# Computation of String Tension in Lattice QCD

There is a quantity called String Tension in lattice QCD calculation.

How is this quantity (String Tension) defined and computed in Lattice QCD? Are there some useful formulas to define it both in the continuum as well as on the lattice?

The attached figure shows an example about the result of computations in gauge theories :

Thanks for the answer, if you can help please!

## 1 Answer

Quick answer: The string tension $\sigma$ as shown in that plot (which looks like an older version of Fig. 11 in arXiv:1004.3206) is the dimension-2 coefficient of the linear term in the static potential $V(r)$ (the energy of two infinitely heavy probes separated by spatial distance $r$). So one computes $V(r)$, typically from the exponential decay of rectangular $r\times T$ Wilson loops oriented along the temporal direction, $W(r,T) \propto e^{-V(r)\cdot T}$, and then fits $V(r) = -\frac{C}{r} + A + \sigma r$ to determine $\sigma$.

The stuff above is in continuum language, implicitly working in "lattice units" where the dimensionful lattice spacing is set to $a=1$. Non-zero lattice spacing leads to discretization artifacts in the predictions for the dimensionless combination $a^2 \sigma$, which are removed by extrapolating $a \to 0$. One can play games with lattice perturbation theory to reduce these artifacts.

This is one of the simplest non-trivial calculations one can do in lattice gauge theory, so it should be covered comprehensively in the standard textbooks/lecture notes. Silvia Necco's PhD thesis, hep-lat/0306005, might be a good starting point.