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Corrigendum to “The Gribov problem in presence of background field for SU(2) Yang–Mills theory” [Phys. Lett. B 763 (2016) 94–101](S0370269316305950)(10.1016/j.physletb.2016.10.024))

par Canfora, Fabrizio F.E.;Hidalgo, Diego;Pais Hirigoyen, Pablo
Référence Physics letters. Section B, 763, page (94-101)
Publication Publié, 2016-12

Article révisé par les pairs

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Résumé :

We report a relevant conceptual issue behind Eq. (44). In Section 3, we consider necessary to study the gap equation at non-zero, but small enough temperature, i.e., our background field written as [Formula presented] where r is a dimensionless parameter related to the background field [Formula presented] defined in Equation (23) as [Formula presented], with T the temperature. At a first glance, the above background field should have no physical at all as it is pure gauge. However, the gauge transformation which would remove it is not periodic in Euclidean time. Thus, when [Formula presented] (no matter how small), such a gauge transformation is improper and it is not allowed. That is why it is conceptually important to include a non-vanishing temperature in the analysis of the gap equation. This approach is very close to the Polyakov-loop treatment [32], but in this case we will focus only on the dependence of the Gribov parameter with respect to the background parameter, keeping the temperature constant. The procedure to find the gap equation is pretty the same as before, the only difference is the Equation (44), which is written as [Formula presented] In the last definition, we expanded in the Fourier space the zero-component momentum in the Matsubara bosonic frequencies [Formula presented], and [Formula presented] denotes the spatial momentum vector. We will compute [Formula presented] function using similar techniques which were already applied in GZ approach, which lead us the result (see details in Appendix of the new arXiv version and in [32]), [Formula presented] where [Formula presented] is the modified Bessel function of the second kind extended to the complex plane. In order to normalize the gap equation (43), we shall choose [Formula presented] such that for [Formula presented] the solution is [Formula presented]. Thus, [Formula presented] where [Formula presented] In Fig. 1 (which replaces the Figure 2 of the original text) it is plotted the left hand side of the above gap equation as a function of [Formula presented] for different values of r in (a). In Fig. 1(b) it is plotted the zeros of the gap equation as a function of r at [Formula presented]. The present expression of the Gribov parameter is only valid as long as [Formula presented]. The points where the latter derivative vanishes could signal a change on the phase diagram. Thus, when [Formula presented], the present semi-classical approximation is not valid anymore. Therefore, we have included the plots in the above Figure only the region in which our approximation can be trusted. All the main conclusions remain unchanged. The e-print arXiv version of the manuscript was accordingly updated. The authors would like to thank David Dudal and David Vercauteren for constructive criticism and important suggestions on these relevant issues.