Wilson Loops, Confinement, and the Quark-Antiquark Potential

Hello! Welcome back to our study of computational quantum field theory.

In our last lesson, we established a crucial computational tool: the ability to calculate the vacuum expectation value of a Wilson loop, $⟨ W (C)⟩$ , by averaging over an ensemble of lattice gauge field configurations. We ended with a tantalizing suggestion that the behavior of these loops as a function of their size holds the key to understanding one of the most profound and non-perturbative phenomena in the Standard Model: quark confinement.

Today, we will unlock that connection. Our goal is to relate the area law behavior of Wilson loops to the static quark-antiquark potential and the concept of confinement. We will move from the "how-to" of the calculation to the "why" of its physical significance, laying the complete theoretical groundwork for numerically measuring the force that binds quarks together.

1. The Wilson Loop as a Quark-Antiquark Propagator

First, let's establish the physical interpretation of the rectangular Wilson loop. Consider a rectangular loop $C$ of spatial extent $R$ and temporal extent $T$ in Euclidean spacetime. This object can be understood as tracing the worldline of a static quark-antiquark pair.

At Euclidean time $τ = 0$ , a quark-antiquark pair is created from the vacuum at a separation $R$ .
The pair propagates for a time $T$ without moving spatially. The quark's worldline is a straight line from $(0, 0)$ to $(T, 0)$ , and the antiquark's worldline is from $(0, R)$ to $(T, R)$ .
At time $τ = T$ , the pair annihilates back into the vacuum.

The Wilson loop $W (R, T)$ is precisely the operator that describes this process. Its vacuum expectation value, $⟨ W (R, T)⟩$ , represents the quantum mechanical amplitude for this entire history to occur.

In quantum mechanics, the amplitude for a system to evolve for a Euclidean time $T$ is dominated by its ground state energy $E_{0}$ for large $T$ : $⟨ ψ_{f} ∣ e^{- H T} ∣ ψ_{i} ⟩ \approx e^{- E_{0} T}$ . In our case, the "system" is the static $q q ˉ$ pair, and its ground state energy is the static potential energy between them, $V (R)$ . Therefore, for large temporal extent $T$ , we have the fundamental relation:

⟨ W (R, T)⟩ \sim e^{- V (R) T}

This equation is our bridge from a numerically calculable quantity, $⟨ W (R, T)⟩$ , to a deeply physical one, the potential $V (R)$ .

2. Confinement and the Area Law

The defining feature of Quantum Chromodynamics (QCD) is confinement: quarks are never observed as free particles. No matter how much energy you pump into a hadron, you can't knock a single quark out; you just create more quark-antiquark pairs that form new hadrons.

What does this imply for the potential $V (R)$ ? It means the energy required to separate a quark and antiquark must grow infinitely with their separation $R$ . The simplest potential with this property is a linear one:

V (R) = σ R

Here, $σ$ is a constant known as the string tension. The force between the quarks is $F = - \frac{d V}{d R} = - σ$ , a constant, much like the force required to stretch an idealized elastic string. This gives rise to the physical picture of quarks being connected by a "flux tube" of gluonic fields that has a constant energy per unit length.

Now, let's substitute this confining potential into our fundamental relation for the Wilson loop:

⟨ W (R, T)⟩ \sim e^{- (σ R) T} = e^{- σ (R T)}

Since $A = R T$ is the area of the loop, we arrive at the famous Area Law:

⟨ W (R, T)⟩ \sim e^{- σ A}

The observation of an exponential decay of the Wilson loop's expectation value with the area of the loop is the definitive signature of confinement in a lattice gauge theory simulation.

In contrast, a theory like QED is not confining. The potential between an electron and positron is the Coulomb potential, $V (R) \sim - 1/ R$ , which goes to a constant at large $R$ . This leads to $⟨ W (R, T)⟩$ decaying with the perimeter of the loop, $P = 2 R + 2 T$ , a behavior known as the Perimeter Law.

4. Lattice Gauge Theory

The Area Law can be derived analytically in a specific limit of lattice QCD known as the strong coupling expansion (eta o 0). The following notes from David Tong's lectures on Gauge Theory demonstrate this derivation.

Please read the subsection titled 'Confinement' (pages 213-215). You don't need to follow every step of the strong coupling expansion in detail. The key is to appreciate the final result on page 215: the leading-order contribution to the Wilson loop is shown to be proportional to e^{-\sigma A}, with an explicit formula for the string tension \sigma in terms of the lattice spacing a and coupling eta. This provides a firm theoretical justification for the Area Law.

3. Extracting the Static Potential and String Tension

We now have a clear objective: calculate $⟨ W (R, T)⟩$ for various $R$ and $T$ , and check if the data follows an area law. If it does, we can extract the string tension $σ$ . The following practical procedure is standard in lattice QCD studies.

Step 1: Extracting the Potential V(R)

From the relation $⟨ W (R, T)⟩ \approx C e^{- V (R) T}$ , we can isolate $V (R)$ . A robust way to do this is to take ratios of Wilson loops that have the same spatial size $R$ but differ in temporal extent by one lattice spacing, $a$ .

\frac{⟨ W ( R , T + a )⟩}{⟨ W ( R , T )⟩} \approx \frac{C e ^{- V (R) (T + a)}}{C e ^{- V (R) T}} = e^{- aV (R)}

Taking the natural logarithm, we can solve for the potential (in lattice units):

aV (R) = - ln (\frac{⟨ W ( R , T + a )⟩}{⟨ W ( R , T )⟩})

In practice, one computes this ratio for several values of $T$ (for a fixed $R$ ) and looks for a "plateau" where the result becomes independent of $T$ , indicating that the large- $T$ limit has been reached.

Alternatively, one can take the logarithm of the Wilson loop itself: $ln ⟨ W (R, T)⟩ \approx ln C - V (R) T$ . For a fixed $R$ , a plot of $ln ⟨ W (R, T)⟩$ versus $T$ should be a straight line for large $T$ , and its slope gives $- V (R)$ .

Step 2: Extracting the String Tension σ

After repeating Step 1 for several different spatial sizes $R$ , you will have a set of data points for the potential, ${(R_{1}, V (R_{1})), (R_{2}, V (R_{2})), \dots}$ . The final step is to plot $V (R)$ as a function of $R$ .

For small $R$ , QCD predicts a Coulomb-like potential, $V (R) \sim - k / R$ , due to one-gluon exchange.
For large $R$ , confinement dominates, and we expect the linear behavior $V (R) \approx σ R$ .

The full potential is often well-described by the Cornell potential:
$V (R) = σ R - \frac{k}{R} + V_{0}$

By plotting your computed values of $V (R)$ versus $R$ , you can perform a linear fit to the points at large $R$ . The slope of this fit is your numerical, non-perturbative measurement of the string tension $σ$ .

Determining the String Tension Using QCD Lattice Simulations

The following excerpts from a bachelor's thesis by Piter Annema provide a clear, practical guide to this entire process, complete with illustrative plots from an actual simulation.

Please study these sections carefully: Skim the introduction (part of Section 1) to see how confinement and string tension are introduced. Review '3.4 Wilson Loops' (page 12) for a concise definition of the rectangular Wilson loop. Read '3.5 Measuring the String Tension' (pages 13-14) in detail. This section explicitly outlines the method of using ratios and logarithmic plots to find V(R), as we just discussed. Pay close attention to Figure 3.7, which shows exactly the kind of linear fit of \ln W_L vs. t used to extract the potential. Finally, look at Figure 3.9 on page 15. This is the payoff: a plot of the extracted potential aV(r) vs. r/a, showing both the short-range Coulomb behavior and the long-range linear confinement. The solid line is the fit used to determine the string tension \sigma.

Test your understanding!

Suppose your simulation yields the following (hypothetical) data for the static potential:

V(R=3a) = 0.5
V(R=4a) = 0.65
V(R=5a) = 0.80
V(R=6a) = 0.95

Assuming these points are in the linear, confining region, what is your estimate for the string tension $σ$ in lattice units (i.e., $σ a^{2}$ )?

Show answer

The string tension $σ$ is the slope of the $V (R)$ vs. $R$ plot. We can estimate it by taking the difference between adjacent points:

From R=3a to 4a: $Δ V /Δ R = (0.65 - 0.5) / (4 a - 3 a) = 0.15/ a$ .
From R=4a to 5a: $Δ V /Δ R = (0.80 - 0.65) / (5 a - 4 a) = 0.15/ a$ .
From R=5a to 6a: $Δ V /Δ R = (0.95 - 0.80) / (6 a - 5 a) = 0.15/ a$ .

The slope is consistently $0.15/ a$ . Therefore, the string tension is $σ = 0.15/ a^{2}$ , or in dimensionless lattice units, $σ a^{2} = 0.15$ . A more robust approach would be to perform a linear regression on all the points.

Conclusion

Today, we have forged the complete conceptual link between the numerical output of a lattice simulation and the physics of quark confinement. This is a cornerstone of non-perturbative QFT.

Key Takeaways:

The expectation value of a large rectangular Wilson loop of size $R \times T$ behaves as $⟨ W (R, T)⟩ \sim e^{- V (R) T}$ , where $V (R)$ is the static potential between a quark and an antiquark.
Confinement implies a linearly rising potential, $V (R) = σ R$ , where $σ$ is the string tension.
A confining potential leads directly to the Area Law for Wilson loops: $⟨ W (R, T)⟩ \sim e^{- σ A}$ . Observing this behavior is a primary goal of lattice QCD simulations.
By calculating ratios of Wilson loop expectation values, we can numerically determine the potential $V (R)$ .
A linear fit to the large-distance part of the $V (R)$ vs. $R$ data yields a quantitative, first-principles prediction for the string tension $σ$ .

Preview of the Next Lesson:

You now have all the theoretical and methodological tools required. In our next lesson, you will put this knowledge into practice. The learning outcome will be to numerically compute the static potential from Wilson loop data and identify the linear confining term. You will write the code to analyze the output of the simulation you built, extract $V (R)$ , plot the results, and perform the fit to measure the string tension yourself.

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