,273 --> 00:28:35,440 This would just not be true. 442 00:28:35,440 --> 00:28:39,220 But once we factored it and put all the ultra soft fields here, 443 00:28:39,220 --> 00:28:41,540 that everything that's going to be contracted together, 444 00:28:41,540 --> 00:28:42,820 then we can make this statement. 445 00:28:42,820 --> 00:28:44,612 We couldn't even really make this statement 446 00:28:44,612 --> 00:28:46,280 when the Y's were on the other side. 447 00:28:46,280 --> 00:28:50,200 We had to move them over here to ensure that this statement is 448 00:28:50,200 --> 00:28:50,980 completely true. 449 00:28:53,650 --> 00:28:54,150 OK. 450 00:28:54,150 --> 00:29:03,365 So color octet operator is color singlet states. 451 00:29:11,800 --> 00:29:13,870 OK, so then you just put things together 452 00:29:13,870 --> 00:29:22,640 and we can multiply these two things to get the final result. 453 00:29:22,640 --> 00:29:27,040 So if we write it as a matching from the electroweak 454 00:29:27,040 --> 00:29:30,680 Hamiltonian, there are some normalization factors. 455 00:29:30,680 --> 00:29:33,640 So grouping together these factors of F pi, 456 00:29:33,640 --> 00:29:38,750 E pi, and M prime, which I'm not worrying so much about, 457 00:29:38,750 --> 00:29:42,550 there's a Isgur-Wise function, and then there's 458 00:29:42,550 --> 00:29:47,390 a single convolutional between the hard coefficient, which 459 00:29:47,390 --> 00:29:51,020 is kind of like our example of the photon pi on form factor. 460 00:29:53,895 --> 00:29:55,770 And then the slight [INAUDIBLE] distribution. 461 00:29:55,770 --> 00:29:57,537 So we see an example where it showed up 462 00:29:57,537 --> 00:29:59,120 in a totally different type of process 463 00:29:59,120 --> 00:30:03,500 from the one we were considering previously, thereby showing us 464 00:30:03,500 --> 00:30:05,873 kind of the universality of that function. 465 00:30:05,873 --> 00:30:07,790 And then there would be some power corrections 466 00:30:07,790 --> 00:30:11,517 to this whole thing that we're neglecting, 467 00:30:11,517 --> 00:30:13,475 that go like lambda QCD over those hard scales. 468 00:30:18,080 --> 00:30:20,870 OK, so this is the Isgur-Wise function. 469 00:30:20,870 --> 00:30:28,400 And actually I did write down what the W would be. 470 00:30:28,400 --> 00:30:35,520 So this W would be that, some-- 471 00:30:35,520 --> 00:30:38,060 you can write it in terms of the meson masses like that, 472 00:30:38,060 --> 00:30:40,100 so it's the Isgur-Wise function at max recoil. 473 00:30:43,960 --> 00:30:47,860 And this function is measured, for example, 474 00:30:47,860 --> 00:30:49,600 in a semi-leptonic transition. 475 00:30:52,330 --> 00:30:54,400 So you can imagine that the pion distribution 476 00:30:54,400 --> 00:30:56,800 function, or properties of it, were measured 477 00:30:56,800 --> 00:30:59,618 in the photon pion transition. 478 00:30:59,618 --> 00:31:01,410 Is Isgur-Wise function is measured in the B 479 00:31:01,410 --> 00:31:03,145 to DL new transition. 480 00:31:03,145 --> 00:31:05,620 And then you can make predictions for this B to D pi. 481 00:31:09,180 --> 00:31:11,910 OK, so that gives you an example of how you would use 482 00:31:11,910 --> 00:31:14,820 these factorization theorems. 483 00:31:14,820 --> 00:31:16,650 So this applies to basically-- 484 00:31:16,650 --> 00:31:20,160 this type of factorization that we just 485 00:31:20,160 --> 00:31:24,150 talked about, it applies to a lot of different things with 486 00:31:24,150 --> 00:31:28,950 charged pi minuses. 487 00:31:28,950 --> 00:31:31,020 Or you can make the pi minus a rho minus, 488 00:31:31,020 --> 00:31:34,530 and that wouldn't really change anything. 489 00:31:34,530 --> 00:31:37,040 So you could have B0. 490 00:31:37,040 --> 00:31:38,415 If you wanted to look at charges, 491 00:31:38,415 --> 00:31:41,400 you could have B0 to D plus pi minus. 492 00:31:41,400 --> 00:31:46,475 Or you could have B minus to D0 pi minus. 493 00:31:46,475 --> 00:31:47,850 And there's a third one, which is 494 00:31:47,850 --> 00:31:51,060 the one we were talking about over here, B0 to D0 pi 0. 495 00:31:51,060 --> 00:31:52,800 So there's three different ways-- 496 00:31:52,800 --> 00:31:55,080 three different B to D pi transitions 497 00:31:55,080 --> 00:31:56,650 depending on the charges. 498 00:31:56,650 --> 00:32:02,130 And what we've derived applies to charge pions or charge rhos. 499 00:32:02,130 --> 00:32:05,070 The neutral ones end up being power suppressed. 500 00:32:05,070 --> 00:32:08,640 And you can see that kind from our discussion 501 00:32:08,640 --> 00:32:10,110 there was kind of never a-- 502 00:32:10,110 --> 00:32:14,130 there was-- it just writing down the leading order operators, 503 00:32:14,130 --> 00:32:16,620 well, maybe you have to work a little harder. 504 00:32:16,620 --> 00:32:19,290 But effectively, the leading order operators 505 00:32:19,290 --> 00:32:22,183 don't make this transition with the two charges being 506 00:32:22,183 --> 00:32:23,850 the same such that you could get a pi 0, 507 00:32:23,850 --> 00:32:28,980 so you have to do something more in order to get that case. 508 00:32:28,980 --> 00:32:30,420 All right. 509 00:32:30,420 --> 00:32:33,406 So questions? 510 00:32:33,406 --> 00:32:35,830 STUDENT: Can you [INAUDIBLE]? 511 00:32:35,830 --> 00:32:37,960 PROFESSOR: You can if you want. 512 00:32:37,960 --> 00:32:39,580 Oh, MB over MC. 513 00:32:39,580 --> 00:32:41,590 So MB and MC, we're treating both of them 514 00:32:41,590 --> 00:32:44,560 as avoiding the same in what we've done. 515 00:32:44,560 --> 00:32:46,660 So if we wanted to some logs of MB over MC, 516 00:32:46,660 --> 00:32:48,910 we'd have to do something a little different than what 517 00:32:48,910 --> 00:32:49,750 we did. 518 00:32:49,750 --> 00:32:52,690 You'd have to first integrate [? O to MV, ?] treat the charm 519 00:32:52,690 --> 00:32:53,830 quark as a light quark. 520 00:32:53,830 --> 00:32:55,150 You could do that. 521 00:32:55,150 --> 00:32:58,010 STUDENT: Would it make the SCET [INAUDIBLE]---- 522 00:32:58,010 --> 00:32:59,260 PROFESSOR: It turns out that-- 523 00:32:59,260 --> 00:32:59,750 STUDENT: --analysis different? 524 00:32:59,750 --> 00:33:00,500 PROFESSOR: --yeah. 525 00:33:00,500 --> 00:33:02,620 So those are single logs, actually, they're 526 00:33:02,620 --> 00:33:04,018 not double logs. 527 00:33:04,018 --> 00:33:05,560 And it's related to the fact that you 528 00:33:05,560 --> 00:33:06,560 have massive particles. 529 00:33:06,560 --> 00:33:08,018 When you have massive particles you 530 00:33:08,018 --> 00:33:09,850 don't get the extra singularity. 531 00:33:09,850 --> 00:33:12,610 And people in HQT worried about something-- 532 00:33:12,610 --> 00:33:14,950 logs of MB over MC for a while-- 533 00:33:14,950 --> 00:33:17,140 and then after a flight of doing enough calculations 534 00:33:17,140 --> 00:33:18,878 they realized it was totally irrelevant, 535 00:33:18,878 --> 00:33:20,170 and you should just not bother. 536 00:33:20,170 --> 00:33:22,360 You should just calculate the alpha S corrections, 537 00:33:22,360 --> 00:33:24,648 treating MB and MC as comparable, 538 00:33:24,648 --> 00:33:26,440 and summing the logs-- if you sort of think 539 00:33:26,440 --> 00:33:28,910 of leading log as being more important than the order FS 540 00:33:28,910 --> 00:33:30,880 calculation, that misleads you. 541 00:33:30,880 --> 00:33:32,390 Sometimes the sign is even wrong. 542 00:33:32,390 --> 00:33:34,900 And so there's sort of a general experience 543 00:33:34,900 --> 00:33:39,440 that something logs of MB over MC and HQT is not even-- 544 00:33:39,440 --> 00:33:41,190 STUDENT: Just because they're single logs? 545 00:33:41,190 --> 00:33:42,730 PROFESSOR: Just because they're single logs. 546 00:33:42,730 --> 00:33:44,800 I mean, that's one thing that makes it different 547 00:33:44,800 --> 00:33:47,230 than, say, the double logs that you 548 00:33:47,230 --> 00:33:50,320 would resum in this process. 549 00:33:50,320 --> 00:33:53,560 Actually, these double logs are also single logs. 550 00:33:53,560 --> 00:33:56,140 So you could decide whether or not to resum of them. 551 00:33:56,140 --> 00:34:00,285 But either-- whether or not you do resummation, 552 00:34:00,285 --> 00:34:01,660 this is still useful, because you 553 00:34:01,660 --> 00:34:04,178 could make a prediction for this decay rate, 554 00:34:04,178 --> 00:34:05,220 and it works really well. 555 00:34:08,909 --> 00:34:11,409 All right. 556 00:34:11,409 --> 00:34:13,679 You can actually also make predictions 557 00:34:13,679 --> 00:34:16,590 for these decay rates. 558 00:34:16,590 --> 00:34:19,590 You can predict actually the relations between the D0 and D 559 00:34:19,590 --> 00:34:22,440 star 0 using the factorization-- subleading factorization 560 00:34:22,440 --> 00:34:24,750 theorem. 561 00:34:24,750 --> 00:34:29,899 OK, so let's move on to our second topic, which 562 00:34:29,899 --> 00:34:32,980 will take the rest of today. 563 00:34:32,980 --> 00:34:35,190 And that's rapidity divergences. 564 00:34:46,480 --> 00:34:46,980 OK. 565 00:34:46,980 --> 00:34:48,989 So when we were talking about SET1, 566 00:34:48,989 --> 00:34:51,179 and we were talking about loop calculations, 567 00:34:51,179 --> 00:34:53,400 we saw that there was a subtlety where 568 00:34:53,400 --> 00:34:55,050 when we were doing our colinear loops, 569 00:34:55,050 --> 00:34:56,889 that could double count the ultra soft loop, 570 00:34:56,889 --> 00:34:57,570 if you remember. 571 00:35:05,640 --> 00:35:07,230 So kind of schematically, I could 572 00:35:07,230 --> 00:35:16,630 say, that this true CN was sort of a CN naive minus a CN 0 bin. 573 00:35:16,630 --> 00:35:18,540 So you could do a calculation ignoring 574 00:35:18,540 --> 00:35:20,130 that, but then you have to be careful, 575 00:35:20,130 --> 00:35:21,240 and there's a subtraction. 576 00:35:21,240 --> 00:35:24,930 And that subtraction avoids the double counting with the ultra 577 00:35:24,930 --> 00:35:25,788 soft. 578 00:35:25,788 --> 00:35:28,080 So if you think about there being ultra soft amplitudes 579 00:35:28,080 --> 00:35:34,785 and colinear amplitudes, this avoids a double counting. 580 00:35:39,590 --> 00:35:41,340 Now, we never talked about whether there's 581 00:35:41,340 --> 00:35:43,200 something analogous to that in SET2, 582 00:35:43,200 --> 00:35:45,330 and we just did a lot of SET2 examples 583 00:35:45,330 --> 00:35:47,970 without ever even saying those words. 584 00:35:47,970 --> 00:35:51,810 So why it's actually-- for what we've talked about so far-- 585 00:35:51,810 --> 00:35:53,210 OK to ignore this issue. 586 00:35:59,530 --> 00:36:01,260 But in general, it's not OK. 587 00:36:12,110 --> 00:36:17,300 So if we go back to our picture of the degrees of freedom 588 00:36:17,300 --> 00:36:24,200 in SET2, have this hyperbola, and you could have softs, 589 00:36:24,200 --> 00:36:25,640 you could have some colinears. 590 00:36:25,640 --> 00:36:27,380 And then the example that we just did, 591 00:36:27,380 --> 00:36:30,440 it's like these were kind of the relevant modes. 592 00:36:30,440 --> 00:36:34,740 And in general, you might have some guy down here as well. 593 00:36:34,740 --> 00:36:37,670 So these are the degrees of freedom in the SET. 594 00:36:37,670 --> 00:36:40,250 And effectively what's happened is, 595 00:36:40,250 --> 00:36:42,350 if you want to think about double counting, 596 00:36:42,350 --> 00:36:44,300 you're sliding down this hyperbola 597 00:36:44,300 --> 00:36:48,740 so this hyperbola is kind of at a constant invariant mass, say, 598 00:36:48,740 --> 00:36:52,880 lambda QCD squared, or it could be lambda QCD. 599 00:36:55,610 --> 00:36:58,485 And unlike the case in SET1, where 600 00:36:58,485 --> 00:37:00,110 this guy lived in a different hyperbola 601 00:37:00,110 --> 00:37:01,700 here, to get between them you would 602 00:37:01,700 --> 00:37:06,770 be sliding down the hyperbola at fixed invariant mass. 603 00:37:06,770 --> 00:37:08,248 So that's a little different. 604 00:37:12,640 --> 00:37:16,320 But in general in SCET2, there are also 0 bins. 605 00:37:21,880 --> 00:37:24,730 So in general, you would have something like, 606 00:37:24,730 --> 00:37:29,890 met me denote it this way, CN minus CN soft. 607 00:37:29,890 --> 00:37:33,310 And what I mean by this, is this is my original amplitude, where 608 00:37:33,310 --> 00:37:37,595 P mu was scaling like Q lambda squared 1 lambda. 609 00:37:37,595 --> 00:37:39,370 This was the original. 610 00:37:39,370 --> 00:37:42,290 And this would be a subtraction where you take that amplitude, 611 00:37:42,290 --> 00:37:45,220 and you'd make it scale like in the soft regime. 612 00:37:45,220 --> 00:37:51,550 So it would be P mu, lambda, lambda, lambda. 613 00:37:51,550 --> 00:37:57,320 So that's different than the example of SET1. 614 00:37:57,320 --> 00:37:57,820 In 615 00:37:57,820 --> 00:38:00,220 The SET1 case, we would really just 616 00:38:00,220 --> 00:38:02,205 be scaling down the 1 in the lambda, 617 00:38:02,205 --> 00:38:04,330 so that they would be both of order lambda squared. 618 00:38:04,330 --> 00:38:05,920 Here we're actually scaling down the 1 619 00:38:05,920 --> 00:38:09,040 and scaling up the lambda squared to a lambda. 620 00:38:09,040 --> 00:38:12,307 And that's the right thing to do to go from here to here. 621 00:38:12,307 --> 00:38:14,390 So we're just taking the amplitudes in this region 622 00:38:14,390 --> 00:38:16,550 and subtracting them in this region. 623 00:38:16,550 --> 00:38:19,610 And in general, we do have that. 624 00:38:19,610 --> 00:38:22,030 But actually that's not the real complication 625 00:38:22,030 --> 00:38:25,720 that shows up in the SET2. 626 00:38:25,720 --> 00:38:28,000 The real complication has to do with whether there's 627 00:38:28,000 --> 00:38:30,970 any divergences associated to that. 628 00:38:30,970 --> 00:38:33,460 If this amplitude here didn't have any divergences, 629 00:38:33,460 --> 00:38:36,208 it wasn't kind of log singular, then 630 00:38:36,208 --> 00:38:38,500 you wouldn't really care about doing this subtractions, 631 00:38:38,500 --> 00:38:40,810 because then there would be no infrared singularities 632 00:38:40,810 --> 00:38:43,780 that you're double counting and it would just be effectively 633 00:38:43,780 --> 00:38:44,660 a constant. 634 00:38:44,660 --> 00:38:46,360 And the constants are always ambiguous. 635 00:38:46,360 --> 00:38:49,000 So whatever mistake you make in constants here 636 00:38:49,000 --> 00:38:51,680 you just make up by changing your hard matching. 637 00:38:51,680 --> 00:38:54,370 So you don't have to worry if when the colinear goes down 638 00:38:54,370 --> 00:38:57,340 into the soft region there's no divergences. 639 00:38:57,340 --> 00:38:59,680 And that is actually what's happened in all the examples 640 00:38:59,680 --> 00:39:01,150 we've treated so far. 641 00:39:01,150 --> 00:39:04,180 That when the colinear goes into some region where it's not 642 00:39:04,180 --> 00:39:06,760 supposed to have singularities, that you just 643 00:39:06,760 --> 00:39:08,710 end up with no singularities. 644 00:39:08,710 --> 00:39:13,040 There's no log singularities. 645 00:39:13,040 --> 00:39:24,038 OK, so, so far there's no log singularities 646 00:39:24,038 --> 00:39:25,205 from the overlapped regions. 647 00:39:29,570 --> 00:39:31,520 But that's not in general the situation. 648 00:39:31,520 --> 00:39:32,978 And we'll do an example in a minute 649 00:39:32,978 --> 00:39:36,760 where there are singularities that overlap. 650 00:39:36,760 --> 00:39:40,130 And the true difficulty here is the following. 651 00:39:40,130 --> 00:39:43,120 If you think about what's separating these modes, 652 00:39:43,120 --> 00:39:46,240 you might draw lines like this. 653 00:39:46,240 --> 00:39:49,060 Just to draw some straight lines separating the modes. 654 00:39:49,060 --> 00:39:50,710 And remember that we're plotting here 655 00:39:50,710 --> 00:39:53,650 in the P minus, P plus plain. 656 00:39:53,650 --> 00:39:57,670 And that fixed P squared is like fixed product of P minus P 657 00:39:57,670 --> 00:39:59,090 plus. 658 00:39:59,090 --> 00:40:02,980 So P squared is P plus P minus, up to the P perp 659 00:40:02,980 --> 00:40:04,880 squared piece, which we're ignoring. 660 00:40:04,880 --> 00:40:07,930 So you can think of these lines as lines of constant P 661 00:40:07,930 --> 00:40:08,935 plus over P minus. 662 00:40:12,940 --> 00:40:15,570 And if I-- this is the-- 663 00:40:19,840 --> 00:40:24,920 so something orthogonal to P squared. 664 00:40:24,920 --> 00:40:25,810 All right. 665 00:40:25,810 --> 00:40:28,357 So that would be one way of thinking about-- 666 00:40:28,357 --> 00:40:30,190 so you need something that's orthogonal to P 667 00:40:30,190 --> 00:40:32,350 squared in order to distinguish these modes. 668 00:40:32,350 --> 00:40:36,010 And the real issue with that is related to the regulators. 669 00:40:36,010 --> 00:40:38,230 When you use dimensional regularization 670 00:40:38,230 --> 00:40:40,480 it turns out the dimensional regularization is not 671 00:40:40,480 --> 00:40:43,480 sufficient to regulate a divergence that would happen 672 00:40:43,480 --> 00:40:46,680 when the CN comes down on top of the S. 673 00:40:46,680 --> 00:40:49,000 And the reason is, because dimensional regularization 674 00:40:49,000 --> 00:40:50,830 regulates P squared. 675 00:40:50,830 --> 00:40:55,540 It regulates-- remember, it's a Lorentz invariant regulator. 676 00:40:55,540 --> 00:40:58,150 So it's regulating Lorentz invariant things 677 00:40:58,150 --> 00:41:01,330 like P squared, not something like the rapidity, which 678 00:41:01,330 --> 00:41:04,000 is this P plus over P minus that you would need 679 00:41:04,000 --> 00:41:06,560 to distinguish these modes. 680 00:41:06,560 --> 00:41:25,190 So invariant mass does not distinguish the low energy 681 00:41:25,190 --> 00:41:25,690 modes. 682 00:41:46,570 --> 00:41:51,570 So rapidity, you could define-- is usually defined this way. 683 00:41:54,820 --> 00:42:01,150 So exponent of 2Y, where Y is the rapidity, 684 00:42:01,150 --> 00:42:03,220 is P minus over P plus. 685 00:42:03,220 --> 00:42:06,100 And if you look at the scaling of that, 686 00:42:06,100 --> 00:42:10,780 that scaling either lambda minus 2, lambda 0, or lambda 687 00:42:10,780 --> 00:42:16,570 squared for the different cases for CN, S, and CN bar. 688 00:42:16,570 --> 00:42:19,510 So it's this variable that's really distinguishing 689 00:42:19,510 --> 00:42:21,200 the different modes. 690 00:42:21,200 --> 00:42:24,670 OK, and that's these lines-- these orange lines 691 00:42:24,670 --> 00:42:29,551 are just putting dividing lines between these in rapidity. 692 00:42:29,551 --> 00:42:30,340 All right. 693 00:42:36,200 --> 00:42:36,700 OK. 694 00:42:49,770 --> 00:42:52,403 So there's a complication that dimensional regularization 695 00:42:52,403 --> 00:42:53,070 doesn't suffice. 696 00:43:05,460 --> 00:43:06,960 So you can think of it as regulating 697 00:43:06,960 --> 00:43:11,300 P Euclidean squared once you do the Wick rotation, for example. 698 00:43:11,300 --> 00:43:13,450 So it regulates-- it separates hyperbolas, 699 00:43:13,450 --> 00:43:17,230 but it does not separate modes along a hyperbola. 700 00:43:17,230 --> 00:43:18,850 It's a way of regulating singularities 701 00:43:18,850 --> 00:43:21,310 between hyperbolas, but not along hyperbola. 702 00:43:24,070 --> 00:43:25,390 So that's one complication. 703 00:43:25,390 --> 00:43:28,480 We'll need an additional regulator. 704 00:43:28,480 --> 00:43:31,840 And we'll see that that regulator will eventually 705 00:43:31,840 --> 00:43:36,370 lead to a new type of a normalization group flow, which 706 00:43:36,370 --> 00:43:38,470 is flow along a hyperbola. 707 00:43:38,470 --> 00:43:41,500 It's not a flow in invariant mass, but a flow in rapidity. 708 00:43:51,390 --> 00:43:51,890 OK. 709 00:43:51,890 --> 00:43:55,430 So let's explore what can happen in an example 710 00:43:55,430 --> 00:44:00,170 where there are these divergences in sort 711 00:44:00,170 --> 00:44:01,625 of the simplest possible example. 712 00:44:10,604 --> 00:44:12,740 So there's enough going on that we 713 00:44:12,740 --> 00:44:14,980 want to make our lives as simple as possible. 714 00:44:14,980 --> 00:44:20,540 So what I'll talk about is something 715 00:44:20,540 --> 00:44:22,385 called the massive Sudakov form factor. 716 00:44:32,670 --> 00:44:37,630 So you should think of it set up as follows. 717 00:44:37,630 --> 00:44:39,810 We're going to consider a form factor, 718 00:44:39,810 --> 00:44:42,890 and it's going to be a space-like form factor. 719 00:44:42,890 --> 00:44:45,210 So it's a space-like quark-quark form factor. 720 00:44:49,020 --> 00:44:53,970 Q of the photon here is space-like. 721 00:44:53,970 --> 00:44:57,690 And we're going to think about, rather than having photons 722 00:44:57,690 --> 00:45:01,560 or gluons, we're going to think about massive gauge bosons. 723 00:45:01,560 --> 00:45:03,715 So this is going to be some kind of Z, if you like. 724 00:45:03,715 --> 00:45:05,010 It could be a Z boson. 725 00:45:09,810 --> 00:45:15,780 And I'll just call the mass M. OK. 726 00:45:15,780 --> 00:45:19,350 So the thing that I'm going to want to iterate is the mass-- 727 00:45:19,350 --> 00:45:22,590 rather than doing QCD, I'm doing electroweak corrections-- 728 00:45:22,590 --> 00:45:24,840 electroweak corrections from a massive gauge boson. 729 00:45:31,180 --> 00:45:35,080 So this is relevant if we have electroweak corrections 730 00:45:35,080 --> 00:45:36,730 in a situation where Q squared is 731 00:45:36,730 --> 00:45:39,713 much greater than at M squared. 732 00:45:39,713 --> 00:45:41,380 And we're not going to be having gluons. 733 00:45:41,380 --> 00:45:44,530 Instead we'll be talking about these massive gauge bosons-- 734 00:45:44,530 --> 00:45:47,410 multiple Z bosons, if you like. 735 00:45:47,410 --> 00:45:51,130 We could also put the W's in, but let's make it simple 736 00:45:51,130 --> 00:45:53,680 and just talk about Z's. 737 00:45:53,680 --> 00:45:55,570 OK, so let's do this example. 738 00:45:58,310 --> 00:46:01,380 So in the full theory, you would start with a vector current, 739 00:46:01,380 --> 00:46:05,950 say, and you'd want to match that onto SET. 740 00:46:05,950 --> 00:46:08,557 Before we do that, let's just-- 741 00:46:08,557 --> 00:46:10,390 let me just write down a kind of full theory 742 00:46:10,390 --> 00:46:13,180 object using Lorentz invariance. 743 00:46:13,180 --> 00:46:19,720 So you could think about the quark form factor. 744 00:46:19,720 --> 00:46:24,250 And four massive gauge bosons, this is just some form factor 745 00:46:24,250 --> 00:46:28,480 that you can calculate that's a function of Q squared and M 746 00:46:28,480 --> 00:46:32,350 squared, and then there's some spinners. 747 00:46:32,350 --> 00:46:34,840 And so, really kind of the dependence 748 00:46:34,840 --> 00:46:37,690 is encoded in this F, which is a function of Q squared and M 749 00:46:37,690 --> 00:46:38,950 squared. 750 00:46:38,950 --> 00:46:41,650 M squared is acting kind of like an infrared regulator. 751 00:46:41,650 --> 00:46:44,170 So this is Z boson, there's not a soft singularity 752 00:46:44,170 --> 00:46:45,030 associated to it. 753 00:46:48,650 --> 00:46:51,340 And so, what you'd like to do-- and in this process-- 754 00:46:51,340 --> 00:46:53,890 is factorize Q squared and M squared, i.e. 755 00:46:53,890 --> 00:46:55,910 expand this thing in Q squared and M 756 00:46:55,910 --> 00:46:59,590 squared, and maybe some logs of Q squared and M squared. 757 00:46:59,590 --> 00:47:13,480 So we want to factorize some logs, et cetera. 758 00:47:13,480 --> 00:47:15,580 OK, so what are the type of degrees of freedom 759 00:47:15,580 --> 00:47:17,440 that we could have here? 760 00:47:17,440 --> 00:47:21,050 So lambda is going to be M over Q, massive Z 761 00:47:21,050 --> 00:47:24,220 boson over the energy scale of the collision, 762 00:47:24,220 --> 00:47:27,280 of the gamma star. 763 00:47:27,280 --> 00:47:31,000 And if you just look at the Z boson, 764 00:47:31,000 --> 00:47:34,045 then it could be colinear or it could be soft. 765 00:47:36,770 --> 00:47:44,113 So it could be actually three different possibilities. 766 00:47:44,113 --> 00:47:46,030 And I could think about doing this effectively 767 00:47:46,030 --> 00:47:46,960 in a bright frame. 768 00:47:46,960 --> 00:47:48,650 Just like we did earlier. 769 00:47:48,650 --> 00:47:52,060 And then what you would have is that the colinear guy 770 00:47:52,060 --> 00:47:56,860 is like the quark, and then the anti-colinear guys 771 00:47:56,860 --> 00:47:59,700 is like the outgoing quark. 772 00:47:59,700 --> 00:48:00,700 [INAUDIBLE] 773 00:48:00,700 --> 00:48:04,440 So you'd be making a transition in this diagram from N colinear 774 00:48:04,440 --> 00:48:06,580 of objects to N bar colinear of objects. 775 00:48:06,580 --> 00:48:09,400 And you could likewise have a Z boson which could be colinear. 776 00:48:18,850 --> 00:48:22,060 Or you could have a Z boson that's soft. 777 00:48:22,060 --> 00:48:26,020 And you have a soft rather than ultra soft because of the mass. 778 00:48:26,020 --> 00:48:30,730 If Q times lambda is M, and so if we want a propagator that's 779 00:48:30,730 --> 00:48:34,370 like P squared minus M squared, P squared better be of order 780 00:48:34,370 --> 00:48:34,870 M squared. 781 00:48:34,870 --> 00:48:37,750 That happens for softs, not for ultra softs. 782 00:48:37,750 --> 00:48:38,980 So that's why we have softs. 783 00:48:38,980 --> 00:48:41,140 And the same thing for colinears, P squared of order M 784 00:48:41,140 --> 00:48:42,265 squared for these colinear. 785 00:48:42,265 --> 00:48:45,520 So it's really an SET2 type situation, 786 00:48:45,520 --> 00:48:49,030 where the hyperbola is just set by P squared of order M 787 00:48:49,030 --> 00:48:50,280 squared. 788 00:48:50,280 --> 00:48:54,630 We have our mode sitting on that hyperbola. 789 00:48:54,630 --> 00:48:57,040 OK, so it's exactly of this type over there. 790 00:48:57,040 --> 00:48:59,080 And the thing that's new in this example 791 00:48:59,080 --> 00:49:02,536 is that we're going to encounter these rapidity divergences. 792 00:49:02,536 --> 00:49:06,310 STUDENT: You mentioned there's the only form for vertical 793 00:49:06,310 --> 00:49:07,443 in your theories, is that? 794 00:49:07,443 --> 00:49:08,110 PROFESSOR: Yeah. 795 00:49:13,460 --> 00:49:16,520 Make things as simple as possible. 796 00:49:16,520 --> 00:49:20,120 So you could talk about mixed sort of electroweak in QCD, 797 00:49:20,120 --> 00:49:22,160 but yeah, we don't. 798 00:49:22,160 --> 00:49:24,037 Let's make it-- in some sense that's 799 00:49:24,037 --> 00:49:26,120 kind of just like mixing a problem that we already 800 00:49:26,120 --> 00:49:28,080 would know how to deal with, with this one. 801 00:49:28,080 --> 00:49:32,340 So let's just deal with this one. 802 00:49:32,340 --> 00:49:32,840 All right. 803 00:49:32,840 --> 00:49:36,920 So in terms of the external court momenta, 804 00:49:36,920 --> 00:49:40,760 we can therefore kind of treat them as follows. 805 00:49:40,760 --> 00:49:45,020 Let's just let them be of a large component. 806 00:49:45,020 --> 00:49:46,370 They're massless particles. 807 00:49:52,510 --> 00:49:56,600 So this is P, and this is PR. 808 00:49:56,600 --> 00:49:58,603 And these guys are massless-- 809 00:49:58,603 --> 00:49:59,520 should have said that. 810 00:50:04,670 --> 00:50:06,220 And if you go through the kinematics, 811 00:50:06,220 --> 00:50:09,970 Q squared, which is minus P, minus P prime squared. 812 00:50:09,970 --> 00:50:13,150 If you square that, you just find it to P minus P plus. 813 00:50:13,150 --> 00:50:15,670 And we're just effectively, if we pick the bright frame-- 814 00:50:19,420 --> 00:50:21,520 which is what we're going to do-- 815 00:50:21,520 --> 00:50:25,930 each one of those is separately keep going at prime. 816 00:50:25,930 --> 00:50:27,470 Call it bar. 817 00:50:27,470 --> 00:50:31,270 Each one of those is separately Q. The large-- 818 00:50:31,270 --> 00:50:36,310 these guys are just fixed-- both would be Q. All right. 819 00:50:36,310 --> 00:50:39,220 So we could factor is this current 820 00:50:39,220 --> 00:50:40,900 with these degrees of freedom. 821 00:50:40,900 --> 00:50:42,310 The quarks are colinear. 822 00:50:42,310 --> 00:50:46,180 So at lowest order they've just become a CN and a CN bar. 823 00:50:46,180 --> 00:50:50,090 And then we have to address that with Wilson lines. 824 00:50:50,090 --> 00:50:51,270 And we know how to do that. 825 00:50:51,270 --> 00:50:52,812 So let me just break down the answer. 826 00:50:58,080 --> 00:50:59,940 We could again follow our procedure 827 00:50:59,940 --> 00:51:01,500 of going through SCET1, but it's now 828 00:51:01,500 --> 00:51:05,610 so familiar we just know what to write down. 829 00:51:05,610 --> 00:51:06,110 OK. 830 00:51:06,110 --> 00:51:07,630 So the current would look like that. 831 00:51:07,630 --> 00:51:10,222 And I'm not to worry too much about the Dirac structure. 832 00:51:13,390 --> 00:51:16,030 So I won't worry, for example, about gamma fives. 833 00:51:16,030 --> 00:51:23,794 And we could put that in, it's easy. 834 00:51:27,970 --> 00:51:29,470 So that's the leading order current, 835 00:51:29,470 --> 00:51:31,480 and then we'd have leading order Lagrangians, 836 00:51:31,480 --> 00:51:33,515 and we need to start calculating. 837 00:51:36,630 --> 00:51:39,300 And if we calculate what you would expect from a major 838 00:51:39,300 --> 00:51:43,860 from that is, you'd expect that F of Q squared and M squared 839 00:51:43,860 --> 00:51:45,570 is going to split up-- 840 00:51:45,570 --> 00:51:47,220 given the degrees of freedom we have 841 00:51:47,220 --> 00:51:51,090 into some kind of hard function-- 842 00:51:51,090 --> 00:51:54,570 and then some kind of amplitude for the colinear parts, 843 00:51:54,570 --> 00:51:58,080 and then some kind of amplitude for the soft part. 844 00:51:58,080 --> 00:52:02,130 OK, so you'd expect some hard times colinear factorization 845 00:52:02,130 --> 00:52:03,000 of the form factor. 846 00:52:03,000 --> 00:52:05,541 And this is what we'll be after. 847 00:52:05,541 --> 00:52:07,740 So it's always good to sort of have an idea 848 00:52:07,740 --> 00:52:08,850 where you're going. 849 00:52:08,850 --> 00:52:11,080 And that's where we're going. 850 00:52:11,080 --> 00:52:14,310 So let's consider just one loop diagrams. 851 00:52:14,310 --> 00:52:17,190 And it suffices, in order to make the point, 852 00:52:17,190 --> 00:52:19,680 just to consider the most singular one. 853 00:52:19,680 --> 00:52:21,254 So I'm going to consider-- 854 00:52:38,775 --> 00:52:40,150 so there's various loop intervals 855 00:52:40,150 --> 00:52:42,483 that you could have to do when you're doing the diagrams 856 00:52:42,483 --> 00:52:44,080 if they're fermions. 857 00:52:44,080 --> 00:52:45,670 Let's just take the simplest, which 858 00:52:45,670 --> 00:52:48,370 is a scalar loop interval. 859 00:52:48,370 --> 00:52:51,010 And I'm going to contrast how that scalar loop interval would 860 00:52:51,010 --> 00:52:52,593 look if you were doing the full theory 861 00:52:52,593 --> 00:52:55,450 calculation with how it would look in the effective theory. 862 00:52:55,450 --> 00:52:59,140 And then we'll see where the divergences come from. 863 00:53:02,700 --> 00:53:04,820 So you can think about this as kind of-- 864 00:53:04,820 --> 00:53:08,280 the piece where the numerator is independent of the loop 865 00:53:08,280 --> 00:53:10,860 momenta, so the numerator just factors out. 866 00:53:21,000 --> 00:53:25,110 OK, so if I took our vertex triangle diagram over there, 867 00:53:25,110 --> 00:53:27,345 then a piece of it-- where the numerator is trivial 868 00:53:27,345 --> 00:53:28,950 and factors out-- 869 00:53:28,950 --> 00:53:30,310 would look like this interval. 870 00:53:30,310 --> 00:53:32,220 So let's just study this guy. 871 00:53:32,220 --> 00:53:34,380 If we did this interval in the full theory, 872 00:53:34,380 --> 00:53:37,700 this would be both UV and IR finite. 873 00:53:37,700 --> 00:53:40,840 So this is just giving us some result 874 00:53:40,840 --> 00:53:44,310 that involves logs of Q squared over M squared. 875 00:53:48,840 --> 00:53:51,830 So it does have double logs of Q squared over M squared, 876 00:53:51,830 --> 00:53:55,440 and single logs of Q squared over M squared. 877 00:53:55,440 --> 00:53:59,150 But it's perfectly-- there's no 1 over epsilons. 878 00:54:03,280 --> 00:54:04,360 So now let's see what-- 879 00:54:04,360 --> 00:54:08,840 let's think about what would happen in the effective theory. 880 00:54:08,840 --> 00:54:12,910 So we have a kind of analogous loop interval for colinear, 881 00:54:12,910 --> 00:54:14,680 where the gauge boson is colinear. 882 00:54:18,340 --> 00:54:20,860 There's some numerator that again I'm 883 00:54:20,860 --> 00:54:22,920 not going to worry about. 884 00:54:22,920 --> 00:54:25,780 If this numerator is constant, it doesn't-- it's effectively 885 00:54:25,780 --> 00:54:27,867 the same constant. 886 00:54:27,867 --> 00:54:29,950 In the case of the-- if you take the leading order 887 00:54:29,950 --> 00:54:31,408 numerator in the full theory, it'll 888 00:54:31,408 --> 00:54:33,910 be the leading order numerator the effective theory as well. 889 00:54:33,910 --> 00:54:37,030 But the denominators do change. 890 00:54:37,030 --> 00:54:39,340 And so, if we took the N colinear, 891 00:54:39,340 --> 00:54:46,230 then yeah, so this guy doesn't change. 892 00:54:46,230 --> 00:54:48,990 Because that's just like saying P minus and K are-- 893 00:54:48,990 --> 00:54:51,960 P minus and K minus are the same size. 894 00:54:51,960 --> 00:54:54,630 But this guy does change. 895 00:54:54,630 --> 00:54:59,820 OK, so this guy would be K minus P bar plus, because K minus is 896 00:54:59,820 --> 00:55:02,260 big and B plus is big. 897 00:55:02,260 --> 00:55:04,860 Both of these are big. 898 00:55:04,860 --> 00:55:10,690 So both of those are big in this diagram. 899 00:55:10,690 --> 00:55:17,040 And so that's effectively the Wilson line diagram. 900 00:55:17,040 --> 00:55:19,253 OK, where the propagator here was off shell, 901 00:55:19,253 --> 00:55:21,045 got integrated out, and just became iconal. 902 00:55:24,040 --> 00:55:26,560 And the K squared is smaller, so we don't keep it 903 00:55:26,560 --> 00:55:28,801 in a leading order term. 904 00:55:28,801 --> 00:55:33,940 And then analogously, for IN bar, it's the other way. 905 00:55:41,230 --> 00:55:49,300 So both of these are big, and this one remains. 906 00:55:49,300 --> 00:55:52,690 And then they're soft. 907 00:55:57,860 --> 00:56:00,470 And in the soft case, what happens 908 00:56:00,470 --> 00:56:03,470 is that both of the propagators end up being iconal. 909 00:56:12,180 --> 00:56:17,280 And in our SET operator, that's a diagram where 910 00:56:17,280 --> 00:56:20,910 we have our colinear lines, and then we have kind 911 00:56:20,910 --> 00:56:24,600 of a self contraction of the S. 912 00:56:24,600 --> 00:56:26,815 But we're taking an SN with an SN-- 913 00:56:26,815 --> 00:56:28,440 we have a contraction that's like this. 914 00:56:30,965 --> 00:56:32,340 We have two of the Wilson lines-- 915 00:56:32,340 --> 00:56:34,507 soft Wilson lines that are sitting in that operator. 916 00:56:34,507 --> 00:56:36,883 That's a non-zero contraction. 917 00:56:36,883 --> 00:56:38,550 That would lead to a diagram like-- that 918 00:56:38,550 --> 00:56:41,580 would lead to this amplitude. 919 00:56:41,580 --> 00:56:42,175 All right. 920 00:56:42,175 --> 00:56:43,800 I'm going to leave a little space here, 921 00:56:43,800 --> 00:56:45,758 because I'm going to add something in a minute. 922 00:56:48,900 --> 00:56:49,680 All right. 923 00:56:49,680 --> 00:56:53,220 So how do we see that there's a problem with these intervals 924 00:56:53,220 --> 00:56:55,352 that they're not regulated by dim reg? 925 00:56:55,352 --> 00:56:57,060 Well, you could look at the soft integral 926 00:56:57,060 --> 00:56:58,770 and you could just do the perp. 927 00:56:58,770 --> 00:57:01,742 The perp is only showing up in this K squared minus M squared. 928 00:57:01,742 --> 00:57:03,450 So you would get something by doing that. 929 00:57:08,030 --> 00:57:13,000 And so, if we do the perp with dim reg and IS, 930 00:57:13,000 --> 00:57:15,010 we would end up proportional to something 931 00:57:15,010 --> 00:57:23,260 that's DK plus DK minus K plus K minus, minus M squared 932 00:57:23,260 --> 00:57:26,860 to the some power of epsilon, divided still 933 00:57:26,860 --> 00:57:30,310 by the factors of K plus and K minus. 934 00:57:30,310 --> 00:57:34,540 So you see that the invariant mass is being regulated. 935 00:57:34,540 --> 00:57:35,680 We just did the perp. 936 00:57:35,680 --> 00:57:37,090 Perp is gone. 937 00:57:37,090 --> 00:57:39,610 Plus times minus is being regulated, 938 00:57:39,610 --> 00:57:42,460 because plus times minus-- if plus times minus grows larger, 939 00:57:42,460 --> 00:57:45,910 or goes small, and regulated by this epsilon-- 940 00:57:45,910 --> 00:57:50,230 but either one, plus going large or minus going large, or plus 941 00:57:50,230 --> 00:57:54,790 going small minus going small, with plus and times minus fixed 942 00:57:54,790 --> 00:57:56,410 is not regulated. 943 00:57:56,410 --> 00:57:58,120 And that's the rapidity divergence. 944 00:57:58,120 --> 00:58:02,330 If the invariant mass is fixed and K plus over K 945 00:58:02,330 --> 00:58:03,610 minus goes large or small. 946 00:58:11,610 --> 00:58:15,320 So let me write it as K minus over K plus going to 0, 947 00:58:15,320 --> 00:58:16,580 or going to infinity. 948 00:58:20,750 --> 00:58:26,236 Let me say, with K plus times K minus fixed, then 949 00:58:26,236 --> 00:58:28,925 it diverges as these things happen. 950 00:58:28,925 --> 00:58:30,550 And if you think about what's happening 951 00:58:30,550 --> 00:58:34,870 in our picture over there within these limits, 952 00:58:34,870 --> 00:58:37,600 it's exactly a situation where this X here would 953 00:58:37,600 --> 00:58:41,650 be sliding up or sliding down. 954 00:58:41,650 --> 00:58:49,230 So in one of these limits, this one's going towards the CN bar, 955 00:58:49,230 --> 00:58:52,690 and this one would be going towards CN. 956 00:58:52,690 --> 00:58:55,220 So it's exactly a region where you would be overlapping-- 957 00:58:55,220 --> 00:58:56,720 sliding down the hyperbola. 958 00:58:56,720 --> 00:59:00,740 And the interval has log singularities. 959 00:59:00,740 --> 00:59:02,510 So this is exactly a situation where 960 00:59:02,510 --> 00:59:06,680 we can't ignore the overlaps and we have to worry about them. 961 00:59:27,390 --> 00:59:30,060 OK so we need another regulator. 962 00:59:30,060 --> 00:59:31,200 Dim reg is not enough. 963 00:59:31,200 --> 00:59:33,210 We have to do something else. 964 00:59:33,210 --> 00:59:34,130 So what could we do? 965 00:59:34,130 --> 00:59:36,380 So there's lots of different things that you could do. 966 00:59:36,380 --> 00:59:38,360 One thing is, you could just sort of put it 967 00:59:38,360 --> 00:59:41,300 in some plus something in these denominators-- that's 968 00:59:41,300 --> 00:59:43,190 called the delta regulated, K plus, 969 00:59:43,190 --> 00:59:46,490 plus delta-- that's one choice. 970 00:59:46,490 --> 00:59:50,243 We'll do something a little bit more dim reg-like. 971 00:59:50,243 --> 00:59:52,910 Which makes it sort of easier to think about the renormalization 972 00:59:52,910 --> 00:59:54,470 group. 973 00:59:54,470 --> 01:00:05,520 So one choice for an additional regulator is the following. 974 01:00:05,520 --> 01:00:08,417 So if you think about where these divergences came from, 975 01:00:08,417 --> 01:00:09,750 they came from the Wilson lines. 976 01:00:09,750 --> 01:00:12,380 So what you really need to do is regulate the Wilson lines. 977 01:00:15,110 --> 01:00:18,150 And you can do that as follows. 978 01:00:18,150 --> 01:00:19,730 Let me write out the Wilson lines 979 01:00:19,730 --> 01:00:21,830 in our kind of momentum space notation. 980 01:00:24,840 --> 01:00:30,980 So we have some N dot P type momentum for the soft Wilson 981 01:00:30,980 --> 01:00:35,690 line, and an N dot AS field. 982 01:00:35,690 --> 01:00:38,570 And really, it's this one over this iconal denominator that's 983 01:00:38,570 --> 01:00:40,880 giving rise to these denominators here 984 01:00:40,880 --> 01:00:43,070 that are giving rise to the singularity. 985 01:00:43,070 --> 01:00:44,870 So if we want to regulate that singularity 986 01:00:44,870 --> 01:00:48,900 we need to add something, and we could do that as follows. 987 01:00:48,900 --> 01:00:53,040 So this is the regulator we'll pick. 988 01:00:56,232 --> 01:01:02,940 And I'll just write everything as kind of a momentum operator. 989 01:01:06,620 --> 01:01:08,570 So I've just tucked the Z momentum in 990 01:01:08,570 --> 01:01:09,920 and raised it to some power. 991 01:01:12,560 --> 01:01:18,040 So PZ is the difference between P minus and P plus. 992 01:01:18,040 --> 01:01:19,450 And that seems kind of arbitrary, 993 01:01:19,450 --> 01:01:21,880 but that'll do the job for us. 994 01:01:21,880 --> 01:01:24,070 You can motivate why you want to do PZ-- 995 01:01:24,070 --> 01:01:26,590 so this is 2PZ actually. 996 01:01:26,590 --> 01:01:29,110 You can motivate why you want to do PZ rather than something 997 01:01:29,110 --> 01:01:31,190 else in the following way. 998 01:01:31,190 --> 01:01:34,480 And it's a true fact, that once you have enough experience 999 01:01:34,480 --> 01:01:39,965 you realize it's good to use PZ, because PZ doesn't involve P0. 1000 01:01:39,965 --> 01:01:42,340 And the softs don't really make a distinction between any 1001 01:01:42,340 --> 01:01:43,970 of the different components. 1002 01:01:43,970 --> 01:01:47,770 And if you put in P0, something that involved P0, 1003 01:01:47,770 --> 01:01:49,180 that would be dangerous. 1004 01:01:53,052 --> 01:02:00,880 So this is nice, because there is no P0. 1005 01:02:00,880 --> 01:02:04,090 So it's the combination of P plus and P 1006 01:02:04,090 --> 01:02:05,590 minus that you can form that doesn't 1007 01:02:05,590 --> 01:02:07,790 have the P0, which is energy. 1008 01:02:07,790 --> 01:02:09,610 And remember that the polls in P0 1009 01:02:09,610 --> 01:02:12,520 are related to things like quarks and anti-quarks. 1010 01:02:12,520 --> 01:02:14,350 They're related to unitarity. 1011 01:02:14,350 --> 01:02:17,620 So not messing up the structure in P0 1012 01:02:17,620 --> 01:02:21,610 means that you'll be fine with unitarity, fine with causality, 1013 01:02:21,610 --> 01:02:26,280 you're not messing up a lot of nice things about the theory. 1014 01:02:26,280 --> 01:02:32,935 So if you do put P0 in, then you have 1015 01:02:32,935 --> 01:02:34,310 to be careful about those things. 1016 01:02:42,340 --> 01:02:45,090 So if you just arbitrarily put in some power of P0, 1017 01:02:45,090 --> 01:02:47,005 then you'd have more trouble. 1018 01:02:47,005 --> 01:02:49,380 And so that's kind of why we're avoiding and just putting 1019 01:02:49,380 --> 01:02:49,992 in PZ. 1020 01:02:52,950 --> 01:02:54,960 For the colinears, we can do something similar. 1021 01:02:54,960 --> 01:02:57,420 But for the colinears we also have a power counting 1022 01:02:57,420 --> 01:02:59,530 between the minus and the plus. 1023 01:02:59,530 --> 01:03:05,460 So for the colinear, we can still 1024 01:03:05,460 --> 01:03:10,050 make the power counting OK by thinking about putting in PZ, 1025 01:03:10,050 --> 01:03:16,000 but then just expanding it to be a P minus. 1026 01:03:16,000 --> 01:03:18,210 And that's true up to power corrections. 1027 01:03:18,210 --> 01:03:20,790 And we don't really need to worry about power corrections 1028 01:03:20,790 --> 01:03:22,740 when we're regulating these divergences. 1029 01:03:26,630 --> 01:03:28,700 So it's just putting in the large momentum. 1030 01:03:28,700 --> 01:03:33,035 And so W written in a similar notation. 1031 01:03:43,432 --> 01:03:45,640 So I'll explain what the other things in this formula 1032 01:03:45,640 --> 01:03:48,180 are in a minute. 1033 01:03:48,180 --> 01:03:50,890 But the important thing for regulating 1034 01:03:50,890 --> 01:03:52,240 is that we have some factor. 1035 01:03:52,240 --> 01:03:55,756 In this case, it would be a factor of N bar dot P. 1036 01:03:55,756 --> 01:03:58,372 STUDENT: Dot [INAUDIBLE]? 1037 01:03:58,372 --> 01:03:59,080 PROFESSOR: Sorry? 1038 01:03:59,080 --> 01:04:00,807 STUDENT: Dot eta to bar dot P? 1039 01:04:00,807 --> 01:04:01,390 PROFESSOR: No. 1040 01:04:01,390 --> 01:04:05,710 It's supposed to be an N. Looks like eta. 1041 01:04:05,710 --> 01:04:07,000 Too many variables. 1042 01:04:07,000 --> 01:04:08,196 There's the eta. 1043 01:04:14,755 --> 01:04:16,630 So there's some factor raising of the-- again 1044 01:04:16,630 --> 01:04:19,105 the iconal propagator sort of mixing up 1045 01:04:19,105 --> 01:04:20,230 with the iconal propagator. 1046 01:04:20,230 --> 01:04:21,730 In this case, it's even more obvious 1047 01:04:21,730 --> 01:04:25,130 that it's just regulating that iconal propagator. 1048 01:04:25,130 --> 01:04:25,630 OK. 1049 01:04:25,630 --> 01:04:28,390 So if we were to do that, and go back over here, 1050 01:04:28,390 --> 01:04:30,790 and put the regulators into these integrals, 1051 01:04:30,790 --> 01:04:32,430 what would happen? 1052 01:04:32,430 --> 01:04:38,990 So here, we'd get an extra factor-- 1053 01:04:38,990 --> 01:04:40,255 K minus to the eta. 1054 01:04:40,255 --> 01:04:42,340 And so, that would regulate this K minus. 1055 01:04:42,340 --> 01:04:45,310 And these integrals will also have polls from the 1 1056 01:04:45,310 --> 01:04:46,570 over K minus. 1057 01:04:46,570 --> 01:04:47,650 You could think about-- 1058 01:04:47,650 --> 01:04:48,203 well, OK. 1059 01:04:48,203 --> 01:04:49,870 If we did those integrals, we would also 1060 01:04:49,870 --> 01:04:54,073 have rapidity divergences that are kind of the analog ones, 1061 01:04:54,073 --> 01:04:55,990 and the colinear SECT are still the soft ones. 1062 01:04:59,020 --> 01:05:00,700 Here we have two propagators. 1063 01:05:00,700 --> 01:05:04,360 And so if I have two soft Wilson lines, and so 1064 01:05:04,360 --> 01:05:06,010 I get two factors. 1065 01:05:06,010 --> 01:05:09,470 But I've conveniently chose it to be the square root. 1066 01:05:09,470 --> 01:05:11,590 So it comes out kind of looking the same here. 1067 01:05:16,970 --> 01:05:19,998 So one thing that is just a part of this regulator-- which 1068 01:05:19,998 --> 01:05:21,790 actually I don't know a good argument for-- 1069 01:05:21,790 --> 01:05:24,670 is kind of a priority from the symmetries of the theory, 1070 01:05:24,670 --> 01:05:30,190 you might like to argue that that should be eta over 2, 1071 01:05:30,190 --> 01:05:33,430 and this should be eta. 1072 01:05:33,430 --> 01:05:34,960 But it's really just part of-- it's 1073 01:05:34,960 --> 01:05:39,668 just a choice, a convention that we've made, as far as I know. 1074 01:05:39,668 --> 01:05:41,710 There probably is some nice deep argument for it, 1075 01:05:41,710 --> 01:05:44,420 but I don't know it. 1076 01:05:44,420 --> 01:05:45,920 So what are these other factors? 1077 01:05:45,920 --> 01:05:49,810 So nu is going to play the role of mu. 1078 01:05:49,810 --> 01:05:52,210 We've changed the dimension of the operator. 1079 01:05:52,210 --> 01:05:54,730 We've compensated it back with nu, 1080 01:05:54,730 --> 01:05:58,190 just like we were doing with mu. 1081 01:05:58,190 --> 01:06:01,450 We're going to get 1 over eta divergences, which 1082 01:06:01,450 --> 01:06:04,120 are like our 1 over epsilon divergences. 1083 01:06:04,120 --> 01:06:06,850 And we're going to get logs of nu, which 1084 01:06:06,850 --> 01:06:09,568 are the analogs of logs of mu. 1085 01:06:09,568 --> 01:06:11,110 And that's the sense in which there's 1086 01:06:11,110 --> 01:06:16,000 kind of an analog of this M up with our usual dim reg setup. 1087 01:06:16,000 --> 01:06:18,250 In order to have a full analog, we 1088 01:06:18,250 --> 01:06:21,710 should think about having a coupling. 1089 01:06:21,710 --> 01:06:23,830 And so, that's what this W factor is. 1090 01:06:26,265 --> 01:06:27,640 You can think about it like there 1091 01:06:27,640 --> 01:06:30,680 was some bare pseudo coupling, which is really just 1. 1092 01:06:30,680 --> 01:06:32,680 But just imagine that you're switching from bare 1093 01:06:32,680 --> 01:06:35,800 to renormalized, in order to set up a renormalization group 1094 01:06:35,800 --> 01:06:36,880 equation. 1095 01:06:36,880 --> 01:06:42,460 And then this guy here, which is in eta dimensions, if you like, 1096 01:06:42,460 --> 01:06:44,260 would have a renormalization group which 1097 01:06:44,260 --> 01:06:51,910 would say nu D by D nu of this W of eta and nu 1098 01:06:51,910 --> 01:06:55,030 is minus eta over 2. 1099 01:06:55,030 --> 01:06:58,300 Sorry, this is eta over 2. 1100 01:07:04,070 --> 01:07:07,790 So that's the analog of saying mu by D mu of alpha 1101 01:07:07,790 --> 01:07:10,585 is minus 2 epsilon alpha. 1102 01:07:10,585 --> 01:07:12,905 So an analog statement. 1103 01:07:15,695 --> 01:07:17,750 I think this is OK. 1104 01:07:17,750 --> 01:07:20,030 So this guy here is like a dummy coupling. 1105 01:07:20,030 --> 01:07:22,130 And the boundary condition for it 1106 01:07:22,130 --> 01:07:24,000 after you've carried out these-- 1107 01:07:24,000 --> 01:07:26,780 this is just to set it to back to 1. 1108 01:07:26,780 --> 01:07:30,210 So it's identically 1, it's really just a bookkeeping 1109 01:07:30,210 --> 01:07:30,710 device. 1110 01:07:33,300 --> 01:07:42,930 It's just--e it's a dummy coupling once you go to the eta 1111 01:07:42,930 --> 01:07:43,460 dimensions. 1112 01:07:43,460 --> 01:07:45,627 But you just set it always the renormalized coupling 1113 01:07:45,627 --> 01:07:47,840 is just identically set to 1. 1114 01:07:47,840 --> 01:07:49,520 And identically setting it to 1 is 1115 01:07:49,520 --> 01:07:53,210 what you need to keep gauge invariance in these Wilson 1116 01:07:53,210 --> 01:07:53,840 lines. 1117 01:07:53,840 --> 01:07:55,820 It turns out actually that this regulator here 1118 01:07:55,820 --> 01:07:58,752 is gauge invariant, though it doesn't look like it. 1119 01:07:58,752 --> 01:08:00,710 We've modified the structure of the Wilson line 1120 01:08:00,710 --> 01:08:06,230 in some kind of way that looks like it might be drastic. 1121 01:08:06,230 --> 01:08:08,480 But actually these factors here are gauge-- 1122 01:08:08,480 --> 01:08:10,410 still leave a gauge invariant object. 1123 01:08:10,410 --> 01:08:12,970 So-- 1124 01:08:12,970 --> 01:08:15,560 STUDENT: Can you write [INAUDIBLE] space, I assume? 1125 01:08:15,560 --> 01:08:18,180 PROFESSOR: Not that I know of. 1126 01:08:18,180 --> 01:08:18,908 Yeah. 1127 01:08:18,908 --> 01:08:20,116 STUDENT: I think [INAUDIBLE]. 1128 01:08:20,116 --> 01:08:21,680 PROFESSOR: Maybe you can. 1129 01:08:21,680 --> 01:08:22,819 Yeah. 1130 01:08:22,819 --> 01:08:25,515 But it's not-- since it's not-- 1131 01:08:25,515 --> 01:08:26,640 yeah, I don't know how to-- 1132 01:08:26,640 --> 01:08:28,182 I don't know what it would look like. 1133 01:08:28,182 --> 01:08:32,744 You could probably transform that power, and it-- 1134 01:08:32,744 --> 01:08:35,753 STUDENT: [INAUDIBLE]. 1135 01:08:35,753 --> 01:08:36,420 PROFESSOR: Yeah. 1136 01:08:42,187 --> 01:08:43,770 I'm sure you can probably just try out 1137 01:08:43,770 --> 01:08:45,145 before you [? transplant ?] that. 1138 01:08:45,145 --> 01:08:47,090 I'm just not sure if it would look nice. 1139 01:08:47,090 --> 01:08:47,754 Yeah. 1140 01:08:47,754 --> 01:08:49,029 It might not look too bad. 1141 01:08:51,502 --> 01:08:53,210 Yeah, and it might actually be a nice way 1142 01:08:53,210 --> 01:08:57,380 of saying what I'm about to say in a less nice way, which 1143 01:08:57,380 --> 01:08:59,569 is, if you look at the gauge symmetry, 1144 01:08:59,569 --> 01:09:02,035 why is this not messing it up? 1145 01:09:02,035 --> 01:09:03,410 So one way of thinking about that 1146 01:09:03,410 --> 01:09:06,410 is just to look at general covariant gauge. 1147 01:09:06,410 --> 01:09:19,470 So note, the 1 over eta and eta 0 terms are gauge invariant. 1148 01:09:19,470 --> 01:09:21,750 And you can think about that by just going 1149 01:09:21,750 --> 01:09:24,420 to a general covariant gauge and seeing the parameter dependence 1150 01:09:24,420 --> 01:09:26,100 drop out. 1151 01:09:26,100 --> 01:09:31,020 So for example, at 1 loop, you would 1152 01:09:31,020 --> 01:09:33,640 take [? g mu ?] nu in the contractions 1153 01:09:33,640 --> 01:09:35,520 and replace it in general covariant gauge 1154 01:09:35,520 --> 01:09:39,195 by some gauge parameter-- 1155 01:09:39,195 --> 01:09:44,760 of general covariant gauge, K mu, K nu over K squared. 1156 01:09:44,760 --> 01:09:47,460 And you'd like to see it independent of this. 1157 01:09:47,460 --> 01:09:50,700 But this eta to the 0 piece is kind of independent 1158 01:09:50,700 --> 01:09:52,590 of that for the usual reasons. 1159 01:09:52,590 --> 01:09:55,470 And the 1 over eta term is independent of that, 1160 01:09:55,470 --> 01:09:58,500 because this guy actually doesn't deuce any rapidity 1161 01:09:58,500 --> 01:09:59,130 singularities. 1162 01:09:59,130 --> 01:10:03,240 What happens is that if you have an N dot K, then 1163 01:10:03,240 --> 01:10:05,620 you have a corresponding N mu in the numerator. 1164 01:10:05,620 --> 01:10:08,745 And so, basically what happens is, you get an extra N dot 1165 01:10:08,745 --> 01:10:10,840 K in the numerator. 1166 01:10:10,840 --> 01:10:14,070 So any time you have 1 over N dot K, 1167 01:10:14,070 --> 01:10:18,690 you would get for this piece multiplied by an N dot K 1168 01:10:18,690 --> 01:10:21,480 upstairs. 1169 01:10:21,480 --> 01:10:24,360 And so this is cancelling, you don't have a rapidity 1170 01:10:24,360 --> 01:10:27,730 divergence in the C-dependent part. 1171 01:10:27,730 --> 01:10:31,980 So that's why this is invariant under the gauge symmetry. 1172 01:10:31,980 --> 01:10:36,990 And then, because of this boundary condition, 1173 01:10:36,990 --> 01:10:38,850 the kind of cancellation of the C-dependence 1174 01:10:38,850 --> 01:10:41,142 in the order [? A ?] to the 0 piece, this kind of works 1175 01:10:41,142 --> 01:10:45,262 out in the standard way. 1176 01:10:45,262 --> 01:10:46,720 So it gives you an idea of why it's 1177 01:10:46,720 --> 01:10:49,680 gauge invariant without giving you a kind of full proof 1178 01:10:49,680 --> 01:10:50,240 or anything. 1179 01:10:53,210 --> 01:10:55,720 So we have both 1 over epsilon polls 1180 01:10:55,720 --> 01:10:57,550 and 1 over eta polls in general, and we 1181 01:10:57,550 --> 01:10:59,835 have to understand what to do with them. 1182 01:10:59,835 --> 01:11:01,210 So here's what we're going to do. 1183 01:11:03,880 --> 01:11:07,815 For any fixed invariant mass, it turns out 1184 01:11:07,815 --> 01:11:09,565 that we can have these one over eta polls. 1185 01:11:18,650 --> 01:11:20,690 And the right procedure for dealing with them 1186 01:11:20,690 --> 01:11:22,140 is as follows. 1187 01:11:22,140 --> 01:11:25,160 First you take eta goes to 0 and deal with these new polls 1188 01:11:25,160 --> 01:11:28,730 that you have introduced in your amplitude. 1189 01:11:28,730 --> 01:11:31,010 In order to deal with them, because you can have them 1190 01:11:31,010 --> 01:11:33,613 for any invariant mass, you actually 1191 01:11:33,613 --> 01:11:35,030 have to add counter terms that can 1192 01:11:35,030 --> 01:11:36,920 be a whole function of epsilon, where 1193 01:11:36,920 --> 01:11:39,800 you have an expanded in epsilon and then divide it by eta. 1194 01:11:43,027 --> 01:11:47,900 So let me abbreviate counterterm as CT dot. 1195 01:11:47,900 --> 01:11:52,040 Then, after you've done that, you take epsilon goes to 0, 1196 01:11:52,040 --> 01:11:54,834 and you find your 1 over epsilon counterterms. 1197 01:12:00,650 --> 01:12:04,050 And this is the correct way of doing it. 1198 01:12:04,050 --> 01:12:07,780 And we'll see how that works in practice in a minute. 1199 01:12:07,780 --> 01:12:10,690 So let's go back to our integrals that I've now erased 1200 01:12:10,690 --> 01:12:12,750 and just write out the answers. 1201 01:12:12,750 --> 01:12:15,480 We're doing those integrals with this regulator. 1202 01:12:15,480 --> 01:12:17,100 And I'll also make them fermions, 1203 01:12:17,100 --> 01:12:19,610 so I'm putting in the numerators. 1204 01:12:19,610 --> 01:12:21,540 We wrote them down for scalars. 1205 01:12:21,540 --> 01:12:24,748 The scalars where the most divergent integrals actually. 1206 01:12:24,748 --> 01:12:26,790 I can include the numerators, that doesn't really 1207 01:12:26,790 --> 01:12:28,950 change the story. 1208 01:12:28,950 --> 01:12:31,050 And I can include the pre-factors as well. 1209 01:12:41,520 --> 01:12:44,340 And I'll kind of write things in a QCD type notation, 1210 01:12:44,340 --> 01:12:48,210 even we can imagine that it's a non-abelian group, just so CF 1211 01:12:48,210 --> 01:12:50,040 is the whatever group it is, it's 1212 01:12:50,040 --> 01:12:52,575 the Casimir of the fundamental. 1213 01:12:58,030 --> 01:13:01,450 Whatever group our gauge boson's in. 1214 01:13:01,450 --> 01:13:04,360 So here's the eta poll. 1215 01:13:04,360 --> 01:13:06,980 It has a whole function of epsilon in the numerator, 1216 01:13:06,980 --> 01:13:09,070 and it's even divergent. 1217 01:13:09,070 --> 01:13:11,413 So this is 2 eta. 1218 01:13:11,413 --> 01:13:12,955 And then the rest of it I can expand. 1219 01:13:23,960 --> 01:13:27,830 So there's going to be 1 over epsilon times the log. 1220 01:13:27,830 --> 01:13:29,930 When the log replaces that 1 over epsilon 1221 01:13:29,930 --> 01:13:33,080 then I can start to expand, and I get another 1 over-- 1222 01:13:33,080 --> 01:13:35,930 when the log nu replaces the one over eta, 1223 01:13:35,930 --> 01:13:41,360 I can expand this gamma, and it gives me 1 over epsilon. 1224 01:13:41,360 --> 01:13:44,681 And there's also some other pieces. 1225 01:13:44,681 --> 01:13:49,496 So over 2 epsilon there's a log mu over M. 1226 01:13:49,496 --> 01:13:50,548 And there's a constant. 1227 01:13:50,548 --> 01:13:52,340 And I'm never going to write the constants. 1228 01:13:55,437 --> 01:13:58,020 So let me read all the results and then we'll talk about them. 1229 01:13:58,020 --> 01:14:01,300 So ICN bar is the same. 1230 01:14:01,300 --> 01:14:04,080 The only difference between this is that P minus 1231 01:14:04,080 --> 01:14:05,790 close to P bar plus. 1232 01:14:05,790 --> 01:14:08,340 It was really symmetric. 1233 01:14:08,340 --> 01:14:11,742 And then IS is different. 1234 01:14:36,360 --> 01:14:37,615 So the 1 over epsilon-- 1235 01:14:37,615 --> 01:14:39,490 1 over eta poll comes with the opposite sign, 1236 01:14:39,490 --> 01:14:41,795 and it also comes to a factor of 2 different. 1237 01:14:49,543 --> 01:14:50,960 And in this case, there's actually 1238 01:14:50,960 --> 01:14:53,450 1 over 2 epsilons squared term. 1239 01:14:56,520 --> 01:15:02,568 So there's also a double log of mu or M. 1240 01:15:02,568 --> 01:15:03,860 And then there's plus constant. 1241 01:15:07,415 --> 01:15:07,915 OK? 1242 01:15:10,470 --> 01:15:12,940 And so, you could think about adding them up. 1243 01:15:12,940 --> 01:15:15,300 And what happens when you add them up is, 1244 01:15:15,300 --> 01:15:18,150 you have 1 over 2 eta, 1 over 2 eta, minus 1 over eta, 1245 01:15:18,150 --> 01:15:21,040 and so that 1 over eta polls cancel. 1246 01:15:21,040 --> 01:15:22,620 And that's exactly what you'd expect, 1247 01:15:22,620 --> 01:15:26,315 because in the full theory the eta 1248 01:15:26,315 --> 01:15:28,440 was something we introduced in order to distinguish 1249 01:15:28,440 --> 01:15:29,670 these effective theory modes. 1250 01:15:29,670 --> 01:15:32,040 It wasn't something that was there needed 1251 01:15:32,040 --> 01:15:34,020 for the full theory integral. 1252 01:15:34,020 --> 01:15:35,830 And so you don't really-- 1253 01:15:35,830 --> 01:15:38,560 you'd expect that it's sort of-- 1254 01:15:38,560 --> 01:15:41,550 that there's a corresponding regulator 1255 01:15:41,550 --> 01:15:42,550 between the two sectors. 1256 01:15:42,550 --> 01:15:44,008 So that when you add them together, 1257 01:15:44,008 --> 01:15:46,810 that the dependence on that parameter is canceling away, 1258 01:15:46,810 --> 01:15:49,630 because it was just an artificial separation, 1259 01:15:49,630 --> 01:15:52,970 if you like, or separation that we're doing. 1260 01:15:52,970 --> 01:15:58,885 So if I add them up, 1 over eta is cancelled, 1261 01:15:58,885 --> 01:16:02,386 and so do all the logs of nu. 1262 01:16:02,386 --> 01:16:03,195 We have alpha-- 1263 01:16:12,160 --> 01:16:17,515 I'm left with a log of mu over Q, 1 over epsilon poll, 1264 01:16:17,515 --> 01:16:24,410 double log, some types of single logs, 1265 01:16:24,410 --> 01:16:26,567 and some other type of double log. 1266 01:16:38,260 --> 01:16:39,400 So it would look like that. 1267 01:16:39,400 --> 01:16:42,700 All the nu dependence is canceling away. 1268 01:16:42,700 --> 01:16:53,020 So sort of various things which we'll start talking about now, 1269 01:16:53,020 --> 01:16:56,000 and we'll continue talking about next time. 1270 01:16:56,000 --> 01:17:00,380 So the rapidity 1 over eta divergence, 1271 01:17:00,380 --> 01:17:07,510 which we can call a rapidity divergence, cancels in sum. 1272 01:17:07,510 --> 01:17:09,340 And of course, so does the log nu's. 1273 01:17:14,070 --> 01:17:15,045 And that's as expected. 1274 01:17:22,067 --> 01:17:23,650 And if you add an overall counterterm, 1275 01:17:23,650 --> 01:17:27,190 for the entire thing it just involves the hard scale log 1276 01:17:27,190 --> 01:17:34,670 mu over Q. So if you were to think about there being 1277 01:17:34,670 --> 01:17:37,800 some Wilson coefficient, which is sort of C 1278 01:17:37,800 --> 01:17:47,770 bare is ZC minus 1, Z bare is ZC, C renormalized, 1279 01:17:47,770 --> 01:17:55,640 then ZC and the C renormalized only involve logs of mu 1280 01:17:55,640 --> 01:17:58,510 over Q, which is the hard scale. 1281 01:17:58,510 --> 01:17:59,870 OK? 1282 01:17:59,870 --> 01:18:03,110 And that means that our hard function, which 1283 01:18:03,110 --> 01:18:05,870 is the Wilson coefficient squared, or just the Wilson 1284 01:18:05,870 --> 01:18:10,970 coefficient in this case, is only a function of Q and mu. 1285 01:18:10,970 --> 01:18:13,010 OK, so integrating out the hard scale physics 1286 01:18:13,010 --> 01:18:14,522 didn't know about the separation. 1287 01:18:14,522 --> 01:18:15,980 The separation was really something 1288 01:18:15,980 --> 01:18:18,800 that we needed to do in the effective theory 1289 01:18:18,800 --> 01:18:22,640 to distinguish the CN and S modes. 1290 01:18:22,640 --> 01:18:24,420 And you can see why we needed to do it 1291 01:18:24,420 --> 01:18:29,570 if you look at these answers, because if you look 1292 01:18:29,570 --> 01:18:34,080 at the types of logs that are showing up here, in this case, 1293 01:18:34,080 --> 01:18:35,930 we have a nu over P minus. 1294 01:18:35,930 --> 01:18:37,460 And in this case, we have a-- 1295 01:18:37,460 --> 01:18:38,740 is it nu over mu? 1296 01:18:42,013 --> 01:18:44,710 Just make sure I got that right. 1297 01:18:44,710 --> 01:18:46,130 I guess it is. 1298 01:18:46,130 --> 01:18:50,415 In this case, we have a nu over M. 1299 01:18:50,415 --> 01:18:53,620 And we also have a mu over nu. 1300 01:18:53,620 --> 01:18:56,668 And so, the sort of right scale to-- in order 1301 01:18:56,668 --> 01:18:58,210 to minimize the logarithms here we're 1302 01:18:58,210 --> 01:18:59,793 going to have to, again, as usual take 1303 01:18:59,793 --> 01:19:02,500 different values of mu and nu. 1304 01:19:02,500 --> 01:19:04,180 Well, it's the same value of mu. 1305 01:19:04,180 --> 01:19:07,720 All of them are M. But it's a different value of nu, 1306 01:19:07,720 --> 01:19:11,380 because it's the nu that would need to be of order Q here. 1307 01:19:11,380 --> 01:19:15,940 P minus is Q. And the nu would need to be of order M here. 1308 01:19:15,940 --> 01:19:19,280 So it's the nu that distinguishes the modes. 1309 01:19:19,280 --> 01:19:25,600 So the logs NCN are minimized. 1310 01:19:29,870 --> 01:19:33,200 Or mu of order M, which says being on the hyperbola, 1311 01:19:33,200 --> 01:19:37,730 but the nu should be of order P minus, which is Q. 1312 01:19:37,730 --> 01:19:39,500 And that's precisely actually where 1313 01:19:39,500 --> 01:19:48,890 we put the X in our picture, if you think about it. 1314 01:19:48,890 --> 01:19:54,840 OK, so that's saying that you have a large P minus momentum, 1315 01:19:54,840 --> 01:19:57,623 and we have-- and we're on this hyperbola where P squared is 1316 01:19:57,623 --> 01:19:59,928 an order M squared. 1317 01:19:59,928 --> 01:20:06,750 So this is-- and it's likewise for the other pieces. 1318 01:20:06,750 --> 01:20:08,160 So for the soft piece-- 1319 01:20:12,520 --> 01:20:20,030 so for the-- say for the anti- for the other colinear piece, 1320 01:20:20,030 --> 01:20:21,170 we need the same thing. 1321 01:20:25,840 --> 01:20:35,200 And then for the soft we need a different value 1322 01:20:35,200 --> 01:20:37,140 for this new parameter. 1323 01:20:37,140 --> 01:20:38,837 So having this regulator is behaving 1324 01:20:38,837 --> 01:20:40,920 like dim reg, where we needed different mu's, when 1325 01:20:40,920 --> 01:20:42,087 we had different hyperbolas. 1326 01:20:42,087 --> 01:20:44,122 Now we have different places on the hyperbola, 1327 01:20:44,122 --> 01:20:46,080 and we're tracking that with the new parameter, 1328 01:20:46,080 --> 01:20:47,922 and that's showing up in the logarithms. 1329 01:20:47,922 --> 01:20:50,130 And if you think about what the logarithms are doing, 1330 01:20:50,130 --> 01:20:57,420 you can see that when you combine terms, let's see-- 1331 01:20:57,420 --> 01:20:59,077 if you look at the 1 over epsilon, 1332 01:20:59,077 --> 01:21:00,660 and the mu's are canceling out, you're 1333 01:21:00,660 --> 01:21:04,530 getting a mu over Q. Yeah, that's 1334 01:21:04,530 --> 01:21:06,660 maybe not the best example. 1335 01:21:06,660 --> 01:21:08,070 Look over here. 1336 01:21:08,070 --> 01:21:12,180 You have this log of M over Q In this kind 1337 01:21:12,180 --> 01:21:13,860 of complete decomposition. 1338 01:21:13,860 --> 01:21:16,950 The way that that logarithm here gets made up 1339 01:21:16,950 --> 01:21:21,960 is by having m over nu and nu over Q. All right? 1340 01:21:21,960 --> 01:21:24,650 So in order to get this log that doesn't have any mu's in it, 1341 01:21:24,650 --> 01:21:26,950 mu is not telling you that there's that large log. 1342 01:21:26,950 --> 01:21:28,630 But there is a large log. 1343 01:21:28,630 --> 01:21:32,460 So there's large logs associated to these rapidity divergences. 1344 01:21:32,460 --> 01:21:35,760 And what we'll talk about next time 1345 01:21:35,760 --> 01:21:38,820 is how you do the renormalization group 1346 01:21:38,820 --> 01:21:40,500 with the diagrams like this. 1347 01:21:40,500 --> 01:21:42,570 How you write down in almost dimension equations. 1348 01:21:42,570 --> 01:21:45,180 There'll be an almost dimension equations in both mu and nu 1349 01:21:45,180 --> 01:21:46,190 space. 1350 01:21:46,190 --> 01:21:47,940 So we'll have to move around in that space 1351 01:21:47,940 --> 01:21:50,982 and see how it works to sum of the large logarithms. 1352 01:21:50,982 --> 01:21:54,300 But I'll postpone that to next time. 1353 01:21:57,770 --> 01:21:58,790 So any questions? 1354 01:22:06,430 --> 01:22:08,440 So the general idea is really, as usual, 1355 01:22:08,440 --> 01:22:10,630 it's just that now we're dealing with a situation 1356 01:22:10,630 --> 01:22:12,490 where there's two regulators. 1357 01:22:12,490 --> 01:22:15,310 And they're actually independent regulators. 1358 01:22:15,310 --> 01:22:18,250 One's, if you like, is regulating invariant mass, 1359 01:22:18,250 --> 01:22:21,310 and the other is regulating these extra divergences. 1360 01:22:21,310 --> 01:22:24,580 And so, we'll be able to move around in the space 1361 01:22:24,580 --> 01:22:28,920 without any worrying about path dependence, for example. 1362 01:22:28,920 --> 01:22:31,690 We'll talk a little bit about that next time-- 1363 01:22:31,690 --> 01:22:34,600 in this two dimensional space of mu and nu. 1364 01:22:34,600 --> 01:22:37,370 But you can see just from looking at the logs, as usual, 1365 01:22:37,370 --> 01:22:38,872 you can see where you need to be. 1366 01:22:38,872 --> 01:22:40,330 And you can see that you need to be 1367 01:22:40,330 --> 01:22:42,122 in different places for the different modes 1368 01:22:42,122 --> 01:22:44,578 in order to minimize the logs of these amplitudes. 1369 01:22:44,578 --> 01:22:46,120 And if you do that, then there should 1370 01:22:46,120 --> 01:22:52,460 be some renormalization group that would connect these guys. 1371 01:22:52,460 --> 01:22:58,390 So there should do something RGE that goes between these guys, 1372 01:22:58,390 --> 01:23:01,410 and it'll be an RGE in this new parameter.