No-pole condition in Landau gauge: Properties of the Gribov ghost form factor and a constraint on the2dgluon propagator

Abstract

We study general properties of the Landau-gauge Gribov ghost form factor ðp 2 Þ for SUðN c Þ Yang-Mills theories in the d-dimensional case. We find a qualitatively different behavior for d ¼ 3, 4 with respect to the d ¼ 2 case. In particular, considering any (sufficiently regular) gluon propagator Dðp 2 Þ and the one-loop-corrected ghost propagator, we prove in the 2d case that the function ðp 2 Þ blows up in the infrared limit p ! 0 as ÀDð0Þ lnðp 2 Þ. Thus, for d ¼ 2, the no-pole condition ðp 2 Þ < 1 (for p 2 > 0) can be satisfied only if the gluon propagator vanishes at zero momentum, that is, Dð0Þ ¼ 0. On the contrary, in d ¼ 3 and 4, ðp 2 Þ is finite also if Dð0Þ > 0. The same results are obtained by evaluating the ghost propagator Gðp 2 Þ explicitly at one loop, using fitting forms for Dðp 2 Þ that describe well the numerical data of the gluon propagator in two, three and four space-time dimensions in the SU(2) case. These evaluations also show that, if one considers the coupling constant g 2 as a free parameter, the ghost propagator admits a one-parameter family of behaviors ( labeled by g 2 ), in agreement with previous works by Boucaud et al. In this case the condition ð0Þ 1 implies g 2 g 2 c , where g 2 c is a ''critical'' value. Moreover, a freelike ghost propagator in the infrared limit is obtained for any value of g 2 smaller than g 2 c , while for g 2 ¼ g 2 c one finds an infrared-enhanced ghost propagator. Finally, we analyze the Dyson-Schwinger equation for ðp 2 Þ and show that, for infrared-finite ghost-gluon vertices, one can bound the ghost form factor ðp 2 Þ. Using these bounds we find again that only in the d ¼ 2 case does one need to impose Dð0Þ ¼ 0 in order to satisfy the no-pole condition. The d ¼ 2 result is also supported by an analysis of the Dyson-Schwinger equation using a spectral representation for the ghost propagator. Thus, if the no-pole condition is imposed, solving the d ¼ 2 Dyson-Schwinger equations cannot lead to a massive behavior for the gluon propagator. These results apply to any Gribov copy inside the so-called first Gribov horizon; i.e., the 2d result Dð0Þ ¼ 0 is not affected by Gribov noise. These findings are also in agreement with lattice data.

Ghost dissection

“…As ℜ(c − a − b) = 1 − 2e, we have that for e < 1/2, the hypergeometric function is finite at z = 1, with 4 Here, we assume that σ (k 2 ) attains it maximum value at k 2 = 0, which was already done by Gribov [1]. In some cases this can be shown [3], but going into those details would lead us too far now. In fact, even if σ (k 2 ) < 1 would be maximal at k 2 = k 2 * > 0, the following argument would still work out by replacing σ (0) with σ (k 2 * ).…”

“…In this proceeding, based on the work [3], we shall examine, as general as possible, the consequences of the no-pole condition; in particular, we shall focus on the specific case that the number of space-time dimensions is 2. Why our interest in this case?…”

“…A careful analysis based on rigorously derived bounds allowed to show that D(0) = 0 persists in the infinite-volume limit [6] (see also [19]). Here, we present a summary of the detailed analysis presented in [3]. The aim is to unravel a theoretical reason behind the impossibility to find a gluon propagator with…”

“…Ghost dissection David Dudal where we assumed 3 for simplicity that D(x) ∼ 1/x at large x and that also D ′ (x) goes to zero sufficiently fast at large momenta, e.g. as 1/x 2 .…”

“…Therefore, we shall first derive a set of bounds for the one-loop result (2.3), valid for general d. We need that (see [3,23])…”