Calculating Static Potential from Wilson Loops

The Goal: From Numbers to Physics

The end goal of our analysis is to produce a plot of the static quark-antiquark potential, $V(R)$, as a function of their separation, $R$. This potential is a cornerstone of our understanding of the strong force.

The Method: A Practical Refresher

The entire procedure relies on the fundamental relation we derived previously: for large Euclidean time $T$,

where $C$ is an overlap factor.

To eliminate both $C$ and the exponential, we can take a ratio of Wilson loops with temporal extents differing by one lattice spacing, $a$. For our analysis, we will work in lattice units where $a=1$. This gives us the "effective potential":

For large $T$, this effective potential should approach a constant value, which is the static potential we seek:

Once we have $V(R)$ for several values of $R$, we can plot it and fit it to the Cornell potential model, $V(R)=\sigma R-\frac{k}{R}+V_0$, or simply fit the large-$R$ region to a line, $V(R)\approx \sigma R+\text{const}$, to find the string tension $\sigma$.

Let's quickly review a resource that walks through this exact process.

A Numerical Walkthrough

Let's assume our Monte Carlo simulation of a U(1) gauge theory on a 16^4 lattice has produced a data file, wilson_loops.dat. The file contains the averaged real part of the expectation value of planar Wilson loops, $⟨W(R,T)⟩$, for various integer values of $R$ and $T$ (in lattice units).

A sample of the data might look like this:

# R, T, <W(R,T)>
1, 1, 0.654
1, 2, 0.431
1, 3, 0.284
...
2, 1, 0.425
2, 2, 0.183
2, 3, 0.079
...

Your first task would be to load this data into a convenient structure, such as a Python dictionary or a Pandas DataFrame, for easy access. Given your experience with Python and Numpy, this should be straightforward.

Step 1: Visualizing the Raw Data

Before any calculations, it's wise to plot the data to see if it behaves as expected. We anticipate an exponential decay of $⟨W(R,T)⟩$ with $T$. Therefore, a plot of $\ln ⟨W(R,T)⟩$ versus $T$ for several fixed values of $R$ should yield approximately straight lines.

This plot would be the equivalent of Figure 3 in the "Wilson Loop" resource. The slopes of these lines correspond to $-V(R)$.

Step 2: Calculating the Effective Potential

Now, we apply our main formula. For each spatial separation $R$, we compute the effective potential $V_{\text{eff}}(R,T)$ for all available values of $T$.

Here's a sketch of how you might implement this in Python, assuming your data is in a dictionary W_data[R][T]:

import numpy as np

V_eff = {}
# Assuming R_values and T_values are lists of available R and T
for R in R_values:
    V_eff[R] = {}
    for T in T_values:
        # Check if T+1 data exists
        if (T + 1) in W_data[R]:
            # Note the formula from the resource uses W(t)/W(t+1)
            # which gives V(R). My previous formula was -log(W(t+1)/W(t)). They are identical.
            ratio = W_data[R][T] / W_data[R][T+1]
            V_eff[R][T] = np.log(ratio)

Step 3: Identifying the Plateau

The next crucial step is to plot $V_{\text{eff}}(R,T)$ against $T$ for each fixed $R$. This corresponds to Figure 4 in the resource.

For small $T$, the results will be "contaminated" by higher-energy excited states of the gluon field. As $T$ increases, these faster-decaying exponentials die out, and $V_{\text{eff}}(R,T)$ should settle onto a "plateau." This plateau value is our measurement of the static potential $V(R)$.

You would visually inspect these plots to decide the range of $T$ values (e.g., $T\ge 3$) that form the plateau for each $R$. The value of $V(R)$ is then taken as the average of $V_{\text{eff}}(R,T)$ over this plateau region.

Step 4: Extracting the String Tension

After completing Step 3 for all available $R$ values, you will have a set of data points: $(R,V(R))$. This is the final result of the lattice calculation. Plotting $V(R)$ versus $R$ gives us the static potential curve, equivalent to Figure 5 in the resource.

The final step is to analyze this curve to find the string tension. We expect the potential at large $R$ to be dominated by the linear term, $V(R)\approx \sigma R+\text{const}$.

We can fit the data points for large $R$ (e.g., $R\ge 3$) to a straight line. The slope of this line is our non-perturbative measurement of the string tension, $\sigma a^2$.

In Python, using numpy.polyfit is a simple way to perform this linear regression:

import numpy as np

# Assume R_fit and V_fit contain the R and V(R) values from the linear region
# Fit to a 1st degree polynomial (a line)
slope, intercept = np.polyfit(R_fit, V_fit, 1)

string_tension_a2 = slope

print(f"String tension in lattice units (sigma * a^2): {string_tension_a2:.4f}")

Conclusion

Congratulations! You have now walked through the entire process of performing a key non-perturbative calculation in quantum field theory. By starting with the raw output of a lattice simulation, you have seen how to extract a fundamental physical parameter—the string tension—that governs quark confinement.

Key Takeaways:

• The numerical analysis of Wilson loops follows a clear, multi-step process: data ingestion, calculating the effective potential, identifying plateaus to find $V(R)$, and finally fitting $V(R)$ to extract the string tension $\sigma$.
• Plotting the data at each intermediate step ($\ln W$ vs. $T$, and $V_{\text{eff}}$ vs. $T$) is essential for verifying that the data behaves as theoretically expected and for making sound judgments about, for example, the location of the plateau.
• The final fit of the $V(R)$ vs. $R$ curve allows for a quantitative, first-principles determination of the parameters of the quark-antiquark potential, most notably the string tension $\sigma$.

Preview of the Next Lesson:

Having completed our deep dive into computational QFT and the non-perturbative phenomenon of confinement, we will now change gears significantly. We will return to the world of perturbative QFT and Quantum Electrodynamics (QED). Our next module begins with the essential, and often challenging, topic of Radiative Corrections and Renormalization. In the next lesson, you will learn to identify the ultraviolet (UV) divergences in one-loop QED diagrams, such as the electron self-energy and vacuum polarization. This will set the stage for learning the powerful techniques used to tame these infinities and make precision predictions.