Pseudoscalar $q\overline{q}$ mesons and effective QCD coupling enhanced by $⟨A^2⟩$ condensate

Recent developments provided evidence that the dimension 2 gluon condensate $⟨A^2⟩$ is important for the nonperturbative regime of Yang-Mills theories (quantized in the Landau gauge). We show that it may be relevant for the Dyson-Schwinger approach to QCD. In order that this approach leads to a successful hadronic phenomenology, an enhancement of the effective quark-gluon interaction seems to be needed at intermediate ($p^2\sim 0.5{\mathrm{G}\mathrm{e}\mathrm{V}}^2$) momenta. It is shown that the gluon condensate $⟨A^2⟩$ provides such an enhancement. It is also shown that the resulting effective strong running coupling leads to the sufficiently strong dynamical chiral symmetry breaking and successful phenomenology at least in the light sector of pseudoscalar mesons.

Figure 1

The momentum dependence of various strong running couplings mentioned in the text. JM [4] and MRT [3, 6] ${\alpha }_{\mathrm{e}\mathrm{f}\mathrm{f}}(p^2)$ are depicted by, respectively, dashed and dash-dotted curves. The effective coupling (18) proposed and analyzed in the present paper is depicted by the solid curve, and ${\alpha }_{\mathrm{s}}(p^2)$ (3) of Fischer and Alkofer [2] (fit A) by the dotted curve. The long-dashed curve is the fit (extrapolated all the way to $p^2=0$) of the lattice results of Ref. 29.

Figure 2

The momentum dependence of the ghost renormalization function $G(p^2)$. The solid curve is our result (22). The dotted curve is a result of the ab initio DS study [2]. Concretely, it depicts the fit A in Eq. (41) of Ref. 2 evaluated at $\mu =5\mathrm{G}\mathrm{e}\mathrm{V}$, for which value the dotted curve agrees rather well (in the displayed momentum range) with the long-dashed curve, representing Eq. (4) of the lattice Ref. 29, renormalized at $\mu =5\mathrm{G}\mathrm{e}\mathrm{V}$. Our result (22) is therefore plotted for the same value of $\mu$.

Figure 3

The momentum dependence of the gluon renormalization function $Z(p^2)$. The solid curve is our result (21). The densely dotted curve is the ab initio DS result (fit A) from Refs. 2, 32. As in Fig. 2, both the ab initio DS result and our Eq. (21) are plotted for $\mu =5\mathrm{G}\mathrm{e}\mathrm{V}$. The other curves pertain to some recent lattice results [29, 33, 34, 35]. The ones that agree with the ab initio DS result [2, 32] very well are those of Leinweber et al. [33] [their Eq. (5.14), depicted here by the sparsely dotted curve], and Eq. (3) of Iida et al. [34] (dash-dotted curve) fitting the lattice data of Ref. 35. The third lattice result for $Z(p^2)$ is displayed by the long-dashed curve. It corresponds to Eq. (3) of Ref. 29, which is the fit (extrapolated all the way to $p^2=0$) to the lattice data of Ref. 29. For all lattice results, the renormalization condition $Z({\mu }^2)=1$ is imposed at $\mu =5\mathrm{G}\mathrm{e}\mathrm{V}$.

Figure 4

The momentum dependence of the dynamically generated quark mass $M(p^2)$ for $u$ and $d$ quarks. The solid curve is our result for the parameters giving the second line of Table , but our $M(p^2)$ depends in fact very little on the small explicit chiral symmetry breaking mass parameters ${\tilde{m}}_u$ and ${\tilde{m}}_d$ of the very light $u$ and $d$ quarks as long as their values are at all realistic. The dotted curve is the ab initio DS result [2, 32]. The short-dashed curve is the $M(p^2)$ fit of Ref. 48 to the extrapolation of their lattice data to the chiral limit. The dash-dotted curve is the similar result from another lattice calculation, namely, the fit of $M(p^2)$ from Fig. 14 of Ref. 49.