How exactly does the quark gluon plasma prevent protons from forming?

Question

I have read this question:

The difference is that the quark-gluon plasma is governed by the dynamics of the strong interaction, which isn't very well understood, so that's what people are interested in when they talk about it.

How did photons and electrons arise out of the quark-gluon plasma?

And this one:

Is it sensible to talk about protons at the time when the universe was a quark-gluon plasma?

At high temperature there’s so much energy shared among particles that bound states are not likely to occur. Below a certain critical temperature the bound states are suddenly much more likely. This is a phase change similar to freezing.

If quarks can't be isolated in the first place, how did they become confined in the early universe?

Snapshot of a proton -- and imagine all of the quarks (up,down,and strange -- u,d,s), antiquarks (u,d,s with a bar on top), and gluons (g) zipping around near the speed of light, banging into each other, and appearing and disappearing. (M.Strassler 2010)

Why do the quarks constantly change colors?

None of these give satisfactory answers as to how exactly the quark gluon plasma prevents quarks from forming protons. I have two main possibilities:

1. in the early universe, quarks had extremely high kinetic energy and thus the strong force was not able to keep them in confinement. But I cannot explain this any further, as the gluons are massless, however fast the quarks are moving, the gluons are still (as viewed from the massive quarks' frame) moving at the speed of light, so no matter how high the kinetic energy of the quarks is, the strong force is still mediated between them by gluons. As far as I understand, quarks are already moving near the speed of light inside the proton, yet they can still be confined.

2. in the early universe, spacetime distances between quarks was shrunk to very small scales, and as we know, the strong force becomes repulsive at short distances, thus, protons were not able to form. I cannot explain this any further either, because the strong force becomes repulsive (at short distances) at the current state of the universe, but maybe it was different at the early times.

Question:

1. How exactly does the quark gluon plasma prevent protons from forming?

Answer 1

A proton in a quark-gluon plasma would be like a water droplet underwater. The idea of a droplet includes a boundary and a surrounding non-water region. The higher the energy density of quark matter, the less likely it is that all of the quarks but three will separate from three of the quarks by enough distance, and for enough time, that you could reasonably say that a proton existed. It's not absolutely impossible, just very unlikely.

At a lower energy density, it becomes energetically favorable for pockets of vacuum to form, and the quark-gluon plasma breaks apart into hadrons. Before that happens, the quarks are "free" inside what amounts to a giant, universe-spanning hadron, not free in vacuum. It's impossible for quarks to be free in vacuum.

Answer 2

It is true that gluons are always moving faster than quarks, and it is true that the strong force is actively pulling quarks together, but that does not require protons to form. What follows is a lengthy explanation about how the important factor for being "bound" is total energy.

A proton is a bound state. A bound state is not possible with a high enough kinetic energy, because a bound state requires the total energy (kinetic and potential) to be <0. If the energy were 0, a system of two particles would have the same amount of energy as the case where the two particles are at infinite distance apart (0 potential energy) and at rest (0 kinetic energy).

Let's make an analogy with gravity. A fast enough planet passing by a star will not orbit the star. This is despite the force of gravity moving at the speed of light.

Let's make an analogy with atoms. A fast enough electron passing by a proton does not make an atom. This is despite the force of electromagnetic moving at the speed of light.

The fact that gluons move at the speed of light does not force quarks to be bound. So what determines if something is bound?

In the case of a planet orbiting a star, the planet has total negative energy due to its potential energy $-\frac{GMm}{r}$. For a circular orbit, the velocity will obey $\frac{v^{2}}{r} = \frac{GM}{r^{2}}$ so the kinetic energy is $\frac{1}{2}mv^{2} = \frac{1}{2}\frac{GMm}{r}$. So you can see the total energy (potential + kinetic) is negative. This is what defines a bound system, as you would need to put in energy for the planet to increase its distance from the star.

A similar analysis holds for an atom. Electrons have negative energy <=> they are bound to nuclei. Energy must be put in if you want to free the electron from the nucleus, ionizing the atom.

This is also why the Earth has an escape velocity. To become unbound from the Earth, a space traveller must have enough kinetic energy per mass.

Edit: Aside from this confusion in your scenario #1, there is also the physics of asymptotic freedom, which you can read about here. In short, at closer distances, the strong force becomes weaker.

Answer 3

How exactly does the quark gluon plasma prevent protons from forming?

Look at the temperature of the quark gluon plasma stage

Schematic phase diagram of QCD. The vertical axis indicates temperature TT, the horizontal axis indicates baryon density. At low enough temperature quarks and gluons only appear as hadron bound states (confinement). But above a critical temperature these hadron bound states break apart (deconfinement) and quarks and gluons may exist freely. This phase of QCD is the quark-gluon plasma.

In a cosmological model the quark gluon plasma temperature is four orders of magnitude higher, $10^{15}K$, than the MeV values $~1x10^{10}$ of nuclear bound states.

What does the temperature mean in a cosmological model? Thermodynamically it is the black body distribution temperature. There exist tails in the black body distribution, so in the soup of quarks gluons electrons positrons photons etc there will be a tiny probability for a proton to form, and a high probability to disintegrate again in a collision with one of the particles in the soup.