Gribov’s horizon and the ghost dressing function

Abstract

We study a relation recently derived by K. Kondo at zero momentum between the Zwanziger's horizon function, the ghost dressing function and Kugo's functions u and w. We agree with this result as far as bare quantities are considered. However, assuming the validity of the horizon gap equation, we argue that the solution wð0Þ ¼ 0 is not acceptable since it would lead to a vanishing renormalized ghost dressing function. On the contrary, when the cutoff goes to infinity, uð0Þ ! 1, wð0Þ ! À1 such that uð0Þ þ wð0Þ ! À1. Furthermore w and u are not multiplicatively renormalizable. Relaxing the gap equation allows wð0Þ ¼ 0 with uð0Þ ! À1. In both cases the bare ghost dressing function, Fð0; ÃÞ, goes logarithmically to infinity at infinite cutoff. We show that, although the lattice results provide bare results not so different from the Fð0; ÃÞ ¼ 3 solution, this is an accident due to the fact that the lattice cutoffs lie in the range 1-3 GeV À1 . We show that the renormalized ghost dressing function should be finite and nonzero at zero momentum and can be reliably estimated on the lattice up to powers of the lattice spacing; from published data on a 80 4 lattice at ¼ 5:7 we obtain F R ð0; ¼ 1:5 GeVÞ ' 2:2.

Strong Running Coupling atτandZ0Mass Scales from Lattice QCD

“…(2) is dominated by the first correction introduced by the nonvanishing dimension-two Landau-gauge gluon condensate [16][17][18][19][20][21], where the Wilson coefficient is applied at the O(α 4 )-order. In the previous methodological paper [3], we provided with a strong indication that the OPE analysis is indeed in order: it was clearly shown that the lattice data could be only explained by including non-perturbative contributions and that the Wilson coefficient for the Landau-gauge gluon condensate was needed to describe the behaviour of data above p 4 GeV and up to p 7 GeV (see next Fig.…”

Refining the detection of the zero crossing for the three-gluon vertex in symmetric and asymmetric momentum subtraction schemes

“…Both are nonperturbative definitions for the QCD running coupling, that have been extensively studied on the lattice [56][57][58][59][60][61]. Other MOM schemes based on different QCD vertices and kinematical configurations, as that for the ghost-gluon vertex [62][63][64][65], lead to alternative nonperturbative definitions although they can be related at any order in perturbation theory [66,67] …”

Renormalization of quark propagators from twisted-mass lattice QCD atNf=2

“…It turns out that the dominant non-perturbative correction in Landau gauge is due to the non vanishing vacuum expectation value of the only dimension-two operator : A 2 ≡ A a µ A aµ [1], and that it is not small [2][3][4][5][6][7][8]. It is thus necessary to carefully look for the possibility of such a contribution, which appears in the OPE, as a 1/p 2 contribution up to logarithmic corrections.…”

“…The scheme for the anomalous dimension of Z A 2 is imposed through the renormalization of the local operator A 2 , as was done in ref. [2] to obtain its leading logarithm contribution, and it is only known in the MS at the order O(α 4 ) [29]. Then, that is the only possible choice of scheme for γ A 2 in eqs.…”

“…This OPE analysis is analogous to the one performed in refs. [2,3,5,6]. The starting point is the OPE of the inverse of the quark propagator: where only the leading term in p / is kept, the quark mass being assumed to be negligible or to vanish.…”