WEBVTT 00:00:00.000 --> 00:00:07.470 align:middle line:84% [SQUEAKING] [CLICKING] [RUSTLING] [CLICKING] 00:00:07.470 --> 00:00:10.470 align:middle line:90% 00:00:10.470 --> 00:00:13.380 align:middle line:90% PROFESSOR: Let us start. 00:00:13.380 --> 00:00:18.530 align:middle line:84% So last time, we started talking about the following process. 00:00:18.530 --> 00:00:21.380 align:middle line:90% 00:00:21.380 --> 00:00:30.900 align:middle line:90% So you consider e plus e minus-- 00:00:30.900 --> 00:00:35.190 align:middle line:84% so the electron, the positron they scatter and then they 00:00:35.190 --> 00:00:40.110 align:middle line:84% will annihilate and that creates some other particles. 00:00:40.110 --> 00:00:40.890 align:middle line:90% OK? 00:00:40.890 --> 00:00:44.460 align:middle line:84% So b and the b bar-- some final particles 00:00:44.460 --> 00:00:47.830 align:middle line:84% can be different kinds of things. 00:00:47.830 --> 00:00:49.500 align:middle line:84% So if we draw the Feynman diagram 00:00:49.500 --> 00:00:54.285 align:middle line:84% with the time going up, and then the process 00:00:54.285 --> 00:00:55.410 align:middle line:90% will be like the following. 00:00:55.410 --> 00:00:59.760 align:middle line:90% 00:00:59.760 --> 00:01:02.020 align:middle line:90% So the initial particle-- 00:01:02.020 --> 00:01:05.700 align:middle line:84% so this is the e minus and this is e plus. 00:01:05.700 --> 00:01:11.650 align:middle line:84% And then we have b and the b bar, OK? 00:01:11.650 --> 00:01:14.090 align:middle line:90% So we can-- 00:01:14.090 --> 00:01:15.260 align:middle line:90% Yeah. 00:01:15.260 --> 00:01:20.900 align:middle line:84% So such a process is a very important discovery machine 00:01:20.900 --> 00:01:24.530 align:middle line:90% for new particles, OK? 00:01:24.530 --> 00:01:40.690 align:middle line:84% Very important for discovering new particles. 00:01:40.690 --> 00:01:44.180 align:middle line:84% So let me just mention two examples here. 00:01:44.180 --> 00:01:51.310 align:middle line:90% So in 1974 to 1976 at SLAC-- 00:01:51.310 --> 00:01:55.900 align:middle line:84% so SLAC is called Stanford Linear Acceleration Center. 00:01:55.900 --> 00:02:00.580 align:middle line:84% So they have an electron positron collider. 00:02:00.580 --> 00:02:05.440 align:middle line:84% And so there they first discovered the tau particle. 00:02:05.440 --> 00:02:10.370 align:middle line:90% 00:02:10.370 --> 00:02:12.530 align:middle line:90% And this tau particle-- 00:02:12.530 --> 00:02:15.710 align:middle line:90% it's mass is the 1.8 GeV. 00:02:15.710 --> 00:02:27.200 align:middle line:84% So when your total energy is greater than twice the 1.8 GeV, 00:02:27.200 --> 00:02:31.100 align:middle line:84% then you can, in principle, pair create this tau part-- 00:02:31.100 --> 00:02:34.070 align:middle line:90% tau plus and tau minus. 00:02:34.070 --> 00:02:42.710 align:middle line:84% And another example is in 1974, again, at the SLAC. 00:02:42.710 --> 00:02:48.080 align:middle line:84% And the e plus, e minus goes to c, c bar. 00:02:48.080 --> 00:02:50.750 align:middle line:90% So c is the charm quark here. 00:02:50.750 --> 00:02:52.590 align:middle line:90% OK, it's the charm quark here. 00:02:52.590 --> 00:02:57.515 align:middle line:84% So what they observed is the bound state of the c, 00:02:57.515 --> 00:03:00.240 align:middle line:90% c bar-- call it J/psi. 00:03:00.240 --> 00:03:03.630 align:middle line:84% So under the mass of the J/psi, it's about 3.1 GeV. 00:03:03.630 --> 00:03:06.430 align:middle line:90% 00:03:06.430 --> 00:03:09.180 align:middle line:90% OK. 00:03:09.180 --> 00:03:11.790 align:middle line:90% 3.1 GeV. 00:03:11.790 --> 00:03:18.850 align:middle line:84% So this J/psi-- so this particle was discovered at the same time 00:03:18.850 --> 00:03:23.710 align:middle line:84% at Brookhaven through the proton-- 00:03:23.710 --> 00:03:28.100 align:middle line:84% through the collision of the protons by our own colleague 00:03:28.100 --> 00:03:29.350 align:middle line:90% Samuel Ting. 00:03:29.350 --> 00:03:34.540 align:middle line:84% And then they both get the Nobel Prize in 1975, I think. 00:03:34.540 --> 00:03:35.350 align:middle line:90% OK. 00:03:35.350 --> 00:03:39.610 align:middle line:84% So there was actually a funny story regarding 00:03:39.610 --> 00:03:41.870 align:middle line:90% the discovery of this particle. 00:03:41.870 --> 00:03:48.820 align:middle line:84% So at Brookhaven, they collide the protons 00:03:48.820 --> 00:03:51.160 align:middle line:90% on the fixed target. 00:03:51.160 --> 00:03:54.160 align:middle line:84% So that is a much messier machine, 00:03:54.160 --> 00:03:58.240 align:middle line:84% because protons contain lots of quarks and gluons 00:03:58.240 --> 00:04:05.080 align:middle line:84% and so it's with strong interactions. 00:04:05.080 --> 00:04:09.400 align:middle line:84% So if you collide high enough energy, 00:04:09.400 --> 00:04:13.120 align:middle line:90% you can create any particles. 00:04:13.120 --> 00:04:15.190 align:middle line:84% But since it's involving strong interactions, 00:04:15.190 --> 00:04:17.510 align:middle line:84% and strong interactions have many, many particles. 00:04:17.510 --> 00:04:21.840 align:middle line:84% So the so the collider at Brookhaven is very, very messy. 00:04:21.840 --> 00:04:22.340 align:middle line:90% OK. 00:04:22.340 --> 00:04:23.930 align:middle line:84% So in order to find the new particles, 00:04:23.930 --> 00:04:25.560 align:middle line:90% it's very, very difficult. 00:04:25.560 --> 00:04:26.060 align:middle line:90% OK. 00:04:26.060 --> 00:04:31.940 align:middle line:84% But they managed to find this new particle at Brookhaven. 00:04:31.940 --> 00:04:38.180 align:middle line:84% But Samuel Ting was famously extraordinarily meticulous. 00:04:38.180 --> 00:04:41.600 align:middle line:84% OK, so he wanted to be absolutely right, 00:04:41.600 --> 00:04:44.050 align:middle line:90% so he was checking and checking. 00:04:44.050 --> 00:04:45.670 align:middle line:90% So during one of the-- 00:04:45.670 --> 00:04:48.730 align:middle line:90% Yeah, so this is a rumor. 00:04:48.730 --> 00:04:52.870 align:middle line:84% So the rumor said that during one of the checking process-- 00:04:52.870 --> 00:04:57.220 align:middle line:84% and somehow the rumor spread that he 00:04:57.220 --> 00:04:59.770 align:middle line:90% discovered a new particle. 00:04:59.770 --> 00:05:02.650 align:middle line:84% And so the person was doing the experiment at SLAC 00:05:02.650 --> 00:05:04.690 align:middle line:90% and called him up. 00:05:04.690 --> 00:05:07.180 align:middle line:84% Even the rumor-- the rumor even-- 00:05:07.180 --> 00:05:10.150 align:middle line:90% Yeah, so the news leaked. 00:05:10.150 --> 00:05:14.350 align:middle line:84% It even leaked the energy scale of the particle they found. 00:05:14.350 --> 00:05:16.690 align:middle line:84% And then the person was doing the register 00:05:16.690 --> 00:05:20.200 align:middle line:84% at the SLAC then called the Samuel Ting. 00:05:20.200 --> 00:05:25.080 align:middle line:84% He said, oh, I heard you have found a new particle. 00:05:25.080 --> 00:05:27.590 align:middle line:90% Samuel Ting said, no, I didn't. 00:05:27.590 --> 00:05:28.770 align:middle line:90% Yeah. 00:05:28.770 --> 00:05:32.650 align:middle line:90% He didn't want to have the-- 00:05:32.650 --> 00:05:35.080 align:middle line:84% Yeah, actually it's not very good, because I'm recorded. 00:05:35.080 --> 00:05:38.050 align:middle line:90% 00:05:38.050 --> 00:05:39.280 align:middle line:90% I'm recorded. 00:05:39.280 --> 00:05:44.920 align:middle line:84% So I think this maybe it's not a good spread rumors. 00:05:44.920 --> 00:05:49.120 align:middle line:84% If it's not recorded, actually, I can tell the story. 00:05:49.120 --> 00:05:50.590 align:middle line:90% Yeah. 00:05:50.590 --> 00:05:59.020 align:middle line:84% Anyway so accordingly, then the register 00:05:59.020 --> 00:06:01.360 align:middle line:84% said then, oh, you didn't find the particle. 00:06:01.360 --> 00:06:03.010 align:middle line:90% Then I can discover it. 00:06:03.010 --> 00:06:05.350 align:middle line:84% Because they already knew the energy scale. 00:06:05.350 --> 00:06:07.930 align:middle line:84% They already had the energy scale. 00:06:07.930 --> 00:06:10.480 align:middle line:84% And this is a very, very clean machine. 00:06:10.480 --> 00:06:14.120 align:middle line:84% So electron, positron, when they collide only energy created. 00:06:14.120 --> 00:06:15.820 align:middle line:84% So it don't create a lot of junks. 00:06:15.820 --> 00:06:18.250 align:middle line:84% And if you add the right energy, you can immediately 00:06:18.250 --> 00:06:19.720 align:middle line:90% find the new particle. 00:06:19.720 --> 00:06:21.350 align:middle line:90% You don't need to do much work. 00:06:21.350 --> 00:06:25.240 align:middle line:84% And so they put the machine at that energy, immediately, 00:06:25.240 --> 00:06:27.520 align:middle line:90% the new particle was discovered. 00:06:27.520 --> 00:06:29.830 align:middle line:90% And so they were-- 00:06:29.830 --> 00:06:30.430 align:middle line:90% Yeah. 00:06:30.430 --> 00:06:35.360 align:middle line:84% So they were awarded the Nobel Prize jointly. 00:06:35.360 --> 00:06:36.170 align:middle line:90% Anyway. 00:06:36.170 --> 00:06:40.610 align:middle line:84% So this was an important discovery, 00:06:40.610 --> 00:06:45.420 align:middle line:84% because before that, people didn't believe quarks exist, 00:06:45.420 --> 00:06:47.630 align:middle line:90% OK? 00:06:47.630 --> 00:06:49.250 align:middle line:84% People didn't quite believe the quark 00:06:49.250 --> 00:06:53.850 align:middle line:84% existed, even though there are many models, et cetera. 00:06:53.850 --> 00:06:56.300 align:middle line:84% And there are lots of other evidences. 00:06:56.300 --> 00:07:01.760 align:middle line:90% But this was the first-- 00:07:01.760 --> 00:07:06.230 align:middle line:84% yeah, because C-quark is very heavy. 00:07:06.230 --> 00:07:09.960 align:middle line:84% And so c and c bar, they bound not very strongly. 00:07:09.960 --> 00:07:12.740 align:middle line:84% And so this is the first direct evidence 00:07:12.740 --> 00:07:14.340 align:middle line:90% actually quark existed. 00:07:14.340 --> 00:07:16.830 align:middle line:84% And so this actually shocked the community. 00:07:16.830 --> 00:07:19.410 align:middle line:84% So this was discovered in October. 00:07:19.410 --> 00:07:21.320 align:middle line:84% So they call it the October Revolution. 00:07:21.320 --> 00:07:24.260 align:middle line:90% 00:07:24.260 --> 00:07:29.930 align:middle line:84% So yeah, it played a very important role for people 00:07:29.930 --> 00:07:33.040 align:middle line:84% to accept the existence of quarks. 00:07:33.040 --> 00:07:33.880 align:middle line:90% Anyway. 00:07:33.880 --> 00:07:40.890 align:middle line:84% So in particular, by looking at the more general process-- 00:07:40.890 --> 00:07:44.480 align:middle line:90% so this goes to hadrons. 00:07:44.480 --> 00:07:45.960 align:middle line:90% OK, you just collide them. 00:07:45.960 --> 00:07:47.000 align:middle line:90% And then you look at it. 00:07:47.000 --> 00:07:47.960 align:middle line:90% It's going to hadrons. 00:07:47.960 --> 00:07:57.190 align:middle line:84% You can actually show that the quarks have three colors. 00:07:57.190 --> 00:08:06.050 align:middle line:90% 00:08:06.050 --> 00:08:06.620 align:middle line:90% OK. 00:08:06.620 --> 00:08:10.290 align:middle line:84% You can show quark have three colors. 00:08:10.290 --> 00:08:14.650 align:middle line:84% And yeah, so we will see that, OK? 00:08:14.650 --> 00:08:20.330 align:middle line:84% So before doing that, let's first have to do a calculation, 00:08:20.330 --> 00:08:20.830 align:middle line:90% OK? 00:08:20.830 --> 00:08:24.350 align:middle line:84% So we have to calculate this explicitly. 00:08:24.350 --> 00:08:24.850 align:middle line:90% Let me see. 00:08:24.850 --> 00:08:26.440 align:middle line:90% I get my page a little bit. 00:08:26.440 --> 00:08:31.980 align:middle line:90% 00:08:31.980 --> 00:08:32.710 align:middle line:90% Right. 00:08:32.710 --> 00:08:33.210 align:middle line:90% OK. 00:08:33.210 --> 00:08:35.370 align:middle line:84% Yeah, so first we have to do a calculation. 00:08:35.370 --> 00:08:37.740 align:middle line:84% We have to calculate this process, OK? 00:08:37.740 --> 00:08:40.500 align:middle line:84% So after we have calculated this, 00:08:40.500 --> 00:08:45.270 align:middle line:84% then it's obvious how this can tell us why 00:08:45.270 --> 00:08:46.740 align:middle line:90% the quark have three colors. 00:08:46.740 --> 00:08:47.430 align:middle line:90% OK. 00:08:47.430 --> 00:08:48.150 align:middle line:90% Yes. 00:08:48.150 --> 00:08:50.733 align:middle line:84% AUDIENCE: So since you only see the bound state of the quarks, 00:08:50.733 --> 00:08:54.950 align:middle line:84% how can you know for sure that they are a bound state? 00:08:54.950 --> 00:08:57.300 align:middle line:90% PROFESSOR: Yeah, so indeed. 00:08:57.300 --> 00:09:00.140 align:middle line:84% So that's why, for many years, if you 00:09:00.140 --> 00:09:02.450 align:middle line:84% look at the proton, neutron, and the pion, 00:09:02.450 --> 00:09:04.260 align:middle line:90% they're very tightly bound. 00:09:04.260 --> 00:09:07.990 align:middle line:84% And so even though there were models 00:09:07.990 --> 00:09:10.240 align:middle line:90% about the quarks inside there-- 00:09:10.240 --> 00:09:13.150 align:middle line:90% so it's not quite-- 00:09:13.150 --> 00:09:16.690 align:middle line:84% yeah, but in the c case, c are very heavy. 00:09:16.690 --> 00:09:20.380 align:middle line:84% And so they form a weakly bound state. 00:09:20.380 --> 00:09:23.110 align:middle line:84% And so in the sense that you're looking at the-- 00:09:23.110 --> 00:09:26.380 align:middle line:90% yeah, so here, it's-- 00:09:26.380 --> 00:09:28.960 align:middle line:84% in a sense, you can look at the C-quark, 00:09:28.960 --> 00:09:32.104 align:middle line:84% because it's a very weakly bound state of the c and C-quark. 00:09:32.104 --> 00:09:34.920 align:middle line:90% 00:09:34.920 --> 00:09:35.610 align:middle line:90% Other questions? 00:09:35.610 --> 00:09:36.600 align:middle line:90% Yes. 00:09:36.600 --> 00:09:40.240 align:middle line:84% AUDIENCE: Does weakly bound imply it's usually larger? 00:09:40.240 --> 00:09:41.890 align:middle line:90% PROFESSOR: Yeah. 00:09:41.890 --> 00:09:45.130 align:middle line:84% Weakly bound just say you can probe with the-- 00:09:45.130 --> 00:09:47.110 align:middle line:90% Yeah, you can essentially-- 00:09:47.110 --> 00:09:49.720 align:middle line:84% you can probe the internal structure much easier. 00:09:49.720 --> 00:09:55.666 align:middle line:90% 00:09:55.666 --> 00:09:59.890 align:middle line:84% AUDIENCE: If electron collider is much cleaner than Hadron 00:09:59.890 --> 00:10:03.670 align:middle line:84% Collider, then why is like the most important machine right 00:10:03.670 --> 00:10:05.440 align:middle line:90% now is the Hadron Collider? 00:10:05.440 --> 00:10:06.700 align:middle line:90% PROFESSOR: Yeah. 00:10:06.700 --> 00:10:11.890 align:middle line:84% That's good, because electron collider is more difficult, 00:10:11.890 --> 00:10:14.540 align:middle line:84% because you have to build a straight line, 00:10:14.540 --> 00:10:19.000 align:middle line:84% and because the electron have a much stronger synchrotron 00:10:19.000 --> 00:10:22.400 align:middle line:84% radiation if they're moving in the circle. 00:10:22.400 --> 00:10:25.010 align:middle line:84% And so that loses a lot of energy. 00:10:25.010 --> 00:10:26.802 align:middle line:84% And so that's the electron collider. 00:10:26.802 --> 00:10:28.510 align:middle line:84% They should build a strong straight line. 00:10:28.510 --> 00:10:31.120 align:middle line:84% But straight line you cannot have very high energy, 00:10:31.120 --> 00:10:33.050 align:middle line:90% require very long distance. 00:10:33.050 --> 00:10:36.550 align:middle line:84% And so that's why a lot of people 00:10:36.550 --> 00:10:40.750 align:middle line:84% built the proton collider, which is easier. 00:10:40.750 --> 00:10:44.470 align:middle line:84% And yeah, it's easier to get higher energy. 00:10:44.470 --> 00:10:49.330 align:middle line:84% But the next generation of the accelerator of the LHC, people 00:10:49.330 --> 00:10:53.210 align:middle line:84% are talking about building an electron collider again. 00:10:53.210 --> 00:10:53.710 align:middle line:90% Yeah. 00:10:53.710 --> 00:10:56.240 align:middle line:90% 00:10:56.240 --> 00:10:57.020 align:middle line:90% Good. 00:10:57.020 --> 00:10:59.210 align:middle line:84% So now let's try to calculate this. 00:10:59.210 --> 00:11:00.140 align:middle line:90% OK. 00:11:00.140 --> 00:11:03.920 align:middle line:84% So essentially we need to calculate this diagram. 00:11:03.920 --> 00:11:06.180 align:middle line:84% And yeah, this diagram is very easy. 00:11:06.180 --> 00:11:09.350 align:middle line:84% We need to calculate the cross section corresponding to this. 00:11:09.350 --> 00:11:13.710 align:middle line:90% So let me just label a label. 00:11:13.710 --> 00:11:15.230 align:middle line:90% So this is e minus. 00:11:15.230 --> 00:11:17.030 align:middle line:90% So this is e plus. 00:11:17.030 --> 00:11:18.020 align:middle line:90% So this would be b. 00:11:18.020 --> 00:11:20.490 align:middle line:90% And this will be b bar. 00:11:20.490 --> 00:11:22.200 align:middle line:90% So let me label the momentum. 00:11:22.200 --> 00:11:25.700 align:middle line:90% So here, let's take p1, r1. 00:11:25.700 --> 00:11:32.400 align:middle line:84% And then this would be p2, r2 bar. 00:11:32.400 --> 00:11:35.250 align:middle line:84% So tell you this is an antiparticle. 00:11:35.250 --> 00:11:39.950 align:middle line:90% And so this one is k1, s1. 00:11:39.950 --> 00:11:45.020 align:middle line:90% And this one is k2 s2 bar. 00:11:45.020 --> 00:11:46.250 align:middle line:90% OK. 00:11:46.250 --> 00:11:51.160 align:middle line:84% And so the amplitude for this will be M. So the cross 00:11:51.160 --> 00:11:53.610 align:middle line:90% section. 00:11:53.610 --> 00:11:59.130 align:middle line:84% So now, suppose we consider unpolarized process. 00:11:59.130 --> 00:12:02.315 align:middle line:84% As we discussed last time, the cross section -- 00:12:02.315 --> 00:12:06.130 align:middle line:90% 00:12:06.130 --> 00:12:13.850 align:middle line:84% yeah, so d sigma, d omega should be proportional to the spin. 00:12:13.850 --> 00:12:19.090 align:middle line:84% So we should sum over all the final states-- 00:12:19.090 --> 00:12:21.370 align:middle line:84% sum over the polarization of the final states, 00:12:21.370 --> 00:12:23.860 align:middle line:84% because we don't measure individual polarization. 00:12:23.860 --> 00:12:28.060 align:middle line:84% But we also should average over the polarization 00:12:28.060 --> 00:12:30.730 align:middle line:90% of the initial states. 00:12:30.730 --> 00:12:34.660 align:middle line:84% So the result, as we discussed at the end of last lecture, 00:12:34.660 --> 00:12:37.060 align:middle line:84% is that you need to take one quarter, come 00:12:37.060 --> 00:12:43.330 align:middle line:84% from the average over the spin of these two initial particles, 00:12:43.330 --> 00:12:48.190 align:middle line:84% and then, just sum of spins of all the particles, OK? 00:12:48.190 --> 00:12:56.890 align:middle line:90% 00:12:56.890 --> 00:12:57.490 align:middle line:90% OK? 00:12:57.490 --> 00:13:02.610 align:middle line:90% And right. 00:13:02.610 --> 00:13:04.290 align:middle line:90% OK. 00:13:04.290 --> 00:13:06.510 align:middle line:84% So that's the thing we need to-- so let's just 00:13:06.510 --> 00:13:08.860 align:middle line:90% try to first compute the m. 00:13:08.860 --> 00:13:14.910 align:middle line:90% So m is given by-- 00:13:14.910 --> 00:13:17.420 align:middle line:90% OK, so we just follow our rule. 00:13:17.420 --> 00:13:21.620 align:middle line:84% So you follow the fermionic line, OK? 00:13:21.620 --> 00:13:24.240 align:middle line:90% So here there's a vertex. 00:13:24.240 --> 00:13:26.570 align:middle line:90% So here there's a gamma. 00:13:26.570 --> 00:13:32.690 align:middle line:84% So each vertex gives you minus ie gamma mu. 00:13:32.690 --> 00:13:33.500 align:middle line:90% OK? 00:13:33.500 --> 00:13:36.890 align:middle line:84% So if you follow this one, then we 00:13:36.890 --> 00:13:52.830 align:middle line:84% get v bar r2, p2, minus ie gamma mu and ur1 p1. 00:13:52.830 --> 00:13:54.050 align:middle line:90% OK? 00:13:54.050 --> 00:13:58.450 align:middle line:84% So this is coming from this fermionic line. 00:13:58.450 --> 00:14:02.840 align:middle line:84% And from this fermionic line, we start from here. 00:14:02.840 --> 00:14:07.160 align:middle line:90% So this would be the u. 00:14:07.160 --> 00:14:07.880 align:middle line:90% Yeah. 00:14:07.880 --> 00:14:09.740 align:middle line:90% Also, we have this propagator. 00:14:09.740 --> 00:14:14.420 align:middle line:84% Let me just write down the propagator for the photon. 00:14:14.420 --> 00:14:16.490 align:middle line:84% So let's take xi equal to 1 gauge. 00:14:16.490 --> 00:14:19.580 align:middle line:90% 00:14:19.580 --> 00:14:24.220 align:middle line:84% The Lorentz gauge with xi equal to 1. 00:14:24.220 --> 00:14:25.640 align:middle line:90% And then, lets follow this line. 00:14:25.640 --> 00:14:29.620 align:middle line:84% So here, we should have a vertex gamma nu. 00:14:29.620 --> 00:14:30.160 align:middle line:90% OK. 00:14:30.160 --> 00:14:50.930 align:middle line:84% And then we will have u bar s1 k1 minus ie gamma nu v s2 k2. 00:14:50.930 --> 00:14:51.800 align:middle line:90% OK? 00:14:51.800 --> 00:14:55.090 align:middle line:90% So this is the full amplitude. 00:14:55.090 --> 00:14:59.900 align:middle line:90% So this is a number. 00:14:59.900 --> 00:15:04.570 align:middle line:84% OK, so this is a number, because this is a row vector. 00:15:04.570 --> 00:15:05.860 align:middle line:90% This is a matrix. 00:15:05.860 --> 00:15:09.220 align:middle line:84% And this is a column vector, so this will be a number. 00:15:09.220 --> 00:15:11.470 align:middle line:84% And similarly, this will be a number. 00:15:11.470 --> 00:15:15.417 align:middle line:84% And then the mu nu indices are contracted with the propagator 00:15:15.417 --> 00:15:16.000 align:middle line:90% of the photon. 00:15:16.000 --> 00:15:19.790 align:middle line:90% 00:15:19.790 --> 00:15:23.060 align:middle line:84% So actually, if I want to be precise, 00:15:23.060 --> 00:15:26.090 align:middle line:84% there's actually a factor of i here. 00:15:26.090 --> 00:15:29.550 align:middle line:84% But this factor of i is not important, because-- 00:15:29.550 --> 00:15:31.950 align:middle line:90% and then the q-- 00:15:31.950 --> 00:15:35.210 align:middle line:84% so the q is the momentum of the photon here. 00:15:35.210 --> 00:15:38.630 align:middle line:84% So the q should be the sum of the initial momentum. 00:15:38.630 --> 00:15:46.630 align:middle line:84% OK, so q should be equal to p1 plus p2. 00:15:46.630 --> 00:15:53.830 align:middle line:84% And also this means that the q square is equal to minus s, OK? 00:15:53.830 --> 00:16:00.180 align:middle line:84% Remember s is defined to be the minus p1 plus p2 square, OK? 00:16:00.180 --> 00:16:05.580 align:middle line:84% So now we can write this amplitude using a shorthand 00:16:05.580 --> 00:16:08.220 align:middle line:90% notation. 00:16:08.220 --> 00:16:10.740 align:middle line:84% So if I forget all this momentum dependence, 00:16:10.740 --> 00:16:18.150 align:middle line:84% and then this is equal to minus ie square divided by s, OK? 00:16:18.150 --> 00:16:22.170 align:middle line:84% If you cancel all the i's, and then we have two terms-- 00:16:22.170 --> 00:16:36.485 align:middle line:84% v bar r2 gamma nu and ur1 and the u bar s1 gamma nu vs2. 00:16:36.485 --> 00:16:41.910 align:middle line:90% 00:16:41.910 --> 00:16:45.520 align:middle line:90% And now we need to-- 00:16:45.520 --> 00:16:49.690 align:middle line:84% then what we need to do is we need to square it and sum over 00:16:49.690 --> 00:16:51.310 align:middle line:90% all the spin. 00:16:51.310 --> 00:16:54.790 align:middle line:84% And then we plug the resulting expression 00:16:54.790 --> 00:16:57.940 align:middle line:84% into the formula we derived for the cross section 00:16:57.940 --> 00:17:00.280 align:middle line:90% we derived last time, OK? 00:17:00.280 --> 00:17:02.890 align:middle line:90% 00:17:02.890 --> 00:17:07.329 align:middle line:84% So the whole calculation is a little bit tedious. 00:17:07.329 --> 00:17:08.980 align:middle line:84% But I think we should at least go 00:17:08.980 --> 00:17:11.500 align:middle line:84% through one such calculation, because this 00:17:11.500 --> 00:17:19.839 align:middle line:84% is the kind of prime example of QED calculation. 00:17:19.839 --> 00:17:25.569 align:middle line:84% So each of us should at least see once in our life, OK? 00:17:25.569 --> 00:17:27.535 align:middle line:84% So we will do it explicitly here. 00:17:27.535 --> 00:17:30.700 align:middle line:90% 00:17:30.700 --> 00:17:32.920 align:middle line:84% AUDIENCE: The final line needs metric. 00:17:32.920 --> 00:17:34.570 align:middle line:84% PROFESSOR: The final line is a metric. 00:17:34.570 --> 00:17:36.700 align:middle line:84% You are right, but I can do that, 00:17:36.700 --> 00:17:38.290 align:middle line:90% just put the gamma mu here. 00:17:38.290 --> 00:17:41.460 align:middle line:90% 00:17:41.460 --> 00:17:45.000 align:middle line:84% AUDIENCE: Do we to keep the i epsilon? 00:17:45.000 --> 00:17:45.960 align:middle line:90% PROFESSOR: Oh, OK. 00:17:45.960 --> 00:17:50.490 align:middle line:84% Yeah, also here, because the q square is never equal to 0, 00:17:50.490 --> 00:17:52.370 align:middle line:84% so we don't need to care about i epsilon. 00:17:52.370 --> 00:17:55.450 align:middle line:90% 00:17:55.450 --> 00:17:56.690 align:middle line:90% OK? 00:17:56.690 --> 00:17:57.890 align:middle line:90% Good. 00:17:57.890 --> 00:17:59.215 align:middle line:90% So now let's do the square. 00:17:59.215 --> 00:18:01.735 align:middle line:90% 00:18:01.735 --> 00:18:02.735 align:middle line:90% Now let's to the square. 00:18:02.735 --> 00:18:05.390 align:middle line:90% 00:18:05.390 --> 00:18:09.790 align:middle line:84% So square-- we just square this term and square this term, OK? 00:18:09.790 --> 00:18:11.365 align:middle line:90% So when we square this term-- 00:18:11.365 --> 00:18:15.650 align:middle line:90% 00:18:15.650 --> 00:18:17.230 align:middle line:90% so we also square this term. 00:18:17.230 --> 00:18:20.220 align:middle line:84% So here, we just get the e to the power 4 divided by s 00:18:20.220 --> 00:18:21.910 align:middle line:90% square. 00:18:21.910 --> 00:18:27.300 align:middle line:84% And when we square this term, we just get another-- 00:18:27.300 --> 00:18:31.270 align:middle line:84% yeah, so this is gamma nu gamma mu. 00:18:31.270 --> 00:18:36.720 align:middle line:84% So if we square it, then we get another term. 00:18:36.720 --> 00:18:38.085 align:middle line:90% So we just copy this term. 00:18:38.085 --> 00:18:41.530 align:middle line:90% 00:18:41.530 --> 00:18:42.790 align:middle line:90% Yeah, let me just copy here. 00:18:42.790 --> 00:18:50.300 align:middle line:90% 00:18:50.300 --> 00:18:56.930 align:middle line:84% ur1-- Yeah, when we square it, we get another copy of this. 00:18:56.930 --> 00:18:59.260 align:middle line:90% And so let me just write it. 00:18:59.260 --> 00:19:02.020 align:middle line:90% 00:19:02.020 --> 00:19:04.870 align:middle line:90% Yeah, sorry. 00:19:04.870 --> 00:19:07.390 align:middle line:84% So we take the complex conjugate. 00:19:07.390 --> 00:19:10.180 align:middle line:84% So when we take the complex, we need to take the modulus. 00:19:10.180 --> 00:19:12.650 align:middle line:84% So that means we need to multiply this, 00:19:12.650 --> 00:19:15.130 align:middle line:90% with its own complex conjugate. 00:19:15.130 --> 00:19:18.760 align:middle line:84% And the complex conjugate of this 00:19:18.760 --> 00:19:29.290 align:middle line:84% is given by u bar r1, then gamma nu and vr2, 00:19:29.290 --> 00:19:35.820 align:middle line:84% and then, times this one and its complex conjugate. 00:19:35.820 --> 00:19:52.390 align:middle line:84% And so we have us1 bar gamma nu vs2 and vs2 bar gamma nu us1. 00:19:52.390 --> 00:19:53.100 align:middle line:90% OK. 00:19:53.100 --> 00:19:59.160 align:middle line:84% So this is the original copy with the nu contract 00:19:59.160 --> 00:20:00.450 align:middle line:90% with this nu. 00:20:00.450 --> 00:20:03.480 align:middle line:84% And this is the complex conjugate with this nu contract 00:20:03.480 --> 00:20:04.455 align:middle line:90% with this nu, OK? 00:20:04.455 --> 00:20:12.710 align:middle line:90% 00:20:12.710 --> 00:20:13.210 align:middle line:90% Good? 00:20:13.210 --> 00:20:16.530 align:middle line:90% 00:20:16.530 --> 00:20:20.180 align:middle line:90% So this looks like a mess. 00:20:20.180 --> 00:20:23.830 align:middle line:90% OK, this look like a big mess. 00:20:23.830 --> 00:20:25.870 align:middle line:84% And it turns out actually, there's 00:20:25.870 --> 00:20:29.080 align:middle line:84% a series of beautiful tricks we can use 00:20:29.080 --> 00:20:31.240 align:middle line:90% to calculate this quantity, OK? 00:20:31.240 --> 00:20:35.350 align:middle line:84% So naively, you say, oh to calculate this thing, 00:20:35.350 --> 00:20:38.020 align:middle line:84% I have to plug-in the individual wave 00:20:38.020 --> 00:20:43.480 align:middle line:84% functions for this v and u, et cetera we worked out before. 00:20:43.480 --> 00:20:46.120 align:middle line:84% But fortunately, actually, we don't need to do it, OK? 00:20:46.120 --> 00:20:49.780 align:middle line:84% So we can actually go through a series of tricks. 00:20:49.780 --> 00:20:51.860 align:middle line:90% And then we don't have to do it. 00:20:51.860 --> 00:20:52.945 align:middle line:90% So the first trick-- 00:20:52.945 --> 00:20:55.610 align:middle line:90% 00:20:55.610 --> 00:20:59.920 align:middle line:90% you say this is a number, OK? 00:20:59.920 --> 00:21:07.260 align:middle line:84% Since this is a number, I can put a trace outside of it, 00:21:07.260 --> 00:21:11.670 align:middle line:84% because the trace of a number is just a number. 00:21:11.670 --> 00:21:14.110 align:middle line:90% OK? 00:21:14.110 --> 00:21:19.070 align:middle line:84% And now you remember the trace have this cyclic properties, 00:21:19.070 --> 00:21:19.570 align:middle line:90% OK? 00:21:19.570 --> 00:21:20.950 align:middle line:90% Trace have the cyclic property. 00:21:20.950 --> 00:21:24.740 align:middle line:84% So I can actually move this to there, OK? 00:21:24.740 --> 00:21:26.660 align:middle line:90% So I can move it. 00:21:26.660 --> 00:21:31.400 align:middle line:84% So now I can write it as e to the 4 s square. 00:21:31.400 --> 00:21:34.000 align:middle line:90% 00:21:34.000 --> 00:21:39.070 align:middle line:84% I can just move this to there so that the two come together. 00:21:39.070 --> 00:21:48.320 align:middle line:84% So we have vr2, v bar r2, gamma mu, and ur1. 00:21:48.320 --> 00:21:54.200 align:middle line:90% u bar r1 times trace. 00:21:54.200 --> 00:22:02.060 align:middle line:84% Similarly, I put this us to the other side, us1, u bar s1, 00:22:02.060 --> 00:22:04.420 align:middle line:90% gamma nu. 00:22:04.420 --> 00:22:08.665 align:middle line:90% Sorry, I missed the gamma nu. 00:22:08.665 --> 00:22:14.900 align:middle line:90% 00:22:14.900 --> 00:22:19.250 align:middle line:84% Sorry, did I-- so I put this-- oh, sorry. 00:22:19.250 --> 00:22:20.340 align:middle line:90% Oh, no. 00:22:20.340 --> 00:22:22.980 align:middle line:90% Sorry, I just missed that one. 00:22:22.980 --> 00:22:26.710 align:middle line:90% u bar, r1, gamma nu. 00:22:26.710 --> 00:22:27.870 align:middle line:90% OK. 00:22:27.870 --> 00:22:33.810 align:middle line:84% And similarly, with that, so I have u bar s 1 gamma mu. 00:22:33.810 --> 00:22:41.045 align:middle line:84% And then we have vs2, v bar s2, and gamma nu. 00:22:41.045 --> 00:22:41.545 align:middle line:90% OK? 00:22:41.545 --> 00:22:46.330 align:middle line:90% 00:22:46.330 --> 00:22:49.090 align:middle line:90% So I get that. 00:22:49.090 --> 00:22:50.550 align:middle line:90% So the reason I do that-- 00:22:50.550 --> 00:22:53.140 align:middle line:90% 00:22:53.140 --> 00:22:56.170 align:middle line:84% so now, in this form, now this is a matrix. 00:22:56.170 --> 00:22:57.760 align:middle line:90% OK, now this is-- 00:22:57.760 --> 00:22:59.680 align:middle line:90% now this is a column. 00:22:59.680 --> 00:23:01.450 align:middle line:90% This is a row. 00:23:01.450 --> 00:23:02.745 align:middle line:90% And this is a matrix. 00:23:02.745 --> 00:23:03.620 align:middle line:90% And this is a column. 00:23:03.620 --> 00:23:04.370 align:middle line:90% And this is a row. 00:23:04.370 --> 00:23:07.540 align:middle line:84% So this is a matrix times a matrix and times a matrix 00:23:07.540 --> 00:23:09.010 align:middle line:90% and times a matrix, OK? 00:23:09.010 --> 00:23:12.490 align:middle line:84% And then you take the trace and then you get the number, OK-- 00:23:12.490 --> 00:23:13.900 align:middle line:84% similarly with things since here. 00:23:13.900 --> 00:23:17.560 align:middle line:90% 00:23:17.560 --> 00:23:19.860 align:middle line:84% And then we need to sum over all the spins, OK? 00:23:19.860 --> 00:23:21.360 align:middle line:84% Remember, we need to actually-- when 00:23:21.360 --> 00:23:23.250 align:middle line:84% we sum all the spins, it means we 00:23:23.250 --> 00:23:31.820 align:middle line:84% sum over all possible with some order, r2, r1, s1, and s2, OK? 00:23:31.820 --> 00:23:36.290 align:middle line:84% So now, we can use a trick which we had when 00:23:36.290 --> 00:23:38.180 align:middle line:90% we discussed the Dirac spinors. 00:23:38.180 --> 00:23:39.080 align:middle line:90% OK, so now, recall-- 00:23:39.080 --> 00:23:43.630 align:middle line:90% 00:23:43.630 --> 00:23:50.680 align:middle line:84% now, recall for on-shell spinors means they both satisfy 00:23:50.680 --> 00:23:52.540 align:middle line:90% the equation of motion, OK? 00:23:52.540 --> 00:23:57.670 align:middle line:90% Those all the eigenspinors, OK? 00:23:57.670 --> 00:24:00.730 align:middle line:84% They satisfy the following, sum over r-- 00:24:00.730 --> 00:24:04.780 align:middle line:84% if you sum over all the r of the u, urp-- 00:24:04.780 --> 00:24:09.270 align:middle line:90% 00:24:09.270 --> 00:24:19.410 align:middle line:84% ur bar p-- that give you i, ip slash plus m, OK? 00:24:19.410 --> 00:24:37.342 align:middle line:84% And when you sum over v, you get minus i, minus ik slash plus m. 00:24:37.342 --> 00:24:37.842 align:middle line:90% OK? 00:24:37.842 --> 00:24:41.460 align:middle line:90% 00:24:41.460 --> 00:24:43.890 align:middle line:90% So now, we see we have-- 00:24:43.890 --> 00:24:48.630 align:middle line:84% if we sum over spin, then we have these combinations. 00:24:48.630 --> 00:24:51.120 align:middle line:90% Now, we can use those formulas. 00:24:51.120 --> 00:24:53.780 align:middle line:90% Now, we can use those formulas. 00:24:53.780 --> 00:25:01.740 align:middle line:84% So now we find one quarter when we sum over all the indices, 00:25:01.740 --> 00:25:09.420 align:middle line:90% r1, r2, s1, s2, the m square-- 00:25:09.420 --> 00:25:17.650 align:middle line:90% 00:25:17.650 --> 00:25:23.550 align:middle line:84% so we get e4 divided by 4 s squared. 00:25:23.550 --> 00:25:28.250 align:middle line:84% And then we just apply those formulas into here. 00:25:28.250 --> 00:25:38.160 align:middle line:84% OK, so we get trace minus ip2 slash plus m. 00:25:38.160 --> 00:25:41.100 align:middle line:90% 00:25:41.100 --> 00:25:45.330 align:middle line:84% So m is the mass of the initial particle. 00:25:45.330 --> 00:25:49.810 align:middle line:84% And I take the m prime of the mass of the final particle. 00:25:49.810 --> 00:25:55.650 align:middle line:84% So the e plus minus of mass m and b, 00:25:55.650 --> 00:25:59.550 align:middle line:90% b bar have mass m prime, OK? 00:25:59.550 --> 00:26:05.500 align:middle line:90% 00:26:05.500 --> 00:26:11.685 align:middle line:84% So then I have this, then this times, then this gamma mu. 00:26:11.685 --> 00:26:14.450 align:middle line:90% 00:26:14.450 --> 00:26:16.340 align:middle line:84% Yeah, so this is still inside this bracket. 00:26:16.340 --> 00:26:20.220 align:middle line:90% 00:26:20.220 --> 00:26:24.135 align:middle line:90% ip1 slash plus m, gamma nu. 00:26:24.135 --> 00:26:26.810 align:middle line:90% 00:26:26.810 --> 00:26:40.850 align:middle line:84% And then I have trace ik1 slash plus m prime, gamma mu, 00:26:40.850 --> 00:26:48.980 align:middle line:84% and then, minus ik2 slash plus m prime gamma nu. 00:26:48.980 --> 00:26:50.440 align:middle line:90% OK? 00:26:50.440 --> 00:26:53.620 align:middle line:84% So I just-- when I do this-- so the spin sum 00:26:53.620 --> 00:26:56.380 align:middle line:84% is crucial for this simplification. 00:26:56.380 --> 00:27:01.300 align:middle line:84% When I sum over spins, and then without using 00:27:01.300 --> 00:27:07.360 align:middle line:84% explicit expression of those u and v, we can write all of them 00:27:07.360 --> 00:27:10.930 align:middle line:84% just in terms of p slash and k slash, et cetera. 00:27:10.930 --> 00:27:13.930 align:middle line:90% 00:27:13.930 --> 00:27:16.600 align:middle line:90% Any questions on this? 00:27:16.600 --> 00:27:20.190 align:middle line:84% And now we can evaluate the trace. 00:27:20.190 --> 00:27:22.710 align:middle line:90% OK, so evaluate the trace. 00:27:22.710 --> 00:27:25.050 align:middle line:84% Now we need to use the various formula. 00:27:25.050 --> 00:27:29.280 align:middle line:84% I think some of them you have done in your pset. 00:27:29.280 --> 00:27:32.430 align:middle line:90% Is that now recall. 00:27:32.430 --> 00:27:38.640 align:middle line:84% So this is the place, all those exercises, they become useful. 00:27:38.640 --> 00:27:40.910 align:middle line:90% So become mu. 00:27:40.910 --> 00:27:46.080 align:middle line:84% So here, if you look at the structure of these terms-- 00:27:46.080 --> 00:27:48.122 align:middle line:90% so here, there's a p slash. 00:27:48.122 --> 00:27:51.480 align:middle line:84% So there's one gamma matrix here. 00:27:51.480 --> 00:27:53.880 align:middle line:84% And there's another gamma matrix here. 00:27:53.880 --> 00:27:56.002 align:middle line:84% And here, p slash is the gamma matrix here. 00:27:56.002 --> 00:27:57.210 align:middle line:90% There's another gamma matrix. 00:27:57.210 --> 00:28:00.090 align:middle line:84% So at the most, we can have the product of the 4 gamma 00:28:00.090 --> 00:28:02.100 align:middle line:90% matrices inside the trace. 00:28:02.100 --> 00:28:03.780 align:middle line:84% And also, depend on the product, we 00:28:03.780 --> 00:28:06.270 align:middle line:90% can have four, three, and two. 00:28:06.270 --> 00:28:07.350 align:middle line:90% OK? 00:28:07.350 --> 00:28:09.360 align:middle line:90% Three and two. 00:28:09.360 --> 00:28:15.960 align:middle line:84% So for this purpose, we need to delete those formulas. 00:28:15.960 --> 00:28:21.480 align:middle line:84% What are the product of the four matrices inside the integral? 00:28:21.480 --> 00:28:30.330 align:middle line:84% So this is just given by eta mu nu, eta lambda rho, plus eta mu 00:28:30.330 --> 00:28:32.070 align:middle line:90% rho-- 00:28:32.070 --> 00:28:44.580 align:middle line:84% eta mu rho eta mu lambda minus eta mu lambda eta mu rho. 00:28:44.580 --> 00:28:47.160 align:middle line:90% OK? 00:28:47.160 --> 00:28:51.380 align:middle line:84% And also if you have three gamma matrices, 00:28:51.380 --> 00:28:52.380 align:middle line:90% do you know the answer? 00:28:52.380 --> 00:28:54.885 align:middle line:90% 00:28:54.885 --> 00:28:55.770 align:middle line:90% AUDIENCE: Zero. 00:28:55.770 --> 00:28:57.210 align:middle line:90% PROFESSOR: Yes. 00:28:57.210 --> 00:28:58.020 align:middle line:90% Good. 00:28:58.020 --> 00:29:03.960 align:middle line:84% And if I have two gamma matrices, 00:29:03.960 --> 00:29:06.285 align:middle line:90% do you know what is this? 00:29:06.285 --> 00:29:07.230 align:middle line:90% AUDIENCE: 4 eta mu nu. 00:29:07.230 --> 00:29:09.280 align:middle line:84% PROFESSOR: That's right, it's 4 eta mu mu. 00:29:09.280 --> 00:29:09.780 align:middle line:90% OK. 00:29:09.780 --> 00:29:14.040 align:middle line:84% So you can just apply those formulas to here. 00:29:14.040 --> 00:29:18.390 align:middle line:84% So we'll not do the calculation explicitly. 00:29:18.390 --> 00:29:22.500 align:middle line:90% So you can just plug them in. 00:29:22.500 --> 00:29:24.480 align:middle line:90% And then, it's mechanical. 00:29:24.480 --> 00:29:26.100 align:middle line:90% You can just calculate it. 00:29:26.100 --> 00:29:37.800 align:middle line:84% And then you find 1 over 4 sum over spin m squared 00:29:37.800 --> 00:29:40.780 align:middle line:84% is equal to-- let me write down the answer. 00:29:40.780 --> 00:29:42.660 align:middle line:84% So even though this answer is slightly long, 00:29:42.660 --> 00:29:44.580 align:middle line:84% but I think it's instructive to write it 00:29:44.580 --> 00:29:46.660 align:middle line:90% down so that you see it. 00:29:46.660 --> 00:29:52.530 align:middle line:84% So when you plug all this in and do the algebras, 00:29:52.530 --> 00:29:57.730 align:middle line:84% and you find that the expression can be written as this, 00:29:57.730 --> 00:30:14.570 align:middle line:84% k1 dot p1 k2 dot p2 plus p1 dot k2 and p2 dot k1. 00:30:14.570 --> 00:30:24.484 align:middle line:84% Then minus m square k1 dot k2 minus m square p1 00:30:24.484 --> 00:30:28.130 align:middle line:90% dot p2 m prime square. 00:30:28.130 --> 00:30:29.210 align:middle line:90% Sorry. 00:30:29.210 --> 00:30:39.545 align:middle line:84% Here it should be m prime square plus 2m square m prime square. 00:30:39.545 --> 00:30:40.045 align:middle line:90% OK. 00:30:40.045 --> 00:30:43.020 align:middle line:90% 00:30:43.020 --> 00:30:49.020 align:middle line:84% So we discussed before that the amplitude square 00:30:49.020 --> 00:30:51.770 align:middle line:90% must be a Lorentz scalar. 00:30:51.770 --> 00:30:55.410 align:middle line:90% OK, it must be a Lorentz scalar. 00:30:55.410 --> 00:30:58.020 align:middle line:84% And so you see it explicitly, here. 00:30:58.020 --> 00:31:02.570 align:middle line:84% So it only involves either m square, m prime square, 00:31:02.570 --> 00:31:06.170 align:middle line:84% or the product dot product of the initial and final momentum, 00:31:06.170 --> 00:31:06.890 align:middle line:90% OK? 00:31:06.890 --> 00:31:09.380 align:middle line:84% And you see all kinds of combination 00:31:09.380 --> 00:31:14.690 align:middle line:84% of the initial and final momentum appearing here. 00:31:14.690 --> 00:31:18.510 align:middle line:90% And now, so it's convention-- 00:31:18.510 --> 00:31:20.550 align:middle line:84% so as we mentioned before, actually, 00:31:20.550 --> 00:31:23.640 align:middle line:84% if you look at all these different dot 00:31:23.640 --> 00:31:28.090 align:middle line:84% product between initial and final momentum, 00:31:28.090 --> 00:31:34.100 align:middle line:84% actually, because of the momentum conservation-- 00:31:34.100 --> 00:31:41.170 align:middle line:84% And actually, there are only two Lorentz invariant, OK? 00:31:41.170 --> 00:31:45.220 align:middle line:84% So it would be convenient to write 00:31:45.220 --> 00:31:50.650 align:middle line:84% this or this in terms of those two Lorentz invariant, OK? 00:31:50.650 --> 00:31:55.030 align:middle line:84% So that's the two of the stu variables 00:31:55.030 --> 00:31:56.770 align:middle line:90% we defined previously. 00:31:56.770 --> 00:32:00.230 align:middle line:84% So it's often just to use stu the same time. 00:32:00.230 --> 00:32:04.000 align:middle line:90% So we just-- yeah. 00:32:04.000 --> 00:32:08.970 align:middle line:84% So now, if you write them in terms of stu, 00:32:08.970 --> 00:32:10.980 align:middle line:90% and then you find, for example-- 00:32:10.980 --> 00:32:12.910 align:middle line:84% let me just give you some example. 00:32:12.910 --> 00:32:15.438 align:middle line:90% For example, 2k1 p1. 00:32:15.438 --> 00:32:18.060 align:middle line:90% 00:32:18.060 --> 00:32:21.960 align:middle line:84% So this can be written as t minus m 00:32:21.960 --> 00:32:25.830 align:middle line:84% square minus m prime square, which 00:32:25.830 --> 00:32:33.300 align:middle line:90% is the same as 2k2 dot p2, OK? 00:32:33.300 --> 00:32:42.660 align:middle line:84% And 2k1 dot p2, then it's actually equal to 2k2 dot p1, 00:32:42.660 --> 00:32:48.600 align:middle line:84% then equal to u minus m square minus m prime square. 00:32:48.600 --> 00:32:53.130 align:middle line:84% And then, since related to the initial momentum-- 00:32:53.130 --> 00:32:59.280 align:middle line:84% then related to the s equal to s minus 2m square and minus 00:32:59.280 --> 00:33:05.430 align:middle line:84% k1 k2 equal to s minus 2m prime square. 00:33:05.430 --> 00:33:11.210 align:middle line:84% OK, So you just apply those equations in here. 00:33:11.210 --> 00:33:14.120 align:middle line:84% Then, you can write them in terms of stu variables. 00:33:14.120 --> 00:33:22.100 align:middle line:84% And then, you find that one over sum over spin m squared 00:33:22.100 --> 00:33:28.610 align:middle line:84% is equal to the following expression, 2e to the power 4 s 00:33:28.610 --> 00:33:39.930 align:middle line:84% square t minus m square m prime square square 00:33:39.930 --> 00:33:47.670 align:middle line:84% plus u minus m square minus m prime square square plus 2s. 00:33:47.670 --> 00:33:51.350 align:middle line:90% 00:33:51.350 --> 00:33:53.330 align:middle line:90% OK. 00:33:53.330 --> 00:33:59.420 align:middle line:84% So recall that s plus t plus u equal to 2. 00:33:59.420 --> 00:34:02.420 align:middle line:90% 00:34:02.420 --> 00:34:07.000 align:middle line:84% So you can eliminate one of them. 00:34:07.000 --> 00:34:10.090 align:middle line:84% So you can eliminate one of them. 00:34:10.090 --> 00:34:14.989 align:middle line:84% So this concludes the calculation 00:34:14.989 --> 00:34:18.350 align:middle line:90% of the scattering amplitude. 00:34:18.350 --> 00:34:21.080 align:middle line:84% OK, so that's the expression you get, OK? 00:34:21.080 --> 00:34:23.060 align:middle line:84% And you see that it's all expressed 00:34:23.060 --> 00:34:24.620 align:middle line:90% in terms of this stu-- 00:34:24.620 --> 00:34:30.909 align:middle line:84% can be expressed nicely in terms of these stu variables. 00:34:30.909 --> 00:34:32.170 align:middle line:90% Any questions on this? 00:34:32.170 --> 00:34:37.389 align:middle line:90% 00:34:37.389 --> 00:34:39.130 align:middle line:90% Good. 00:34:39.130 --> 00:34:42.730 align:middle line:84% So now, let's calculate the cross section. 00:34:42.730 --> 00:34:45.460 align:middle line:84% And our goal will be to calculate the total cross 00:34:45.460 --> 00:34:47.370 align:middle line:90% section, OK? 00:34:47.370 --> 00:34:50.290 align:middle line:84% And so for this purpose, let's do the center of mass frame, 00:34:50.290 --> 00:34:50.790 align:middle line:90% OK? 00:34:50.790 --> 00:34:54.120 align:middle line:90% 00:34:54.120 --> 00:34:56.699 align:middle line:84% So let's go to consider the center of mass frame. 00:34:56.699 --> 00:34:59.640 align:middle line:90% 00:34:59.640 --> 00:35:01.715 align:middle line:90% Well, this is the simplest. 00:35:01.715 --> 00:35:05.370 align:middle line:90% 00:35:05.370 --> 00:35:08.675 align:middle line:84% So remember, in the center of mass frame we-- 00:35:08.675 --> 00:35:11.900 align:middle line:90% 00:35:11.900 --> 00:35:14.690 align:middle line:90% so first we need to-- 00:35:14.690 --> 00:35:17.120 align:middle line:84% yeah, so remember the center of mass frame 00:35:17.120 --> 00:35:30.085 align:middle line:84% you get that d sigma, d omega cm is given by-- 00:35:30.085 --> 00:35:36.780 align:middle line:90% 00:35:36.780 --> 00:35:37.280 align:middle line:90% yeah. 00:35:37.280 --> 00:35:39.590 align:middle line:90% Let me check my formula. 00:35:39.590 --> 00:35:44.860 align:middle line:84% Suddenly I'm not sure what I'm looking for. 00:35:44.860 --> 00:35:49.330 align:middle line:84% Yeah, so it's given by, say, the M 00:35:49.330 --> 00:36:00.230 align:middle line:84% square divided by p cm prime divided by 64 pi square 00:36:00.230 --> 00:36:03.890 align:middle line:90% s pcm prime. 00:36:03.890 --> 00:36:09.110 align:middle line:84% So pcm prime is the momentum for the final particles. 00:36:09.110 --> 00:36:12.440 align:middle line:84% And pcm is the momentum in the center of mass frame 00:36:12.440 --> 00:36:14.420 align:middle line:90% for the initial particles. 00:36:14.420 --> 00:36:17.040 align:middle line:90% 00:36:17.040 --> 00:36:22.980 align:middle line:84% So now, we also need to find the pcm, et cetera, OK? 00:36:22.980 --> 00:36:28.440 align:middle line:84% We also need to express the m, the amplitude in terms 00:36:28.440 --> 00:36:31.800 align:middle line:84% of the center of mass square, OK? 00:36:31.800 --> 00:36:35.070 align:middle line:84% So in the center of mass frame, then 00:36:35.070 --> 00:36:38.910 align:middle line:84% we have the p1 would be, say, E, pcm. 00:36:38.910 --> 00:36:42.660 align:middle line:90% 00:36:42.660 --> 00:36:49.200 align:middle line:84% OK, the p2 will be equal to the same E, minus pcm. 00:36:49.200 --> 00:36:50.790 align:middle line:84% OK, they have the same E, because they 00:36:50.790 --> 00:36:53.930 align:middle line:90% have the same mass. 00:36:53.930 --> 00:37:08.190 align:middle line:84% Similarly, k1 would be E prime, pcm prime. 00:37:08.190 --> 00:37:17.310 align:middle line:84% And the k2 would be E prime, minus pcm prime. 00:37:17.310 --> 00:37:22.490 align:middle line:90% 00:37:22.490 --> 00:37:25.310 align:middle line:84% So now, from energy conservation, 00:37:25.310 --> 00:37:28.730 align:middle line:84% immediately concluded that E must be equal to E prime. 00:37:28.730 --> 00:37:33.200 align:middle line:84% So I can forget about E prime, because the total energy 00:37:33.200 --> 00:37:34.340 align:middle line:90% will be 2E. 00:37:34.340 --> 00:37:36.510 align:middle line:84% And the final energy will also be 2e. 00:37:36.510 --> 00:37:40.540 align:middle line:84% So e have to be equal to e prime. 00:37:40.540 --> 00:37:42.620 align:middle line:90% And then we can find-- 00:37:42.620 --> 00:37:49.940 align:middle line:84% And so E would be equal to one half square root of s, OK? 00:37:49.940 --> 00:37:51.620 align:middle line:90% So this is the-- 00:37:51.620 --> 00:37:54.360 align:middle line:90% 00:37:54.360 --> 00:38:01.110 align:middle line:84% or s is equal to 4E square, because the p1 plus p2 00:38:01.110 --> 00:38:02.550 align:middle line:90% is just 2E. 00:38:02.550 --> 00:38:05.070 align:middle line:84% You square it, it becomes 4E square. 00:38:05.070 --> 00:38:13.500 align:middle line:84% And then we find the pcm square equal to s 00:38:13.500 --> 00:38:16.700 align:middle line:90% divided by 4 minus m square. 00:38:16.700 --> 00:38:23.165 align:middle line:84% And the pcm prime square equal to 00:38:23.165 --> 00:38:26.360 align:middle line:84% s divided by 4 minus s prime square. 00:38:26.360 --> 00:38:39.320 align:middle line:90% 00:38:39.320 --> 00:38:42.330 align:middle line:90% Any questions on this? 00:38:42.330 --> 00:38:43.573 align:middle line:90% Yes. 00:38:43.573 --> 00:38:47.810 align:middle line:84% AUDIENCE: We know that [INAUDIBLE] first component 00:38:47.810 --> 00:38:50.012 align:middle line:84% of your 4 vector, they have some [INAUDIBLE].. 00:38:50.012 --> 00:38:51.470 align:middle line:84% But how do you know that it exactly 00:38:51.470 --> 00:38:55.070 align:middle line:90% has to split [INAUDIBLE]? 00:38:55.070 --> 00:38:55.790 align:middle line:90% PROFESSOR: Sorry? 00:38:55.790 --> 00:38:58.332 align:middle line:84% AUDIENCE: How do you know that the final state particles have 00:38:58.332 --> 00:38:59.647 align:middle line:90% to have the same energy? 00:38:59.647 --> 00:39:00.230 align:middle line:90% PROFESSOR: Oh. 00:39:00.230 --> 00:39:02.060 align:middle line:84% AUDIENCE: But you know that there's some that have to be-- 00:39:02.060 --> 00:39:03.290 align:middle line:84% PROFESSOR: Yeah, because they have the same mass. 00:39:03.290 --> 00:39:03.980 align:middle line:90% AUDIENCE: I see. 00:39:03.980 --> 00:39:04.430 align:middle line:90% PROFESSOR: Yeah. 00:39:04.430 --> 00:39:06.138 align:middle line:84% Yeah, because they have momentum opposite 00:39:06.138 --> 00:39:08.180 align:middle line:84% to each other in the center of mass frame. 00:39:08.180 --> 00:39:09.350 align:middle line:90% And they have the same mass. 00:39:09.350 --> 00:39:11.433 align:middle line:84% AUDIENCE: So you can allow for your final particle 00:39:11.433 --> 00:39:13.910 align:middle line:84% to have-- final particles to have different masses and then 00:39:13.910 --> 00:39:14.510 align:middle line:90% [INAUDIBLE]. 00:39:14.510 --> 00:39:16.040 align:middle line:90% PROFESSOR: Yeah, yeah. 00:39:16.040 --> 00:39:17.600 align:middle line:84% Yeah, but in this process, we are 00:39:17.600 --> 00:39:20.660 align:middle line:84% considering the-- always the particle and the antiparticle. 00:39:20.660 --> 00:39:21.890 align:middle line:90% It's an annihilation process. 00:39:21.890 --> 00:39:26.500 align:middle line:84% We always create the particle and the antiparticle together. 00:39:26.500 --> 00:39:29.640 align:middle line:90% Other questions? 00:39:29.640 --> 00:39:35.530 align:middle line:84% OK, So you can similarly express the t. 00:39:35.530 --> 00:39:37.660 align:middle line:84% OK, you can similarly express the t 00:39:37.660 --> 00:39:43.810 align:middle line:84% and u in terms of this center of mass, the momentum, et cetera. 00:39:43.810 --> 00:39:48.400 align:middle line:90% and So you can just do that. 00:39:48.400 --> 00:39:53.223 align:middle line:84% So let me just write down the final answer. 00:39:53.223 --> 00:39:54.890 align:middle line:84% Let me just write down the final answer. 00:39:54.890 --> 00:40:00.640 align:middle line:84% You can also rewrite 1/4 sum spin, 00:40:00.640 --> 00:40:07.160 align:middle line:84% m squared equal to, in terms of the center of mass. 00:40:07.160 --> 00:40:11.840 align:middle line:84% And then you find it's equal to e to the power 4, 00:40:11.840 --> 00:40:17.600 align:middle line:84% 1 plus m squared plus m prime squared 00:40:17.600 --> 00:40:28.860 align:middle line:84% divided by E squared plus pcm squared, pcm 00:40:28.860 --> 00:40:30.840 align:middle line:90% prime squared cosine theta. 00:40:30.840 --> 00:40:34.850 align:middle line:90% 00:40:34.850 --> 00:40:39.590 align:middle line:90% OK, and the theta is the-- 00:40:39.590 --> 00:40:45.860 align:middle line:84% so you have the scattering of the initial particle with pcm. 00:40:45.860 --> 00:40:51.980 align:middle line:84% And then you have the finite particle with the pcm prime. 00:40:51.980 --> 00:40:54.770 align:middle line:90% And so this angle is theta, OK? 00:40:54.770 --> 00:40:55.730 align:middle line:90% This angle is theta. 00:40:55.730 --> 00:40:59.800 align:middle line:90% 00:40:59.800 --> 00:41:04.920 align:middle line:84% And so the theta comes from when you calculate using the-- 00:41:04.920 --> 00:41:07.950 align:middle line:90% involving t and u, OK? 00:41:07.950 --> 00:41:10.670 align:middle line:90% Involving t and u. 00:41:10.670 --> 00:41:14.060 align:middle line:90% And so now we have everything. 00:41:14.060 --> 00:41:15.340 align:middle line:90% So we have the pcm. 00:41:15.340 --> 00:41:18.890 align:middle line:90% 00:41:18.890 --> 00:41:21.640 align:middle line:90% It's all in terms of s. 00:41:21.640 --> 00:41:26.800 align:middle line:84% And then your amplitude is also expressed in terms of either s 00:41:26.800 --> 00:41:29.545 align:middle line:90% or E. Yeah, E is also the s. 00:41:29.545 --> 00:41:31.920 align:middle line:90% And then we have the theta, OK? 00:41:31.920 --> 00:41:35.350 align:middle line:90% Then we have the theta. 00:41:35.350 --> 00:41:38.170 align:middle line:84% So you just plug the whole thing into here. 00:41:38.170 --> 00:41:39.880 align:middle line:84% Just plug the whole thing into here. 00:41:39.880 --> 00:41:42.490 align:middle line:84% And then you find the total cross-section. 00:41:42.490 --> 00:41:46.300 align:middle line:90% 00:41:46.300 --> 00:41:53.830 align:middle line:84% You just do d sigma, d omega, center of mass frame, 00:41:53.830 --> 00:41:58.540 align:middle line:84% and then sine theta d theta d phi. 00:41:58.540 --> 00:42:00.370 align:middle line:90% OK, nothing here depends on phi. 00:42:00.370 --> 00:42:03.220 align:middle line:84% So the phi integral, you can just done trivially. 00:42:03.220 --> 00:42:05.230 align:middle line:84% And then you just need to do the theta integral. 00:42:05.230 --> 00:42:06.380 align:middle line:90% And then you have this. 00:42:06.380 --> 00:42:09.370 align:middle line:90% 00:42:09.370 --> 00:42:14.880 align:middle line:84% So again, let me just write down the final answer. 00:42:14.880 --> 00:42:22.070 align:middle line:84% So in the end, you get 4 pi alpha squared divided by s. 00:42:22.070 --> 00:42:26.340 align:middle line:84% So the alpha is the fine-structure constant. 00:42:26.340 --> 00:42:32.160 align:middle line:84% Yeah, so the alpha is e squared divided by 4 pi, 00:42:32.160 --> 00:42:34.300 align:middle line:90% is the fine-structure constant. 00:42:34.300 --> 00:42:40.080 align:middle line:84% And this is 4 pi alpha squared, 3s, 00:42:40.080 --> 00:42:49.130 align:middle line:84% and then 1 minus m prime squared divided by e squared, 1 00:42:49.130 --> 00:42:52.675 align:middle line:84% minus m squared divided by e squared. 00:42:52.675 --> 00:42:55.580 align:middle line:90% 00:42:55.580 --> 00:43:02.590 align:middle line:84% And the whole thing times 1 plus m squared divided by 2e 00:43:02.590 --> 00:43:03.090 align:middle line:90% squared. 00:43:03.090 --> 00:43:05.700 align:middle line:90% 00:43:05.700 --> 00:43:09.330 align:middle line:84% And the 1 plus m prime squared divided by 2e squared. 00:43:09.330 --> 00:43:13.910 align:middle line:90% 00:43:13.910 --> 00:43:16.460 align:middle line:90% So this is the final answer. 00:43:16.460 --> 00:43:20.220 align:middle line:84% And this is the answer you can compare with experiments. 00:43:20.220 --> 00:43:22.577 align:middle line:84% So this is also you can compare with experiments. 00:43:22.577 --> 00:43:28.660 align:middle line:90% 00:43:28.660 --> 00:43:31.310 align:middle line:90% Any questions? 00:43:31.310 --> 00:43:35.910 align:middle line:84% So now, if you look at this answer, 00:43:35.910 --> 00:43:40.590 align:middle line:84% it's almost completely symmetric in terms of m and m prime 00:43:40.590 --> 00:43:43.980 align:middle line:84% except that for this ratio, the m is downstairs 00:43:43.980 --> 00:43:46.230 align:middle line:90% and m prime is upstairs. 00:43:46.230 --> 00:43:48.624 align:middle line:90% Otherwise, it's the-- yeah. 00:43:48.624 --> 00:43:53.640 align:middle line:90% 00:43:53.640 --> 00:43:55.590 align:middle line:90% So this m prime is upstairs. 00:43:55.590 --> 00:43:58.830 align:middle line:90% Of course, it's the-- 00:43:58.830 --> 00:44:01.160 align:middle line:90% do I have the-- 00:44:01.160 --> 00:44:02.000 align:middle line:90% yeah, yeah. 00:44:02.000 --> 00:44:04.880 align:middle line:90% 00:44:04.880 --> 00:44:05.380 align:middle line:90% Good. 00:44:05.380 --> 00:44:10.280 align:middle line:90% Yeah, so this factor is key. 00:44:10.280 --> 00:44:12.080 align:middle line:84% OK, you have to have such kind of factor 00:44:12.080 --> 00:44:15.260 align:middle line:84% in the upstairs because of when E, 00:44:15.260 --> 00:44:19.620 align:middle line:84% you have to be E to be greater than m prime squared in order 00:44:19.620 --> 00:44:22.120 align:middle line:90% to produce the particle. 00:44:22.120 --> 00:44:26.770 align:middle line:84% If your energy is not big enough to produce-- 00:44:26.770 --> 00:44:30.130 align:middle line:84% in order to produce a b and b bar, your energy at least-- 00:44:30.130 --> 00:44:33.850 align:middle line:90% 00:44:33.850 --> 00:44:37.630 align:middle line:84% so each E has to be large enough to be greater 00:44:37.630 --> 00:44:43.180 align:middle line:84% than m, because your 2E, total energy, 00:44:43.180 --> 00:44:44.740 align:middle line:90% must be greater than 2m prime. 00:44:44.740 --> 00:44:47.530 align:middle line:90% 00:44:47.530 --> 00:44:53.740 align:middle line:84% And so you see that when E is smaller than m prime 00:44:53.740 --> 00:44:57.550 align:middle line:84% and when E equals to m prime, then this factor becomes 0. 00:44:57.550 --> 00:44:59.170 align:middle line:84% And then the cross-section becomes 0. 00:44:59.170 --> 00:45:02.700 align:middle line:90% 00:45:02.700 --> 00:45:03.960 align:middle line:90% Any questions on this? 00:45:03.960 --> 00:45:07.510 align:middle line:90% 00:45:07.510 --> 00:45:11.230 align:middle line:84% OK, so now let's consider some specific cases. 00:45:11.230 --> 00:45:45.670 align:middle line:90% 00:45:45.670 --> 00:45:57.480 align:middle line:84% So now let's consider e plus, e minus going to muon. 00:45:57.480 --> 00:46:06.350 align:middle line:84% So the muon is the next massive lepton after the electron. 00:46:06.350 --> 00:46:14.010 align:middle line:84% So in this case, say me -- m square will be m e squared. 00:46:14.010 --> 00:46:19.530 align:middle line:84% And the m prime squared will be m mu squared. 00:46:19.530 --> 00:46:22.820 align:middle line:90% And the m mu-- 00:46:22.820 --> 00:46:26.420 align:middle line:90% so the muon mass is 207 MeV. 00:46:26.420 --> 00:46:29.880 align:middle line:90% 00:46:29.880 --> 00:46:32.020 align:middle line:90% Oh, no. 00:46:32.020 --> 00:46:33.105 align:middle line:90% No, it's the 100-- 00:46:33.105 --> 00:46:36.702 align:middle line:90% 00:46:36.702 --> 00:46:38.160 align:middle line:84% actually, I didn't code the number. 00:46:38.160 --> 00:46:44.105 align:middle line:84% So the muon mass is 207 times m, the mass of the electron. 00:46:44.105 --> 00:46:46.870 align:middle line:90% 00:46:46.870 --> 00:46:49.840 align:middle line:84% So does anybody remember the electron mass? 00:46:49.840 --> 00:46:53.067 align:middle line:90% 00:46:53.067 --> 00:46:54.735 align:middle line:90% AUDIENCE: [INAUDIBLE]? 00:46:54.735 --> 00:46:55.360 align:middle line:90% PROFESSOR: Hmm? 00:46:55.360 --> 00:46:56.790 align:middle line:90% AUDIENCE: 0.511 MeV. 00:46:56.790 --> 00:46:58.040 align:middle line:90% PROFESSOR: Yeah, that's right. 00:46:58.040 --> 00:47:00.880 align:middle line:90% Yeah, yeah, 0.5-- 00:47:00.880 --> 00:47:01.750 align:middle line:90% 1? 00:47:01.750 --> 00:47:02.727 align:middle line:90% Did you say 1? 00:47:02.727 --> 00:47:03.310 align:middle line:90% AUDIENCE: Yes. 00:47:03.310 --> 00:47:04.360 align:middle line:90% PROFESSOR: Yeah, yeah. 00:47:04.360 --> 00:47:07.210 align:middle line:90% Yeah, it's 511 KeV. 00:47:07.210 --> 00:47:12.010 align:middle line:90% Yeah, 0.511 MeV. 00:47:12.010 --> 00:47:16.090 align:middle line:84% And the muon is the 207 times the mass of the electron. 00:47:16.090 --> 00:47:18.730 align:middle line:90% 00:47:18.730 --> 00:47:22.240 align:middle line:84% So if you want to create the muon, then 00:47:22.240 --> 00:47:23.905 align:middle line:90% the initial total energy-- 00:47:23.905 --> 00:47:27.730 align:middle line:90% 00:47:27.730 --> 00:47:32.170 align:middle line:84% so the square root s, which is 2E, 00:47:32.170 --> 00:47:36.370 align:middle line:84% has to be greater than the 2 m mu. 00:47:36.370 --> 00:47:39.340 align:middle line:84% So that means that the E has to be greater than m mu. 00:47:39.340 --> 00:47:42.010 align:middle line:90% 00:47:42.010 --> 00:47:44.920 align:middle line:84% And then this is much, much larger than the electron mass. 00:47:44.920 --> 00:47:50.230 align:middle line:90% 00:47:50.230 --> 00:47:52.870 align:middle line:84% So this is about 200 times larger 00:47:52.870 --> 00:47:57.936 align:middle line:84% than m E. So that means actually that, 00:47:57.936 --> 00:48:01.935 align:middle line:84% for all purposes, that you can just neglect the electron mass. 00:48:01.935 --> 00:48:06.610 align:middle line:90% 00:48:06.610 --> 00:48:09.870 align:middle line:90% So this ratio is tiny. 00:48:09.870 --> 00:48:10.680 align:middle line:90% It's tiny. 00:48:10.680 --> 00:48:15.203 align:middle line:84% So we can treat essentially the electron as massless. 00:48:15.203 --> 00:48:16.786 align:middle line:84% AUDIENCE: [INAUDIBLE] the denominator, 00:48:16.786 --> 00:48:22.060 align:middle line:84% and we added those to the [INAUDIBLE].. 00:48:22.060 --> 00:48:25.508 align:middle line:84% PROFESSOR: Yeah, just not well defined, yeah. 00:48:25.508 --> 00:48:28.760 align:middle line:84% AUDIENCE: [INAUDIBLE] makes sense that it would. 00:48:28.760 --> 00:48:30.990 align:middle line:84% PROFESSOR: No, you don't even-- yeah, in this case, 00:48:30.990 --> 00:48:32.115 align:middle line:90% you don't even go to that-- 00:48:32.115 --> 00:48:34.940 align:middle line:90% 00:48:34.940 --> 00:48:38.930 align:middle line:90% so in the case that the-- 00:48:38.930 --> 00:48:40.220 align:middle line:90% yeah. 00:48:40.220 --> 00:48:40.730 align:middle line:90% Let me see. 00:48:40.730 --> 00:48:46.260 align:middle line:90% 00:48:46.260 --> 00:48:48.885 align:middle line:90% Let me think a little bit. 00:48:48.885 --> 00:48:49.385 align:middle line:90% Yeah. 00:48:49.385 --> 00:48:53.150 align:middle line:90% 00:48:53.150 --> 00:48:55.010 align:middle line:90% Right. 00:48:55.010 --> 00:48:58.200 align:middle line:84% So yeah, for this case, of course it's not well defined. 00:48:58.200 --> 00:49:00.230 align:middle line:84% But in the opposite case, let's imagine 00:49:00.230 --> 00:49:05.000 align:middle line:84% if you have the two very massive particles to collide 00:49:05.000 --> 00:49:07.610 align:middle line:90% to create a lighter particle. 00:49:07.610 --> 00:49:10.918 align:middle line:90% And then in that case-- 00:49:10.918 --> 00:49:12.710 align:middle line:84% yeah, actually, I don't remember the answer 00:49:12.710 --> 00:49:13.668 align:middle line:90% from my top of my head. 00:49:13.668 --> 00:49:15.030 align:middle line:90% Yeah, it's a good question. 00:49:15.030 --> 00:49:19.510 align:middle line:84% I will try to find it afterwards. 00:49:19.510 --> 00:49:22.350 align:middle line:90% Any other questions? 00:49:22.350 --> 00:49:24.680 align:middle line:90% OK, good. 00:49:24.680 --> 00:49:28.550 align:middle line:84% So in this case, so when the electron 00:49:28.550 --> 00:49:30.260 align:middle line:90% mass is much larger than-- 00:49:30.260 --> 00:49:35.030 align:middle line:84% when the muon mass is much larger than the electron mass, 00:49:35.030 --> 00:49:44.630 align:middle line:84% then we can set essentially m e equal to 0. 00:49:44.630 --> 00:49:46.790 align:middle line:84% And in this case, the formula becomes simpler. 00:49:46.790 --> 00:49:49.900 align:middle line:90% 00:49:49.900 --> 00:49:52.000 align:middle line:84% In this case, it becomes simpler. 00:49:52.000 --> 00:49:55.890 align:middle line:84% The downstairs is just equal to 1. 00:49:55.890 --> 00:49:59.830 align:middle line:90% And this factor also becomes 1. 00:49:59.830 --> 00:50:03.420 align:middle line:90% This factor also becomes 1. 00:50:03.420 --> 00:50:05.385 align:middle line:84% So then there are two interesting regimes. 00:50:05.385 --> 00:50:15.510 align:middle line:90% 00:50:15.510 --> 00:50:18.260 align:middle line:84% There are two interesting regimes. 00:50:18.260 --> 00:50:20.980 align:middle line:84% So the first is you just near the threshold. 00:50:20.980 --> 00:50:36.980 align:middle line:90% 00:50:36.980 --> 00:50:39.380 align:middle line:84% OK, just near the threshold means 00:50:39.380 --> 00:50:46.645 align:middle line:84% that the pcm prime is much, much smaller than m mu. 00:50:46.645 --> 00:50:50.150 align:middle line:90% 00:50:50.150 --> 00:50:55.010 align:middle line:84% So in this regime, then the E would be approximately just m 00:50:55.010 --> 00:50:57.560 align:middle line:90% mu. 00:50:57.560 --> 00:51:01.020 align:middle line:84% And then the s would be approximately just for m mu 00:51:01.020 --> 00:51:01.520 align:middle line:90% squared. 00:51:01.520 --> 00:51:04.440 align:middle line:90% 00:51:04.440 --> 00:51:09.030 align:middle line:84% So now, if you just look at that expression-- 00:51:09.030 --> 00:51:11.580 align:middle line:84% yeah, so let's look at that expression. 00:51:11.580 --> 00:51:13.713 align:middle line:90% Then, you find the-- 00:51:13.713 --> 00:51:20.240 align:middle line:90% 00:51:20.240 --> 00:51:28.170 align:middle line:84% yeah, then you find that the differential cross-section 00:51:28.170 --> 00:51:36.400 align:middle line:84% is equal to alpha squared over 8 m mu squared and then 00:51:36.400 --> 00:51:42.890 align:middle line:84% pcm prime divided by m mu plus higher-order corrections. 00:51:42.890 --> 00:51:47.170 align:middle line:84% OK, so you expand in pcm divided by m mu-- 00:51:47.170 --> 00:51:48.850 align:middle line:90% pcm prime divided by m mu. 00:51:48.850 --> 00:51:53.240 align:middle line:84% And the total cross-section is just 00:51:53.240 --> 00:52:00.350 align:middle line:84% given by pi alpha squared, 2 mu squared 00:52:00.350 --> 00:52:05.320 align:middle line:90% and pcm prime divided by m mu. 00:52:05.320 --> 00:52:08.900 align:middle line:84% OK, so here, you see explicitly that when 00:52:08.900 --> 00:52:12.740 align:middle line:84% you reach the threshold, the threshold corresponding 00:52:12.740 --> 00:52:15.080 align:middle line:90% to this quantity goes to 0. 00:52:15.080 --> 00:52:23.420 align:middle line:84% So when this quantity goes to 0, then you, at the cross-section, 00:52:23.420 --> 00:52:25.050 align:middle line:90% go to 0. 00:52:25.050 --> 00:52:30.695 align:middle line:84% So another regime of interest is the ultra-relativistic regime. 00:52:30.695 --> 00:52:41.660 align:middle line:90% 00:52:41.660 --> 00:52:44.000 align:middle line:90% So ultra-relativistic regime. 00:52:44.000 --> 00:52:46.640 align:middle line:84% So in this case, we can take E to be 00:52:46.640 --> 00:52:52.400 align:middle line:84% much, much both than m e and m mu, so also much greater 00:52:52.400 --> 00:52:53.690 align:middle line:90% than m mu. 00:52:53.690 --> 00:52:57.300 align:middle line:84% So of course, it's already much greater than m e. 00:52:57.300 --> 00:53:00.890 align:middle line:84% So in this case, then you can, just here-- 00:53:00.890 --> 00:53:04.520 align:middle line:90% 00:53:04.520 --> 00:53:06.500 align:middle line:84% you can just essentially take the both mass 00:53:06.500 --> 00:53:08.180 align:middle line:90% to be 0 in this case. 00:53:08.180 --> 00:53:14.420 align:middle line:84% And then you find, in this case, that d sigma, d omega 00:53:14.420 --> 00:53:19.400 align:middle line:84% cm just given by alpha squared divided by 4s-- 00:53:19.400 --> 00:53:22.345 align:middle line:90% 00:53:22.345 --> 00:53:25.670 align:middle line:90% squared theta. 00:53:25.670 --> 00:53:27.320 align:middle line:90% OK. 00:53:27.320 --> 00:53:29.570 align:middle line:84% And then you can find the total cross-section. 00:53:29.570 --> 00:53:46.900 align:middle line:90% 00:53:46.900 --> 00:53:51.640 align:middle line:84% So you can find the total cross-section 00:53:51.640 --> 00:54:01.610 align:middle line:84% given by 4 pi alpha squared divided by 3s. 00:54:01.610 --> 00:54:04.350 align:middle line:90% 00:54:04.350 --> 00:54:08.520 align:middle line:84% So now, this final formula is very simple. 00:54:08.520 --> 00:54:12.150 align:middle line:84% So you could have written down this formula 00:54:12.150 --> 00:54:17.420 align:middle line:84% without doing any calculation, of course up to the prefactor, 00:54:17.420 --> 00:54:20.120 align:middle line:90% just on a dimensional grounds. 00:54:20.120 --> 00:54:24.920 align:middle line:84% So in the ultra-relativistic regime, then essentially 00:54:24.920 --> 00:54:27.620 align:middle line:84% you can neglect both the mass of the initial particle 00:54:27.620 --> 00:54:29.870 align:middle line:90% and the final product. 00:54:29.870 --> 00:54:35.120 align:middle line:84% And then in this whole process, if the initial particle 00:54:35.120 --> 00:54:37.960 align:middle line:84% is massless and the final particle is massless, then 00:54:37.960 --> 00:54:42.320 align:middle line:84% for this whole process, the only scale is, what? 00:54:42.320 --> 00:54:45.660 align:middle line:90% 00:54:45.660 --> 00:54:47.430 align:middle line:90% The only scale is s, OK? 00:54:47.430 --> 00:54:49.290 align:middle line:90% That's the only scale. 00:54:49.290 --> 00:54:53.080 align:middle line:90% So the only scale is s. 00:54:53.080 --> 00:54:55.040 align:middle line:90% And then that means-- 00:54:55.040 --> 00:55:00.460 align:middle line:84% and this should have dimension area. 00:55:00.460 --> 00:55:04.035 align:middle line:84% And then that means that there should be 1 over s. 00:55:04.035 --> 00:55:05.410 align:middle line:84% And then alpha squared just comes 00:55:05.410 --> 00:55:10.785 align:middle line:84% from because we have two vertex from a two-level process. 00:55:10.785 --> 00:55:15.765 align:middle line:90% 00:55:15.765 --> 00:55:16.265 align:middle line:90% Good? 00:55:16.265 --> 00:55:23.080 align:middle line:90% 00:55:23.080 --> 00:55:24.610 align:middle line:90% Good. 00:55:24.610 --> 00:55:26.125 align:middle line:90% So if you draw the plot-- 00:55:26.125 --> 00:55:32.130 align:middle line:90% 00:55:32.130 --> 00:55:36.125 align:middle line:84% so often, if we draw the total cross-section times s-- 00:55:36.125 --> 00:55:39.580 align:middle line:90% 00:55:39.580 --> 00:55:42.160 align:middle line:84% so s times the total cross-section 00:55:42.160 --> 00:55:45.320 align:middle line:84% as a function of the square root of s, 00:55:45.320 --> 00:55:50.340 align:middle line:84% then the threshold is at 2 mu, 2 m mu. 00:55:50.340 --> 00:55:58.070 align:middle line:84% And then here it goes to 0, like a square root. 00:55:58.070 --> 00:56:03.360 align:middle line:84% And then you're going to, say, approach your constant. 00:56:03.360 --> 00:56:09.320 align:middle line:84% This is 4 pi alpha squared divided by 3. 00:56:09.320 --> 00:56:11.150 align:middle line:90% So you have a curve like this. 00:56:11.150 --> 00:56:20.390 align:middle line:90% 00:56:20.390 --> 00:56:26.830 align:middle line:84% So now let's consider using this process 00:56:26.830 --> 00:56:31.060 align:middle line:84% to show that actually the quarks have three colors. 00:56:31.060 --> 00:56:33.370 align:middle line:90% The quarks have three colors. 00:56:33.370 --> 00:56:41.560 align:middle line:84% So for this purpose, let's consider e plus plus e minus 00:56:41.560 --> 00:56:42.395 align:middle line:90% goes to hadrons. 00:56:42.395 --> 00:56:49.400 align:middle line:90% 00:56:49.400 --> 00:56:54.210 align:middle line:84% So this seemingly is a very complicated process. 00:56:54.210 --> 00:56:57.560 align:middle line:84% Seemingly, this is a very complicated process. 00:56:57.560 --> 00:57:05.820 align:middle line:84% It's because the-- so suppose if you have e plus, 00:57:05.820 --> 00:57:10.290 align:middle line:84% you create a quark and an antiquark pair. 00:57:10.290 --> 00:57:17.630 align:middle line:84% But you remember, the quarks they are confined, OK? 00:57:17.630 --> 00:57:20.390 align:middle line:84% So we don't observe individual quarks. 00:57:20.390 --> 00:57:23.720 align:middle line:84% It's because the QCD is strongly interacting. 00:57:23.720 --> 00:57:27.260 align:middle line:84% And so the quarks are confined inside the proton, pions, 00:57:27.260 --> 00:57:29.380 align:middle line:90% et cetera. 00:57:29.380 --> 00:57:35.040 align:middle line:84% And so in this process, even though it originally created 00:57:35.040 --> 00:57:38.670 align:middle line:84% a quark, but each individual quark and antiquark, 00:57:38.670 --> 00:57:43.170 align:middle line:84% they cannot survive on their own. 00:57:43.170 --> 00:57:45.960 align:middle line:84% So their presence will polarize the vacuum, 00:57:45.960 --> 00:57:49.640 align:middle line:90% create many other particles. 00:57:49.640 --> 00:57:53.700 align:middle line:84% So what happens is that you create-- 00:57:53.700 --> 00:57:58.070 align:middle line:84% so each of them will create lots of particles along with them. 00:57:58.070 --> 00:58:00.740 align:middle line:84% So in the end, it's a complicated product. 00:58:00.740 --> 00:58:02.630 align:middle line:90% You observe many hadrons. 00:58:02.630 --> 00:58:04.880 align:middle line:90% You observe many hadrons. 00:58:04.880 --> 00:58:07.370 align:middle line:84% And then, of course, in the detector, 00:58:07.370 --> 00:58:09.620 align:middle line:84% we don't directly observe those quarks and antiquarks. 00:58:09.620 --> 00:58:10.870 align:middle line:90% We just observe those hadrons. 00:58:10.870 --> 00:58:14.340 align:middle line:90% 00:58:14.340 --> 00:58:20.250 align:middle line:84% So then you say, how do we have any hope 00:58:20.250 --> 00:58:23.700 align:middle line:84% to be able to understand this process just by measuring 00:58:23.700 --> 00:58:27.000 align:middle line:84% this very complicated product, because this product can 00:58:27.000 --> 00:58:31.700 align:middle line:84% have protons, neutrons, pions, can also have other things? 00:58:31.700 --> 00:58:36.010 align:middle line:84% And then this is a very complicated process. 00:58:36.010 --> 00:58:39.240 align:middle line:84% But now, the key is the following. 00:58:39.240 --> 00:58:41.530 align:middle line:84% But now the key is the following. 00:58:41.530 --> 00:58:45.350 align:middle line:84% Indeed, if you look at the question 00:58:45.350 --> 00:58:49.380 align:middle line:84% of going to each individual hadron is complicated. 00:58:49.380 --> 00:58:55.920 align:middle line:84% But if you can see the so-called inclusive process, which 00:58:55.920 --> 00:58:59.850 align:middle line:84% goes to all possible hadrons in this process, 00:58:59.850 --> 00:59:01.830 align:middle line:84% then that probability is just given 00:59:01.830 --> 00:59:04.520 align:middle line:90% by the total cross-section. 00:59:04.520 --> 00:59:06.200 align:middle line:84% Remember, the total cross-section 00:59:06.200 --> 00:59:10.710 align:middle line:84% just matches the total probability 00:59:10.710 --> 00:59:12.870 align:middle line:90% to create such particles. 00:59:12.870 --> 00:59:17.970 align:middle line:84% And then when they are created, they 00:59:17.970 --> 00:59:19.980 align:middle line:84% go through a complicated process. 00:59:19.980 --> 00:59:22.830 align:middle line:90% But that process is unitary. 00:59:22.830 --> 00:59:27.210 align:middle line:84% Will not-- will be, say, all probability 1 00:59:27.210 --> 00:59:30.640 align:middle line:84% to go to all possible final products. 00:59:30.640 --> 00:59:32.980 align:middle line:84% So if we include the cross-section going 00:59:32.980 --> 00:59:35.430 align:middle line:84% into all possible final products, 00:59:35.430 --> 00:59:38.810 align:middle line:84% that probability just captures by the total cross-section. 00:59:38.810 --> 00:59:48.080 align:middle line:84% So the total cross-section for this process, for the e plus, 00:59:48.080 --> 00:59:50.660 align:middle line:84% e minus going to q, q bar, and then 00:59:50.660 --> 00:59:54.790 align:middle line:84% just including the total cross-section, 00:59:54.790 --> 01:00:13.110 align:middle line:84% so inclusive section for all hadron products, this way, 01:00:13.110 --> 01:00:18.307 align:middle line:84% we can actually measure this process indirectly, 01:00:18.307 --> 01:00:20.390 align:middle line:84% even though you don't directly measure the quarks. 01:00:20.390 --> 01:00:24.580 align:middle line:90% 01:00:24.580 --> 01:00:29.370 align:middle line:84% So now let's calculate the total cross-section. 01:00:29.370 --> 01:00:31.710 align:middle line:84% So now let's calculate the total cross-section. 01:00:31.710 --> 01:00:35.470 align:middle line:90% 01:00:35.470 --> 01:00:38.170 align:middle line:84% And now we also don't have a choice 01:00:38.170 --> 01:00:41.270 align:middle line:90% which quark will be produced. 01:00:41.270 --> 01:00:47.780 align:middle line:84% So we need to sum over the total cross-section for e plus, e 01:00:47.780 --> 01:00:52.860 align:middle line:84% minus goes to, say, different kind of quarks. 01:00:52.860 --> 01:00:55.155 align:middle line:84% So i corresponding to different types of quarks. 01:00:55.155 --> 01:01:10.030 align:middle line:90% 01:01:10.030 --> 01:01:11.650 align:middle line:84% So each of them, we can just write it 01:01:11.650 --> 01:01:14.180 align:middle line:90% down using our formula. 01:01:14.180 --> 01:01:16.180 align:middle line:90% So the only difference-- 01:01:16.180 --> 01:01:21.610 align:middle line:84% the only difference-- so the same diagram applies. 01:01:21.610 --> 01:01:25.490 align:middle line:90% 01:01:25.490 --> 01:01:28.400 align:middle line:84% The only difference is that the quark has a different charge 01:01:28.400 --> 01:01:29.540 align:middle line:90% from the electron. 01:01:29.540 --> 01:01:35.210 align:middle line:90% So here is i e gamma mu. 01:01:35.210 --> 01:01:38.860 align:middle line:84% Here, we should use the charge for the quark. 01:01:38.860 --> 01:01:43.160 align:middle line:90% So that would be i qe gamma mu. 01:01:43.160 --> 01:01:45.400 align:middle line:84% q would be the quark of the charge compared 01:01:45.400 --> 01:01:49.870 align:middle line:90% to the charge of the electron. 01:01:49.870 --> 01:01:53.650 align:middle line:90% So then what do we get? 01:01:53.650 --> 01:02:01.030 align:middle line:84% So suppose qi is the charge for the quark qi. 01:02:01.030 --> 01:02:05.230 align:middle line:90% And then this gives you-- 01:02:05.230 --> 01:02:09.376 align:middle line:84% so this total is going to all hadrons-- 01:02:09.376 --> 01:02:11.048 align:middle line:90% to all hadrons. 01:02:11.048 --> 01:02:15.930 align:middle line:90% 01:02:15.930 --> 01:02:24.360 align:middle line:84% So this will be given by sum over the number of colors. 01:02:24.360 --> 01:02:27.930 align:middle line:84% And c is the number of colors for each quark. 01:02:27.930 --> 01:02:30.520 align:middle line:90% And then i will be the flavor. 01:02:30.520 --> 01:02:36.970 align:middle line:84% Now the i is the flavor of quark, 01:02:36.970 --> 01:02:40.840 align:middle line:84% so whether it's the u quark or d quark or charm quark, 01:02:40.840 --> 01:02:41.560 align:middle line:90% et cetera. 01:02:41.560 --> 01:02:44.840 align:middle line:90% And c is the number of colors. 01:02:44.840 --> 01:02:49.490 align:middle line:90% And then you have qi squared. 01:02:49.490 --> 01:02:56.000 align:middle line:84% And then you have sigma e plus, e minus to mu plus, mu minus. 01:02:56.000 --> 01:02:57.960 align:middle line:90% The rest is the same. 01:02:57.960 --> 01:03:01.890 align:middle line:84% So for each type of quark, this is exactly the same. 01:03:01.890 --> 01:03:04.610 align:middle line:90% You just replace iq-- 01:03:04.610 --> 01:03:06.470 align:middle line:90% this vertex by iqi. 01:03:06.470 --> 01:03:07.820 align:middle line:90% And then you get the qi squared. 01:03:07.820 --> 01:03:13.060 align:middle line:90% 01:03:13.060 --> 01:03:13.560 align:middle line:90% Good? 01:03:13.560 --> 01:03:15.336 align:middle line:90% Yes? 01:03:15.336 --> 01:03:17.170 align:middle line:84% AUDIENCE: So you talked about how when 01:03:17.170 --> 01:03:18.740 align:middle line:90% the quarks hadronize the-- 01:03:18.740 --> 01:03:19.970 align:middle line:90% polarized the vacuum. 01:03:19.970 --> 01:03:20.821 align:middle line:90% PROFESSOR: Yeah. 01:03:20.821 --> 01:03:27.210 align:middle line:84% AUDIENCE: [INAUDIBLE] very small [INAUDIBLE] 01:03:27.210 --> 01:03:29.648 align:middle line:84% calculate out every possible [INAUDIBLE]?? 01:03:29.648 --> 01:03:31.690 align:middle line:84% PROFESSOR: No, that's a very complicated process. 01:03:31.690 --> 01:03:33.280 align:middle line:90% We cannot. 01:03:33.280 --> 01:03:37.180 align:middle line:84% Yeah, that's a very complicated process. 01:03:37.180 --> 01:03:40.148 align:middle line:84% It's a huge field to study that, yeah. 01:03:40.148 --> 01:03:41.690 align:middle line:84% AUDIENCE: But then how do you account 01:03:41.690 --> 01:03:43.922 align:middle line:84% for all possible outcomes outputs? 01:03:43.922 --> 01:03:46.580 align:middle line:84% You're probably accounting for all possible quarks. 01:03:46.580 --> 01:03:47.960 align:middle line:90% PROFESSOR: No, you just measure. 01:03:47.960 --> 01:03:50.518 align:middle line:84% You just measure all possible products, right? 01:03:50.518 --> 01:03:52.143 align:middle line:84% AUDIENCE: But here, you're just summing 01:03:52.143 --> 01:03:55.850 align:middle line:84% over the probability of the flavors and the [INAUDIBLE] 01:03:55.850 --> 01:03:58.450 align:middle line:84% PROFESSOR: No, that's the only thing you can produce. 01:03:58.450 --> 01:04:00.992 align:middle line:84% AUDIENCE: And then you're not worrying about the final states 01:04:00.992 --> 01:04:02.730 align:middle line:90% that happen from these quarks? 01:04:02.730 --> 01:04:04.290 align:middle line:90% PROFESSOR: No, no, no. 01:04:04.290 --> 01:04:07.290 align:middle line:84% No, as I said earlier, we only need 01:04:07.290 --> 01:04:09.150 align:middle line:90% to worry about this process. 01:04:09.150 --> 01:04:13.380 align:middle line:84% And then this process into hadrons is complicated. 01:04:13.380 --> 01:04:17.610 align:middle line:84% But this happens with probability 1, OK? 01:04:17.610 --> 01:04:19.410 align:middle line:84% And so if we calculate this probability, 01:04:19.410 --> 01:04:22.230 align:middle line:84% then we know the total probability to here. 01:04:22.230 --> 01:04:23.670 align:middle line:90% Yeah, yeah, yeah. 01:04:23.670 --> 01:04:27.550 align:middle line:84% Yeah, because this happens with probability 1. 01:04:27.550 --> 01:04:28.050 align:middle line:90% Yes. 01:04:28.050 --> 01:04:29.946 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:04:29.946 --> 01:04:35.170 align:middle line:90% 01:04:35.170 --> 01:04:37.706 align:middle line:90% PROFESSOR: Yeah, yeah, yeah. 01:04:37.706 --> 01:04:39.290 align:middle line:90% Indeed, that can happen. 01:04:39.290 --> 01:04:42.770 align:middle line:90% But this is the leading process. 01:04:42.770 --> 01:04:45.100 align:middle line:84% Yeah, this is the leading process. 01:04:45.100 --> 01:04:47.370 align:middle line:84% Yeah, so this is the leading approximation. 01:04:47.370 --> 01:04:51.300 align:middle line:90% 01:04:51.300 --> 01:04:52.260 align:middle line:90% Other questions? 01:04:52.260 --> 01:04:55.680 align:middle line:90% 01:04:55.680 --> 01:05:01.000 align:middle line:84% OK, so now you're just given by this formula 01:05:01.000 --> 01:05:04.930 align:middle line:84% because you just multiply the qi squared 01:05:04.930 --> 01:05:09.010 align:middle line:84% to the total cross-section of the e plus, e minus to muon. 01:05:09.010 --> 01:05:11.920 align:middle line:84% And then you sum all the flavors and times 01:05:11.920 --> 01:05:13.900 align:middle line:90% the number of the colors. 01:05:13.900 --> 01:05:15.370 align:middle line:90% OK, times the number of colors. 01:05:15.370 --> 01:05:26.730 align:middle line:90% 01:05:26.730 --> 01:05:27.230 align:middle line:90% OK? 01:05:27.230 --> 01:05:36.600 align:middle line:90% 01:05:36.600 --> 01:05:40.460 align:middle line:84% Because all different colors have the same charge. 01:05:40.460 --> 01:05:42.210 align:middle line:84% All different colors have the same charge. 01:05:42.210 --> 01:05:48.150 align:middle line:90% 01:05:48.150 --> 01:05:49.830 align:middle line:90% So now we can define a ratio. 01:05:49.830 --> 01:05:52.770 align:middle line:90% 01:05:52.770 --> 01:05:54.960 align:middle line:84% So this is the same for everybody. 01:05:54.960 --> 01:05:57.640 align:middle line:90% So this is just factorize. 01:05:57.640 --> 01:06:01.765 align:middle line:84% So we can define a ratio for the quantity called r-- 01:06:01.765 --> 01:06:22.320 align:middle line:90% 01:06:22.320 --> 01:06:23.100 align:middle line:90% called r. 01:06:23.100 --> 01:06:28.290 align:middle line:90% 01:06:28.290 --> 01:06:32.370 align:middle line:84% It's defined by sigma e plus, e minus 01:06:32.370 --> 01:06:41.130 align:middle line:84% goes to hadrons divided by sigma e plus, e minus 01:06:41.130 --> 01:06:44.700 align:middle line:90% goes to mu plus, mu minus. 01:06:44.700 --> 01:06:50.820 align:middle line:84% And this is just given by sum nc times sum over i flavor 01:06:50.820 --> 01:06:51.720 align:middle line:90% and the qi squared. 01:06:51.720 --> 01:06:57.840 align:middle line:90% 01:06:57.840 --> 01:07:01.620 align:middle line:84% And this i, of course you should only 01:07:01.620 --> 01:07:05.970 align:middle line:84% sum over i, which is allowed by your energy, OK? 01:07:05.970 --> 01:07:07.890 align:middle line:84% Those quarks which are too heavy will not 01:07:07.890 --> 01:07:09.730 align:middle line:90% be created by your energ-- 01:07:09.730 --> 01:07:11.850 align:middle line:90% You are not including the sum. 01:07:11.850 --> 01:07:25.540 align:middle line:84% So flavor of quarks allowed by energy, by initial energy. 01:07:25.540 --> 01:07:28.450 align:middle line:84% If the quarks are too heavy, then of course you cannot 01:07:28.450 --> 01:07:31.110 align:middle line:90% create them. 01:07:31.110 --> 01:07:38.600 align:middle line:84% So now let's list the quarks we know. 01:07:38.600 --> 01:07:42.600 align:middle line:84% The process-- I will not do the final one. 01:07:42.600 --> 01:07:46.070 align:middle line:84% So we have u quark, d quark, strange quark, charm quark, 01:07:46.070 --> 01:07:47.030 align:middle line:90% bottom quark. 01:07:47.030 --> 01:07:48.290 align:middle line:90% And then there's a top quark. 01:07:48.290 --> 01:07:49.980 align:middle line:90% The top quark is very heavy. 01:07:49.980 --> 01:07:53.460 align:middle line:84% And its process will be complicated. 01:07:53.460 --> 01:07:55.800 align:middle line:90% So let's not worry about that. 01:07:55.800 --> 01:07:58.770 align:middle line:84% So then the charge for each of them. 01:07:58.770 --> 01:08:01.280 align:middle line:90% So this is 2/3. 01:08:01.280 --> 01:08:03.160 align:middle line:90% So this is minus 1/3. 01:08:03.160 --> 01:08:06.430 align:middle line:90% This is minus 1/3. 01:08:06.430 --> 01:08:09.160 align:middle line:90% This is 2/3. 01:08:09.160 --> 01:08:12.060 align:middle line:90% This is minus 1/3. 01:08:12.060 --> 01:08:15.480 align:middle line:84% And then the mass, twice of the mass. 01:08:15.480 --> 01:08:18.090 align:middle line:84% So I'll just give you some rough number. 01:08:18.090 --> 01:08:22.229 align:middle line:90% So u would be about a few MeV. 01:08:22.229 --> 01:08:24.960 align:middle line:90% So this is a few MeV. 01:08:24.960 --> 01:08:28.979 align:middle line:90% So this, again, is a few MeV. 01:08:28.979 --> 01:08:30.560 align:middle line:90% Just a few MeV, OK? 01:08:30.560 --> 01:08:33.330 align:middle line:90% 01:08:33.330 --> 01:08:36.330 align:middle line:90% So the strange will become-- 01:08:36.330 --> 01:08:37.920 align:middle line:90% strange is heavier, twice. 01:08:37.920 --> 01:08:39.720 align:middle line:90% It's 190 MeV. 01:08:39.720 --> 01:08:42.649 align:middle line:90% 01:08:42.649 --> 01:08:49.090 align:middle line:84% And the charm, the threshold is about 3.1 GeV. 01:08:49.090 --> 01:08:52.824 align:middle line:84% And the threshold for the bottom quark is about 9.5 GeV-- 01:08:52.824 --> 01:08:55.979 align:middle line:90% 01:08:55.979 --> 01:08:58.770 align:middle line:90% about 9.5. 01:08:58.770 --> 01:09:07.689 align:middle line:84% So now let's consider the value of R as a function of energy. 01:09:07.689 --> 01:09:18.529 align:middle line:84% So then if you consider for s, smaller than the charm, 01:09:18.529 --> 01:09:26.479 align:middle line:84% say 3 GeV, in this case, we can just produce uds. 01:09:26.479 --> 01:09:36.930 align:middle line:84% So in this case, we have R equal to nc then times, say this. 01:09:36.930 --> 01:09:40.529 align:middle line:90% So there's one 2/3 and two 1/3. 01:09:40.529 --> 01:09:46.380 align:middle line:84% So we have 2/3 squared plus twice 1/3 squared-- 01:09:46.380 --> 01:09:49.790 align:middle line:90% minus 1/3 squared. 01:09:49.790 --> 01:09:53.300 align:middle line:84% So this is giving you nc times 2/3. 01:09:53.300 --> 01:09:56.390 align:middle line:90% 01:09:56.390 --> 01:10:02.240 align:middle line:84% And then the others, you cannot create. 01:10:02.240 --> 01:10:15.220 align:middle line:84% And for square root s, say between 3 GeV 01:10:15.220 --> 01:10:21.970 align:middle line:84% and the 9.5 GeV, the threshold for producing the bottom quark. 01:10:21.970 --> 01:10:25.120 align:middle line:84% Then, now you can create the charm quark. 01:10:25.120 --> 01:10:27.640 align:middle line:90% So you just add another 1/3. 01:10:27.640 --> 01:10:34.270 align:middle line:84% So then now R becomes nc times 10 divided by 9. 01:10:34.270 --> 01:10:38.270 align:middle line:84% Just add another 2/3 squared here. 01:10:38.270 --> 01:10:41.000 align:middle line:84% And then, for now, if you have square root 01:10:41.000 --> 01:10:52.140 align:middle line:84% of s greater than 9.5 GeV, then now you 01:10:52.140 --> 01:10:54.840 align:middle line:90% can create the bottom now. 01:10:54.840 --> 01:10:56.820 align:middle line:84% Then you add another 1/3 squared. 01:10:56.820 --> 01:11:01.470 align:middle line:84% So this R is equal to nc times 10 01:11:01.470 --> 01:11:05.760 align:middle line:84% divided by 9 plus another 1/3 squared. 01:11:05.760 --> 01:11:11.150 align:middle line:84% So that gives you nc, 11 divided by 9. 01:11:11.150 --> 01:11:18.150 align:middle line:84% So now, if you plot the R as a function, 01:11:18.150 --> 01:11:25.070 align:middle line:84% say for square root s as measured by experiment, 01:11:25.070 --> 01:11:31.880 align:middle line:84% you should see a threshold at 3, another threshold around 10-- 01:11:31.880 --> 01:11:36.170 align:middle line:90% yeah, 9.5, say around 10. 01:11:36.170 --> 01:11:39.510 align:middle line:84% And then you see something like this. 01:11:39.510 --> 01:11:42.690 align:middle line:90% Yeah, I'm just drawing roughly. 01:11:42.690 --> 01:11:47.600 align:middle line:84% You create something-- something. 01:11:47.600 --> 01:11:49.220 align:middle line:84% And then you have another threshold. 01:11:49.220 --> 01:11:52.528 align:middle line:84% And then you have another threshold. 01:11:52.528 --> 01:11:57.570 align:middle line:84% So it turns out the first 9 is precisely at the 2. 01:11:57.570 --> 01:12:03.210 align:middle line:84% Yeah, I'd say precisely it's just up to some arrow bar. 01:12:03.210 --> 01:12:06.660 align:middle line:84% And the other one is about, say, a little bit more than 3. 01:12:06.660 --> 01:12:11.640 align:middle line:90% 01:12:11.640 --> 01:12:16.720 align:middle line:90% Yeah, little bit more than 3. 01:12:16.720 --> 01:12:20.890 align:middle line:90% And then your 4 is the one here. 01:12:20.890 --> 01:12:25.520 align:middle line:90% 4 is around here, yeah. 01:12:25.520 --> 01:12:32.660 align:middle line:84% Anyway, so if you compare with these numbers and you conclude 01:12:32.660 --> 01:12:35.280 align:middle line:90% that nc is actually exactly 3. 01:12:35.280 --> 01:12:38.967 align:middle line:84% Yeah, nc is 3, to be compatible with this experiment. 01:12:38.967 --> 01:12:42.170 align:middle line:90% 01:12:42.170 --> 01:12:43.400 align:middle line:90% Any questions on this? 01:12:43.400 --> 01:12:52.620 align:middle line:90% 01:12:52.620 --> 01:12:53.280 align:middle line:90% Yes? 01:12:53.280 --> 01:12:55.172 align:middle line:84% AUDIENCE: [INAUDIBLE] Does this mean 01:12:55.172 --> 01:12:59.600 align:middle line:84% that this is the dominant process for electron, positron? 01:12:59.600 --> 01:13:01.810 align:middle line:90% PROFESSOR: Yeah. 01:13:01.810 --> 01:13:03.420 align:middle line:90% Yeah, going to hadrons, yeah. 01:13:03.420 --> 01:13:07.460 align:middle line:90% 01:13:07.460 --> 01:13:08.420 align:middle line:90% Other questions? 01:13:08.420 --> 01:13:12.210 align:middle line:90% 01:13:12.210 --> 01:13:13.010 align:middle line:90% OK, good. 01:13:13.010 --> 01:13:15.930 align:middle line:90% 01:13:15.930 --> 01:13:20.300 align:middle line:90% So this is a very nice, simple-- 01:13:20.300 --> 01:13:23.810 align:middle line:84% even though this is a very simple process, 01:13:23.810 --> 01:13:30.050 align:middle line:84% actually it has very important physical implications 01:13:30.050 --> 01:13:34.230 align:middle line:84% and, in fact, quite profound physical implications. 01:13:34.230 --> 01:13:38.260 align:middle line:90% 01:13:38.260 --> 01:13:40.920 align:middle line:84% So we only have a few minutes left. 01:13:40.920 --> 01:13:47.895 align:middle line:84% So let's say a few words regarding the next topic. 01:13:47.895 --> 01:14:28.630 align:middle line:90% 01:14:28.630 --> 01:14:33.610 align:middle line:84% OK, so let's again consider this process of e 01:14:33.610 --> 01:14:36.445 align:middle line:90% plus, e minus going to mu-- 01:14:36.445 --> 01:14:39.730 align:middle line:90% 01:14:39.730 --> 01:14:40.360 align:middle line:90% going to mu. 01:14:40.360 --> 01:14:44.580 align:middle line:90% 01:14:44.580 --> 01:14:47.310 align:middle line:90% So we have then this diagram. 01:14:47.310 --> 01:14:53.020 align:middle line:90% 01:14:53.020 --> 01:14:55.720 align:middle line:90% OK? 01:14:55.720 --> 01:15:01.630 align:middle line:84% So suppose this is p1, p2, and k1 and k2. 01:15:01.630 --> 01:15:05.100 align:middle line:90% 01:15:05.100 --> 01:15:09.478 align:middle line:84% So when we draw this diagram, we take the time as going above. 01:15:09.478 --> 01:15:10.770 align:middle line:90% So these are the initial state. 01:15:10.770 --> 01:15:11.853 align:middle line:90% These are the final state. 01:15:11.853 --> 01:15:17.360 align:middle line:90% 01:15:17.360 --> 01:15:21.260 align:middle line:84% So now, if we view this diagram sideways, 01:15:21.260 --> 01:15:25.460 align:middle line:84% say if we view in this direction, 01:15:25.460 --> 01:15:28.620 align:middle line:84% if you try to think that at the time going to right direction, 01:15:28.620 --> 01:15:34.000 align:middle line:84% say if you try to view it sideways, then in this case, 01:15:34.000 --> 01:15:35.670 align:middle line:90% if you view it sideways-- 01:15:35.670 --> 01:15:46.770 align:middle line:90% 01:15:46.770 --> 01:15:47.490 align:middle line:90% OK? 01:15:47.490 --> 01:15:52.108 align:middle line:84% And now this becomes an initial state. 01:15:52.108 --> 01:15:53.400 align:middle line:90% So this is still initial state. 01:15:53.400 --> 01:15:56.430 align:middle line:90% And this is still initial state. 01:15:56.430 --> 01:15:57.960 align:middle line:90% Let me see. 01:15:57.960 --> 01:16:00.930 align:middle line:90% Oh, let me-- oh. 01:16:00.930 --> 01:16:04.160 align:middle line:90% 01:16:04.160 --> 01:16:04.965 align:middle line:90% One second. 01:16:04.965 --> 01:16:05.465 align:middle line:90% Let me see. 01:16:05.465 --> 01:16:10.680 align:middle line:90% 01:16:10.680 --> 01:16:11.820 align:middle line:90% Yeah, yeah, yeah. 01:16:11.820 --> 01:16:12.720 align:middle line:90% Right, right. 01:16:12.720 --> 01:16:13.980 align:middle line:90% Sorry, sorry. 01:16:13.980 --> 01:16:18.960 align:middle line:84% Yeah, view it sideways, and then this is the initial state. 01:16:18.960 --> 01:16:21.855 align:middle line:90% And this is the initial state. 01:16:21.855 --> 01:16:28.140 align:middle line:90% 01:16:28.140 --> 01:16:34.050 align:middle line:84% Now let me draw the arrow this way so that because the-- 01:16:34.050 --> 01:16:36.280 align:middle line:90% yeah, it doesn't matter. 01:16:36.280 --> 01:16:40.260 align:middle line:84% Let's draw the arrow this way so that this is the mu, 01:16:40.260 --> 01:16:41.830 align:middle line:90% this is the mu plus. 01:16:41.830 --> 01:16:44.820 align:middle line:90% 01:16:44.820 --> 01:16:46.830 align:middle line:90% And now let's view this way. 01:16:46.830 --> 01:16:50.430 align:middle line:84% And now this is-- look how you have initial electron. 01:16:50.430 --> 01:16:52.380 align:middle line:84% And then you have initial muon, OK? 01:16:52.380 --> 01:16:56.610 align:middle line:84% And then this becomes a process, e minus plus mu 01:16:56.610 --> 01:17:00.510 align:middle line:84% minus going to e minus and mu minus, 01:17:00.510 --> 01:17:03.300 align:middle line:84% because this is the final particle, now going out 01:17:03.300 --> 01:17:04.380 align:middle line:90% will be the particle. 01:17:04.380 --> 01:17:07.080 align:middle line:84% And this final particle going out will be a particle. 01:17:07.080 --> 01:17:10.950 align:middle line:84% And then the final will be the electron and the muon. 01:17:10.950 --> 01:17:17.060 align:middle line:90% 01:17:17.060 --> 01:17:21.470 align:middle line:90% And then you get the process, b. 01:17:21.470 --> 01:17:23.450 align:middle line:84% So when you view it sideways, then you 01:17:23.450 --> 01:17:24.650 align:middle line:90% have a diagram like this. 01:17:24.650 --> 01:17:26.840 align:middle line:90% If I draw using our-- 01:17:26.840 --> 01:17:30.520 align:middle line:90% again, time going to up-- 01:17:30.520 --> 01:17:32.380 align:middle line:90% then this is like that. 01:17:32.380 --> 01:17:37.240 align:middle line:84% Now this is e minus, e minus, mu minus, and mu minus. 01:17:37.240 --> 01:17:41.270 align:middle line:90% 01:17:41.270 --> 01:17:44.090 align:middle line:84% So now you wonder whether there's 01:17:44.090 --> 01:17:49.370 align:middle line:84% some relation between these two processes, 01:17:49.370 --> 01:17:52.730 align:middle line:84% whether there's some relation between these processes 01:17:52.730 --> 01:17:57.150 align:middle line:84% from you just exchanging initial state to the final state. 01:17:57.150 --> 01:17:59.390 align:middle line:84% So it turns out actually there's a very nice relation 01:17:59.390 --> 01:18:00.770 align:middle line:90% between them. 01:18:00.770 --> 01:18:04.170 align:middle line:84% If you know one of them, you can try to deduce the other. 01:18:04.170 --> 01:18:07.480 align:middle line:84% It can help you deduce the other, which 01:18:07.480 --> 01:18:09.540 align:middle line:90% we are running out of time. 01:18:09.540 --> 01:18:12.160 align:middle line:90% So we will do it the next time. 01:18:12.160 --> 01:18:16.000 align:middle line:90%