QCD: Asymptotic Freedom & Beta Function Explained

[Bobhawke]

In every textbook I've seen, in order to show that QCD is asymptotically free the first coefficient of the beta function is calculated and shown to be negative. However, neglecting the terms that are higher order in the coupling is only allowed if we know the coupling is small, and we don't know when the coupling is small until we've calculated the beta function! This reasoning seems circular to me. Any explanation of this would be appreciated.

 

[Avodyne]

The first term is sufficient if the coupling g is very small. Then we know it gets smaller and smaller as we go to higher and higher energies.

There is no evidence from higher-order corrections (or anything else) that the beta function ever turns around and becomes positive, but it is logically possible. Suppose this happened at g=g*. Then the physics depends on whether g>g* or g<g*. If g<g*, the coupling goes to zero at high energies (asymptotic freedom), but at low energies gets stuck at g=g*. It seems to me this would lead to unconfined quarks, but I'm not sure if this is known to be the case. If g>g*, then the coupling would become arbitrarily large at low energies, but stop at g=g* at high energies. This would seem to be inconsistent with the small measured values of g at high energies; I think these values are within the range where we are confident in the calculation of the beta function.

So, putting all the evidence together, there is a pretty strong case for a beta function that is always negative.

 

[sir-pinski]

The concept of asymptotic freedom in QCD is a crucial aspect of the theory that has been extensively studied and verified through experiments. It is based on the fact that the strong nuclear force, described by QCD, becomes weaker at high energies or short distances. This means that the coupling constant, which characterizes the strength of the force, decreases as the energy increases or the distance decreases.

The beta function, as you mentioned, is a fundamental quantity in QCD that describes how the coupling constant changes with energy or distance. In order to show that QCD is asymptotically free, we need to calculate the first coefficient of the beta function and show that it is negative. This is because a negative coefficient indicates that the coupling constant decreases as the energy increases, which is the hallmark of asymptotic freedom.

You are correct in pointing out that neglecting higher-order terms in the beta function is only allowed if we know that the coupling constant is small. However, the initial calculation of the beta function is done assuming a small coupling constant, and then it is verified through experiments. This is not a circular reasoning, but rather a self-consistent approach. The calculation of the beta function is based on perturbation theory, which assumes a small coupling constant and expands the equations in terms of it. This is a common approach in theoretical physics, where we start with a simplified model and then refine it as we gather more information.

In conclusion, the concept of asymptotic freedom in QCD is not circular reasoning, but rather a self-consistent approach that has been extensively verified through experiments. The calculation of the beta function and its first coefficient is an important step in understanding the behavior of the strong nuclear force and its implications in the subatomic world.