How exactly does a proton form from quarks? What is the exact sequence and mechanism?

Question

What are the steps that lead to the bonding of two up quarks and one down quark into a proton? For instance, does an up quark "bind" with a down quark in quark-gluon plasma, which then binds with another lone up quark? I've been exploring the formation of protons from quarks, and I'm facing challenges in understanding the exact sequence and rules that govern this process.

Background: I've been working on simulations of quark interactions and seeking to understand how protons form from the quark-gluon plasma. I have tried modeling QCD-based interactions between individual quarks of varying types, but that usually just leads to mechanisms for how mesons are created or how annihilations occur. I feel confused about how a baryon forms as a result of color neutral formations of quarks, and how an odd number of quarks can form despite quarks being generated in pairs. Any insights or references would be greatly appreciated!

Edit: As a response to the first comment I received, I've decided to include some context in how I simulated quark interactions. I made a program to simulate the outcome of two quarks interacting according to my understanding of QCD rules. I defined each quark by three properties:

1. The type of quark, which I call the type. e.g. up, down, charm, etc.
2. Whether the quark is an anti particle or not. I call this the family. Anti-particles belong to one family, while normal particles belong to another.
3. The color charge of the particle - red, green, or blue. If I have an anti up-quark with a red charge and a normal down-quark with a blue charge, they'd be defined by these properties. I then use a function to determine the outcome of their interaction:

/**
   * Determines whether the color charge of this quark and another quark is neutral or not in a 1on1 interaction.
   * @param q Another quark.
   * @returns True if neutral, false if not.
   */
  neut(q: Quark)
  {
    if (this.fdiff(q.f)) return this.csame(q.c); // Diff fam, same color in true case.
    else return false; // Not neutral, for example due to having the same fam, e.g. anti and anti 
  }

I am open to, and welcome any corrections in my understanding.

Answer 1

1. What are the steps that lead to the bonding of two up quarks and one down quark into a proton?
2. How do the strong force and color charge play roles in this process?

It is an elaborate theory (QCD), Quantum Chromodynamics, involving "gluon" particles that bind given quarks together into protons. The theory is complicated, but convincingly simulated in lattice gauge theory. You need to take a long advanced course in this to appreciate it.

3. If quarks "spawn" in pairs, and confinement causes them to generate new pairs upon separation, how do baryons get the third quark in the mix?

They don't, really: baryon number is mostly conserved, since the "spawning" is in quark-antiquark pairs; and it is thought to have fixed itself in the early, hot, mysterious universe. If you start with zero baryon number particles, you end up with equal numbers of nucleons and anti nucleons, so no net quark number. In our universe now, there is net quark and baryon number.

4. Are there any recent theories or models that explain this phenomenon in detail?

Why "recent"? These theories, QCD, have been around for half a century now. they are very subtle and difficult, but thousands of theorists are working on them. The final practical steps are addressed in hadronization Monte-Carlo programs.

Answer 2

The answer to your question is unfortunately "you probably need to be more precise with what you are asking". As your question is currently phrased, I would argue there is no such thing as an exact sequence of proton formation from quarks. I'll list some statements below, which may or may not be helpful

• The first thing to mention is that isolated quarks don't exist as actual states in QCD, you can never experimentally isolate a single quark due to confinement. When doing perturbative calculations, you have a choice of working with the underlying QCD lagrangian (only converges at high energies where the coupling is weak, and has little knowledge about hadronization), or working with an Effective Field Theory such as Chiral Perturbation theory, which has little knowledge of the quarks.

• As others have mentioned, the only nonperturbative approach we have to QCD is lattice-QCD. Even in the lattice-QCD framework, its not clear how to write down what you are asking for. Imagining for a second we have supercomputers capable of realtime simulations, you can reasonably imagine preparing an initial quark state with two up quarks and a down quark at some separation, letting them evolve forward in time, and projecting onto a proton state. This involves writing down the QCD hamiltonian and performing the evolution, but its not clear how to phenomenologically extract an 'exact sequence of steps' here.

• The closest thing I know of to what you are asking is what is done for example in showering programs such as Pythia, if you look in their documentation at the Lund String model. I have to emphasize that this is purely a phenomenological fit, these models are not "reality". Their algorithms have many different parameters, that are simply fit to the data we observe at particle colliders.

• Even though it's not clear how to ask about proton formation, it's still meaningful to ask about proton structure. For example, with lattice-QCD you can write down different types of proton operators (e.g. you can have two quarks contracted in a diquark), and see how much these operators overlap onto the actual nonperturbative proton state.