Yang-Mills two-point functions in linear covariant gauges

In this paper we use two different but complementary approaches in order to study the ghost propagator of a pure SU(3) Yang-Mills theory quantized in the linear covariant gauges, focusing on its dependence on the gauge-fixing parameter $\xi$ in the deep infrared. In particular, we first solve the Schwinger-Dyson equation that governs the dynamics of the ghost propagator, using a set of simplifying approximations, and under the crucial assumption that the gluon propagators for $\xi >0$ are infrared finite, as is the case in the Landau gauge $(\xi =0)$. Then we appeal to the Nielsen identities, and express the derivative of the ghost propagator with respect to $\xi$ in terms of certain auxiliary Green’s functions, which are subsequently computed under the same assumptions as before. Within both formalisms we find that for $\xi >0$ the ghost dressing function approaches zero in the deep infrared, in sharp contrast to what happens in the Landau gauge, where it is known to saturate at a finite (nonvanishing) value. The Nielsen identities are then extended to the case of the gluon propagator, and the $\xi$-dependence of the corresponding gluon masses is derived using as input the results obtained in the previous steps. The result turns out to be logarithmically divergent in the deep infrared; the compatibility of this behavior with the basic assumption of a finite gluon propagator is discussed, and a specific Ansatz is put forth, which readily reconciles both features.

Figure 1

The ghost gap equation. White (respectively, black) blobs represent connected (respectively, one-particle irreducible) Green’s functions.

Figure 2

The lattice SU(3) gluon propagator evaluated in the Landau gauge [4] and the corresponding fit used in our calculation [27]. The dashed curve shows a fit featuring an IR maximum which is due to the presence of (divergent) contributions to the gluon (inverse) dressing function [42]. All functions are renormalized at $\mu =4.3\mathrm{GeV}$.

Figure 3

Solution of the SDE (2.14) (upper panel) and the associated ghost propagator (lower panel) for various values of the gauge fixing parameter $\xi$. In the IR the solution obtained is perfectly described by Eq. (2.17) after fitting for determining the value of the arbitrary constant $a$. For comparison we plot also the Landau gauge lattice data of [4].

Figure 4

One-loop diagrams contributing to the auxiliary functions ${\mathrm{\Gamma }}_{\overline{c}\chi A_{\mu }}$ and ${\mathrm{\Gamma }}_{c\chi c^{*}}$ appearing in the ghost two-point Nielsen identity (3.1). Notice the presence of the mixed propagator ${\mathrm{\Delta }}_{bA}$.

Figure 5

Contributions of the one-loop dressed auxiliary functions to the ghost two-point function Nielsen identity. The IR region is perfectly described by the predicted $c_{\mathrm{NI}}\log q^2/{\mu }^2$ behavior yielding $c_{\mathrm{NI}}=0.33$.

Figure 6

One-loop diagrams contributing to the auxiliary function ${\mathrm{\Gamma }}_{A_{\mu }\chi A_{\nu }^{*}}$ appearing in the gluon two-point Nielsen identity (3.12).

Figure 7

Contributions of the one-loop dressed auxiliary functions to the gluon two-point function Nielsen identity. Also in the gluon case the IR region is perfectly described by the predicted $b\log q^2/{\mu }^2$ behavior, now with $c_{\mathrm{NI}}=0.13$.

Figure 8

$\xi$-dependence of the gluon mass (left panels) and gluon propagator (right panels) as predicted by the Ansatz (3.20) for $a_1=-0.2$ (upper panels) and $a_1=0.2$ (lower panels).

Figure 9

$\xi$-dependence of the gluon mass (left) and gluon propagator (right) as predicted by the Ansatz (3.20) for $a_1=0$.

Figure 10

Feynman rules for the $b$-sector. Notice that, due to the $b$-equation (a14), there are no possible quantum correction to these Feynman rules.