The above perturbative calculation of the Wilson loop in two dimensions is very similar to the one performed by Staudacher and Krauth in [13]. There the calculation is in the plane and using the light-cone gauge, while we have the calculation on S2 and employ a diﬀerent gauge. Still we checked for the circular loop that both prescriptions in two dimensions lead to the same result, and as we will show below, in that case it agrees also with the four-dimensional expressions. The important part about the calculation in [13] is not the choice of gauge, but the choice of regularization prescription of a pole in the derivation of the conﬁgurationspace propagator. The one they used was proposed by Wu, Mandelstam and Leibbrandt (WML) [14–16] and it can be used also in Euclidean signature space.
The perturbative calculation of the Wilson loop in two dimensions described here closely resembles the approach taken by Staudacher and Krauth in [13]. While their work was conducted in the plane using the light-cone gauge, our calculation is performed on \\( S^2 \\) with a different gauge choice. Despite these differences, we verified that both methods yield identical results for the circular loop in two dimensions, and as will be shown later, these results also align with the four-dimensional expressions. The key aspect of the calculation in [13] lies not in the gauge choice but in the regularization prescription used to address a pole in the configuration-space propagator. They employed the Wu, Mandelstam, and Leibbrandt (WML) prescription [14–16], which is also applicable in Euclidean signature space.
The perturbative calculation of the Wilson loop in two dimensions described above is quite comparable to the one conducted by Staudacher and Krauth. In their work, the calculation is performed in the plane and uses the light-cone gauge, while our analysis is on S2 and involves a different gauge. Nevertheless, we verified that for the circular loop, both methods in two dimensions yield the same outcome, which we will demonstrate also aligns with the four-dimensional results. A key aspect of the calculation presented by Staudacher and Krauth is not the gauge choice but rather the regularization technique employed for the pole in deriving the configuration space propagator. The regularization they chose was proposed by Wu, Mandelstam, and Leibbrandt, and it is applicable in Euclidean signature space as well.
The perturbative calculation of the Wilson loop in two dimensions above is very similar to the one by Staudacher and Krauth. In their paper the calculation is done in the plane and with the light-cone gauge, whereas our analysis is on S 2 and with a different gauge. However, we checked that for the circular loop, both approaches in two dimensions give the same result, which we will show also corresponds to the four-dimensional results. The most important part of the calculation by Staudacher and Krauth is not the gauge choice but the regularization method used for the pole in the calculation of the configuration space propagator. The regularization they selected was introduced by Wu, Mandelstam and Leibbrandt and is also valid in Euclidean signature space.
The perturbative calculation of the Wilson loop in two dimensions above is very similar to the one by Staudacher and Krauth. In their calculation the calculation is done in the plane and in the light-cone gauge, whereas our analysis is on S2 and in a different gauge. However, we checked that for the circular loop, both approaches in two dimensions give the same result, which we will show also agrees with the four-dimensional results. A crucial part of the calculation of Staudacher and Krauth is not the gauge but the regularization method used to extract the pole in the derivation of the configuration space propagator. The regularization they used was suggested by Wu, Mandelstam, and Leibbrandt, and it works in Euclidean signature space too.