Particles composed of square roots?

Question

There are some particles which require square roots when telling their compositions. Mostly mesons, not baryons.

e.g. $$ρ^0=\frac{u\bar{u}-d\bar{d}}{\sqrt{2}}.$$

Could you explain me how to make sense of it?

Answer 1

There are some particles which are made of a specific composition of quarks and are always made of that composition. For example, as I believe you have heard of, protons and neutrons are always composed of $uud$ and $udd$, respectively.

You can think of this intuitively in the following sense: if you could pick up a proton and look inside it to figure out what it is made of, you would have $100\%$ probability to find $uud$ inside it. Similarly, if you were to pick up a neutron and look inside it, you would always find $udd$ inside it.

This seems a silly way of putting it, perhaps even obvious. They are made like that, what else could happen? Why am I stating what they are made of in terms of probabilities?

Let us consider a different particle then. For example, the $\rho^0$ meson you mentioned. This guy works different: as you noticed, it is composed of a difference of quarks (what is that?) and has that weird square root. What do these things mean?

The tricky bit is that the $\rho^0$ is in a superposition of states (see, e.g., the Wikipedia page on superposition or the Simple English Wikipedia page on Quantum Mechanics). To keep it simple, I'll break one of my personal rules of never talking about Schrödinger's cat and say it is quite similar: if you were to pick up the $\rho^0$ meson and look inside it, there is a $50\%$ chance you will find $u\bar{u}$ and a $50\%$ chance you'll find $d\bar{d}$. It is in a quantum superposition! The minus sign indices that the meson is a superposition of the states $u\bar{u}$ and $d\bar{d}$, and the $\sqrt{2}$ is chosen to ensure the probabilities add to $100\%$ (the rough idea to compute the probabilities is by taking the number in front of the possibility you want to consider and taking its absolute value squared).

It is worth mentioning that the minus sign does have some other implications, and if we changed it to a plus sign it wouldn't be a $\rho^0$ meson, but an $\omega$ meson instead.

In short, the presence of the square roots are just to ensure the sum of all the probabilities are $100\%$. Some particles have well-defined constituents, but some others are superpositions of different possibilities.