Computing Polyakov Loops in Lattice QCD (Basic Question)

[Jaydeep]

Hi PhysicsForums,

I have a pretty basic question about extracting physical parameters from lattice QCD simulations. As described in "Quantum Chromodynamics on the Lattice" by Gattringer and Lang, it seems we should be able to extract the static quark/anti-quark potential by considering the behavior of the Polyakov loop correlator as a function of the distance between loops included in the correlator. Specifically, they give the equation in chapter 3,

[tex] - \log < P(m) P(n) ^\dagger> \propto V(|m-n|), [/tex] where [itex] P(m), V(a) [/itex] are the Polyakov loop from position [itex] m [/itex] and the potential across distance [itex] a [/itex], respectively. It just seems that the Polyakov loop, being a product of traces of group elements, will in general be complex, so the log is throwing me off. Are we to take both real part/absolute value prior to taking the logarithm, or am I missing something? I'm interested both in the SU(3) context, as well as the simpler U(1) case. Apologies for the simple question, but would appreciate any advice/references. Thank you!

 

[AndyW]

Hi there,

Thank you for your question. It's great to see interest in lattice QCD simulations and their applications in extracting physical parameters. To address your question, I'll provide a brief explanation of the equation you mentioned and then discuss the issue with the logarithm.

The equation you mentioned is indeed a key tool in extracting the static quark/anti-quark potential in lattice QCD simulations. The Polyakov loop, denoted as P(m), is a measure of the average color charge of a quark at position m in the lattice. By considering the behavior of the Polyakov loop correlator, <P(m)P(n)^\dagger>, as a function of the distance between the two loops, we can extract the potential between the quark and anti-quark at positions m and n, respectively.

Now, onto your question about the logarithm. You are correct in pointing out that the Polyakov loop, being a product of traces of group elements, can be complex. However, the logarithm in this equation is meant to be taken as the natural logarithm, which is defined for complex numbers. Therefore, you do not need to take the real part or absolute value prior to taking the logarithm.

In the SU(3) context, the Polyakov loop will be complex and the logarithm will give a complex number as well. But when taking the absolute value, we get a real number which is proportional to the potential between the quark and anti-quark. In the simpler U(1) case, the Polyakov loop is real and the logarithm will give a real number directly proportional to the potential.

I hope this helps clarify your question. For further reading on this topic, I recommend checking out "Quantum Chromodynamics on the Lattice" by Gattringer and Lang, as well as "Lattice Gauge Theories: An Introduction" by DeGrand and DeTar.

Best of luck with your research. Keep asking questions and exploring the fascinating world of lattice QCD simulations.