distillery:cpv

# Antiparticles, Charge Conjugation, CP Violation

## Antiparticles

What are antiparticles? First of all, they are solutions to an equation. This is how the concept of antiparticles was introduced. Dirac realized, that his equations had two sets of solutions: To each solution, there existed another solution with the exact same mass but with opposite charge.

An antiparticle always mirrors its corresponding particle in the sense, that it has exactly the same mass and interactions, but all its charges are opposite. For an electromagnetically negatively charged particle like the electron, its antiparticle is an electromagnetically positively charged particle. The amount of charge, as well as the mass of the antiparticle is exactly the same as for the electron. This defining feature of carrying opposing charges while having exactly the same masses allows particles to annihilate with their corresponding antiparticle and produce a photon. Charge conservation dictates that charges cannot be lost, thus if we add a positively charged particle and a negatively charged particle with the same amount of charge, the resulting state needs to be neutral. A photon is neutral. Its light. The whole mass energy of the electron and the positron go into the kinetic energy of the photon since the photon has no mass that would require a part of the energy in the creation.

Now imagine the world we see would not be made of particles but of antiparticles. Could we tell the difference? Naively, you might say 'yeah sure, we know its negatively charged electrons running around the positively charged proton that makes up matter. if this was the other way around we would notice!' But if you think about it, that is not true, we could not tell the difference. The reason is the following: What we call positively charged and negatively charged is just a convention. The only thing we actually know is that there is two kinds of charges where opposites attract and same charges repell each other. We just came historically to define the charge of the electron as negative and the one of the proton as positive, but we could as well define it the other way around, its just labels. The only important point is, that electron and proton have opposite charges, thus they attract each other.

So we cannot tell whether we live in a world of particles or antiparticles? Exactly. Because much like the charges, the terms 'particle' and 'antiparticle' are just labels. Again all that is important is that there is a 'same mass opposite charge' copy for everything flying around. Which one we call particle and which one antiparticle is up to us. Thus, switching all particles for antiparticles and vice versa wont change a thing. What attracted each other before still does and what repelled each other before still does. What annihilates with each other still does and what doesnt still doesnt.

## Charge Conjugation and CP Violation

Physicists believed this 'charge conjugation', the process of swapping particles and antiparticles, to be a universal symmetry of nature. That you could always do it and nothing would change, just as I laid out here. For electrodynamics, this is true. But there is another interaction which is felt only at the subatomic level, which physicists in their unrivaled creativity denoted 'the weak force'. For this 'weak force', it was found in 1964 in an experiment by Cronin and Fitch, that this is actually not true. While observing specific particle decays, they found that the weak force treated matter a little bit different from antimatter. And not in the 'of course, because they have opposite charges, duh!' kind of way, but imagine it more in a way, as if the amount of charge on matter and antimatter was a tiny bit different. Imagine electrons having charge -1 and their antiparticles having charge +1.2. Then, there would be a difference when we exchange matter and antimatter, the repelling of same charges would be stronger in the antiparticle case. This is not a proper analogy to what happens in the case of the weak force, but for the sake of providing an accessible picture how a force could treat particles and antiparticles differently, its ok-ish.

So what did Cronin and Fitch actually do? They created a beam of 'Kaons', a certain funny type of particle, and watched them decay. How this can be used to actually see whether or not the weak force treats particles differently from antiparticles, I will explain now. Kaons are composite particles, similar to a hydrogen atom which consists of a proton and an electron, a Kaon consists of one quark and one antiquark of specific flavors. You get a $K_0$ if you have a down-quark and an anti-strange-quark, and you get a $\overline{K}_0$ if you have an anti-down-quark and a strange-quark (just put the 'anti-' to the other quark). Both Kaons are neutral, as the zero in the name indicates. If you look at this for a few seconds, you realize, that when you swap particles with their antiparticles and vice versa, the $K_0$ will become the $\overline{K}_0$ and vice versa. In this sense, the $\overline{K}_0$ is the antiparticle of the $K_0$, which is what the 'bar' in the name means. But here you can see, that the terms 'particle' and 'antiparticle' are arbitrary. We could have also called the $\overline{K}_0$ the particle and the $K_0$ the antiparticle.

Now Cronin and Fitch prepared their experiment in a way, that they would produce lots of $K_0$s (or $\overline{K}_0$s, it doesnt matter as you will see in a moment) that would fly towards a detector. And now comes the weak force. What the weak force actually does is that it can change quark flavors. Thats its force. Much more superhero or star wars like than the attracting and repelling thing we attribute to electromagnetism as its force. It is though possible (and actually common practice) to change the traditional newtonian notion of force as in electromagnetism and gravity to a more modern concept of a force, which encompasses all the forces we know in a unifying manner. The idea is, that to exert a force, you exchange a particle, a force carrier. Imagine standing on a frozen surface and throwing a heavy object to a friend standing next to you. Throwing it towards your friend will make you drift away, while the process of catching the object will also make your friend drift away. The exchange of the object acted like a force push, if you were constantly throwing balls at each other, a clueless observer that would see the two of you but not the balls could be lead to the conclusion that there is a repulsive force at work between the two of you, pushing you and your friend away from each other. Thats the concept physicists help themselves with when talking about subatomic forces. particles interact with each by throwing other particles back and forth. If you wonder how attraction is possible in this picture, well, you need to exchange particles with negative momentum. That sounds odd since that does not exist in our macroscopic world, but the common interpretation of quantum mechanics, which we anyway employ all the time, allows us to do that on a microscopic level.

So the weak force changes quark flavors. Take a $K_0$, consisting of an anti-strange-quark and a down-quark. Now these two quarks want to interact via the weak force. So they exchange a force carrier of the weak force. Much like you and your friend on the ice surface. But by exchanging this weak force carrier, their quark flavors change in certain ways, the anti-strange-quark becomes say an anti-up-quark (there are several possibilities for this, it just cannot go directly from strange to down) and the down quark becomes an up-quark. In the picture of you and your friend on the ice surface, for convenience we call you bob and your friend hugo, so bob throws a ball at hugo and hugo catches the ball. In a sense, 'bob with ball' became 'ballless bob' while 'ballless hugo' became 'hugo with ball' in the process. For reasons of energy and momentum conservation, however, the former Kaon cannot interact via the weak force with itself ending up in an anti-up up state. It needs to have the same mass like before. Therefore, they need to exchange another weak force carrier to change their flavor again: now the anti-up-quark can become an anti-down-quark, while the up-quark can become a strange-quark. If you followed that (probably not, so much 'anti' and 'quark'), you may see that we started with a $K_0$ and through a double interaction via the weak force with itself, it changed into a $\overline{K}_0$. Since particles and antiparticles have the same mass, this is perfectly fine, not violating anything. You may complain 'but they enter a forbidden state between the two force carrier exchanges, how is this possible?' Well, thats quantum mechanics. One version of the uncertainty principle is that you can take on states which are forbidden by energy conservation, if you do so for only very little time. Like energy conservation is an old grandmother that blinks so slowly that if you are quick enough, she wont notice. Thus the $K_0$ can go into the $\overline{K}_0$ via exchanging two weak force carriers in quick succession. This is a very classical and not entirely quantum mechanically correct interpretation of what is going on, but it helps to understand and physicists use this kind of picture all day. The more correct version is to imagine the double exchange as a single process. Single exchange is forbidden, double exchange is fine. Like youre throwing two balls at hugo. You transition from 'bob with two balls' to 'ballless bob' in an instant without assuming the state of 'bob with one ball' really. Oh boy. Bet Hugo is just glad he walks out of these explanation with two balls.

So a $K_0$ can transform into a $\overline{K}_0$ via double weak exchange. The same goes for the other way around, $\overline{K}_0$ can transform into a $K_0$ via a double weak exchange too. And since they throw these force carriers at each other all the time - its quantum mechanics, its a probability per time to have that kind of exchange - what they do is oscillate back and forth between $K_0$ and $\overline{K}_0$.

So what kind of particle do we actually have here then? When we look it might be a $K_0$, looking again, it might be a $\overline{K}_0$, looking again, it might be a $K_0$ again. We can of course content ourselves with having this state that is oscillating back and forth, but we can also give it a new name. Think of a bowling ball that is painted red on one side and blue on the other side. If you roll it, what you see is red-blue-red-blue-red-blue-… If you even roll it fast enough it might blur into being a purple ball. Same thing here, we just give the whole $K_0$-$\overline{K}_0$-$K_0$-$\overline{K}_0$-… a new name. We just call it $K_S$, spelled 'K short'.

Now we have to accomodate one quantum mechanical feature, for which I just cant find a good everyday analogy. When we combine the $K_0$ and the $\overline{K}_0$ state into the mixture, we can mathematically either just add them, like the bowling ball, just add half of each and you get the purple bowling ball. In the same way we can add half of each $K_0$ and $\overline{K}_0$ and get the $K_S$. But we can also subtract them, for which I cant think of a bowling ball analogy, then we end up with a different combination of $K_0$ and $\overline{K}_0$, this we call $K_L$ for 'K long'. So what we have is

\begin{align} K_S & = K_0 + \overline{K}_0 \\ K_L & = K_0 - \overline{K}_0 \end{align}

Now we're getting close. When we look at how these combinations behave if we swap particles with their antiparticles and vice versa, we find something suprising. Remember $K_0$ becomes $\overline{K}_0$ and $\overline{K}_0$ becomes $K_0$ under this operation. This means, that $K_S$ doesnt change at all. The $K_L$ state, however, becomes minus the $K_L$ state. This doesnt sound like a big deal, and it really isnt, but we can use this to determine what these states can decay into.

Both states are neutral, so they will decay into neutral particles, or at least something thats neutral in the sum. Thats charge conservation. The only neutral particles they can decay in are so called 'pions'. They are like lighter versions of the Kaons. (this decay process also involves the exchange of a weak force carrier, among other stuff) A pion also obtains a minus sign when we do the particle-antiparticle-swap operation. This is important.

And now comes the crucial point of the experiment. If we assume, that the interaction which lets the kaons decay into pions treat particles and antiparticles equally, then a state which obtains a minus sign under the particle-antiparticle-swap can only decay into a state which also obtains a minus sign. While states that stay as they are can also only decay into states that stay as they are. Therefore:

$K_S$, staying as it is, decays into two pions. Because a pion gets a minus and minus times minus is plus. Therefore two pions also stay as they are. $K_L$, obtaining a minus sign, decays into three pions. One isnt possible because of energy momentum conservation.

This, by the way, is where the names come from: $K_L$, the 'K long' decays into three pions, casually speaking, this is comparably hard to do. In terms of quantum mechanics, its really unlikely. But because we assume that the interaction that mediates the decay respects the particle-antiparticle-swapping-symmetry, it is not allowed that the $K_L$ decays into two pions, so it kind of has to sit it out and wait until this unlikely three pion decay happens. Thus 'K long' for having to wait quite long. The $K_S$, on the other hand, can decay into two pions, which is much easier so to say, it has a much larger probability to decay into just two pions, so the name 'K short' is the proper label referring to its lifetime. The difference between the lifetimes is a factor of 600 by the way, so it is indeed huge, allowing to separate the $K_L$ from the $K_S$ by just waiting a bit until all $K_S$ have decayed, then its only $K_L$ remaining.

What Cronin and Fitch did, was to create their beam of Kaons, then place a detector at a suitable distance where they could be sure, that - considering the velocity of the kaons - all $K_S$ would have decayed, so what would reach their detector would be only $K_L$s. Now remember, the $K_L$ obtain that minus sign under particle-antiparticle-swapping, thus they are only allowed to decay into three pions. And then Cronin and Fitch found two pion signatures. Not many, only 1 per mil of the decays, but still statistically significantly enough. Ok, what happened there? Here we remember, that the conclusion, that $K_L$ only decay into three pions relied on the assumption, that the interaction mediating this process would respect the particle-antiparticle-swapping as a symmetry. In other words, it would treat particles and antiparticles absolutely equal. The result of finding '$K_L$ into two pion' signatures suggests, that the interaction does treat particles and antiparticles alike. Being however only about 1 per mil of the total decays, this violation of the particle-antiparticle symmetry in the interaction is very weak, but it is there. Remember that I mentioned, that the weak interaction is also responsible for this decay? So bottom line, they found that the weak interaction does treat particles and antiparticles a tiny bit different.

## Indirect and direct CP violation

The picture commonly employed by particle physicists is much like the one used above, particles flying around, exchanging particles. For lots of stuff this works well and it gives a nice, understandable picture how stuff might be going on. For some effects this is not correct, since it neglects quantum, but some you can build in by hand like the uncertainty relation, just saying that these particles can do weird stuff for short amounts of time. Other effects, like interference, arent that easily built in, but when interference isnt important, youre good. In the same way, I will proceed with this semi-classical picture of taking feynman diagrams (the little doodlings of particles flying around physicists do when they want to calculate something) more serious as it is correct. I will even more bend the interpretation into classical territory, which is in general not correct, but in this case gives a neat and graspable picture of something that is otherwise utterly comlicated and abstract and most people are content with not having a idea of whats going on there at all but resort to just the math. After having understood this not entirely correct picture, it is easier taking a step back and trying to incorporate the classical picture of something step by step happening into a quantum picture, where all possibilities of all intermediate steps have to be combined to yield a probability for the full process and where statements like 'this is happening at this point' do not exist for intermediate processes, but all statements are of statistical nature.

## The not entirely correct classical picture of indirect and direct CP violation

So the weak force violates CP. We can imagine that every time a weak force carrier is emitted or absorbed, there is a tiny chance added to violate CP. Now remember what the Kaons do when they fly: $K_L$ is $K_0$ turns into $\overline{K}_0$ and back again many times per nanosecond via an exchange of two weak force carriers. At the end of the process, when we see Cronin and Fitches signature of the $K_L$ decaying into two pions, there was also a weak force carrier involved in the actual decay. So we have lots of these double exchanges during the flight - we call this 'the mixing' - and another weak force carrier exchange during the decay. Now if each exchange has a tiny chance to screw up treating particle antiparticle equally, we can ask ourselves 'whats the chance of this happening during the mixing?' and 'whats the chance of that happening in the decay?' These two questions coin the terms 'indirect CP violation' and 'direct CP violation', where indirect means that it happens in the mixing and direct means it happens directly in the decay. As I said before, this is a painfully classical picture, it would be more appropriate to say 'indirect CP violation refers to the fraction of the probability to have a two pion final state, which comes from the mixing, while direct CP violation refers to the fraction of the probability which comes from the decay.' As I said, to get more close to a quantum mechanical interpretation, we have to think in terms of probabilities piling up which in the end give a probability for the entire process to happen, instead of having things happen step by step (double exchange of weak carrier, another double exchange, ah in this one happened the CP violation, another double exchange…).

Measurements showed, that while CP violation is a per mil effect, it is almost entirely composed by indirect CP violation. In the picture painted above, this is not suprising at all, since there are tons more force carriers exchanged during the mixing process that add up the probability to screw CP up, as opposed to only a single one in the decay process. The fraction of direct CP violation is about just another per mil of the entire CP violation. And it is usually given that way: epsilon is the variable encoding indirect CP violation, epsilon prime for direct CP violation, however people historically give epsilon prime divided by epsilon, giving the ratio of both.