L6.4 Weak Interactions: Quarks

Key Takeaways

Welcome back to 8701. So, in this lecture we look at the interaction of W bosons with quarks of the charged weak interaction of quarks. Um let's just make a number of observations. So, we observe that the weak interaction respects the lepton generation, meaning that a W couples to an electron and an electron neutrino, but not an electron and the muon neutrino. But in the case of the quarks, there is a violation of this. So, there is um a disrespect of the quark generation when it comes to the interaction with Ws. Um so, when you investigate these two diagrams here, you find that the W couples couples to the U quark, but to the D quark and the U quark, but it also couples to the S quark, S quark and the U quark. All right? Um in order to encapsulate this, we have to make a correction, and the corrections are typically called cosine um theta C and sine theta C. Theta C is the Cabibbo angle, so theta Cabibbo. Turns out this is rather small, 13°. So, it is a correction, a small correction. When we studied the partial decay rates of the kaon leptonic decays, um over the partial decays to a pion over of the pion in leptonic decays, we found that there's a set of form factors, a form factor for the pion decay and a form factor for the kaon decay. It turns out that the form factors are due to this additional correction. So, you find the tangent square of the Cabibbo angle as part of this correction. Good. So far, so good. Now, we have made an observation. We haven't explained anything yet. We can make one more um observation or discuss one, and that's the decay of neutral kaons to a pair of muons. It turns out that those are not very likely. Even so, you would expect that the amplitude has a factor here of um sine and cosine uh Cabibbo, so the amplitude should be in the order of sine theta Cabibbo times cosine theta Cabibbo. Um so, when this was studied, the charm quark hadn't been discovered. Um and the explanation to why uh this decay is suppressed comes from the fact that there is a second diagram here, where we just replace the U quark in this loop with a C quark. This diagram contributes with a minus sign to the amplitude, and therefore those two diagrams, they cancel. Right? They have they have the same magnitude, about the same magnitude, and they have an opposite sign. So, this was the first indication that there must be a a fourth quark contributing to this kind of processes, the charm quark. Um Let me now try to understand what's going on here. Why is the W uh coupling modified? Um or why it's not the full down quark or charm or strange quark participating in the weak interaction. Um we can do the following here. We can rewrite or we note that this the the weak interaction eigenstate, the eigenstate which participates in the weak eigenstate, is not the eigenstate of the particle itself, the so-called mass eigenstate. Um so, we have to write the weak eigenstate as a linear combination of the as a mass eigenstate or weak boson. This can be done in this matrix form here, where we um simply modify multiply the weak eigenstates uh with the uh with a matrix and just basically rotate it into the mass eigenstate. Um so, this was proposed by Cabibbo and rather successful, um but it didn't incorporate the third generation particles. And this was done by uh Kobayashi and Maskawa, who generalized the scheme, uh and proposed the so-called CKM matrix. C is here for Cabibbo, Cabibbo, Kobayashi, and Maskawa. Um because of constraints we'll discuss in in in one of the recitations, um this matrix can be parametrized or has only um three independent angles and one complex phase as independent parameters. So, you can uh choose different parameterization to capture that there's only three four parameters in this uh matrix, which has nine components. Um one is by uh thinking about this matrix as three independent rotations and this complex space here. In terms of numerical values, you see that the diagonal elements of this matrix are very close to one, meaning that this this mixing is for in the quark sector is a rather small effect. You find that those next nearest off-diagonal elements on the order of 20% and the next to next off elements are even smaller. Okay? This leads us into the discussion that we can, you know, use different parameterization in order to capture the effect. You already discussed the standard parameterization, which you can really think about three different rotations. Um and the the values of this uh of those angles are given here, together with the value of this um additional phase. Another way to look at this is the so-called Wolfenstein parameterization, and this captures the fact that it seems like that there is a an a correction being applied to the actual particles. So, you find elements in the order of lambda, lambda is about 22%. Um then you find elements which are in the order of 1 minus a lambda square correction, and then there is elements which are of lambda square and lambda third power. Uh so, this captures the matrix, and then there is higher order correction to that, which are of order lambda to the fourth power. Okay? Um Because there's constraints on this matrix, specifically the uh unitarity constraints, meaning that you know, we have three generations, if I make a mixing of those three uh mass eigenstates to weak eigenstates, then um you know, unitarity the the the total number of particles in this discussion is conserved. Uh this will change if there would be, for example, a fourth generation particle. So, the study of weak charged interaction with quarks helps us to understand whether or not there might be a fourth generation. Um We'll not go into too much detail here, but it also the the the complex phase um explains part of uh our understanding of CP violation, and we might discuss this in a little bit of a later of a later lecture. But nevertheless, what we can actually form here is um those unitarity constraints. Just simply summing over uh the matrix elements, the scalar product of matrix elements, and those where the contribution vanishes vanishes, so those where J and K are not equal, those um can be represented as a triangle. That's quite interesting. You can just rewrite this. You just say that those three elements of the sum are equal to zero, then you normalize by one element. In this case here, normalize by uh VCD uh VCB. And so, then this makes this point being zero, and so we have this nice triangle here, which has three um angles, alpha, beta, and gamma, and this point here, rho and eta. And so, this is a nice way to illustrate actual measurements of the elements of this CKM matrix of the CKM matrix elements. And without actually explaining how we do this experiment, you can assume that, you know, all measurements have or you can understand that all measurements have uh to do with the weak interaction with quark. That's how we have access to the um CKM matrix elements. Uh sometimes this results in modification of um masses or splitting of mass states, and sometimes it's a direct measurement of um of a coupling. Um when you put all of those measurements back together, you can, you know, look at this. So, we see our our triangle here. We see this this point um eta and rho, which is given here in this eta and rho plane. And you see various number of measurements um which corresponds to um which corresponds to uh you know, elements of this of the CKM matrix.