Renormalization group analysis of the gluon mass equation

We carry out a systematic study of the renormalization properties of the integral equation that determines the momentum evolution of the effective gluon mass in pure Yang–Mills theory, without quark effects taken into account. A detailed, all-order analysis of the complete kernel appearing in this particular equation, derived in the Landau gauge, reveals that the renormalization procedure may be accomplished through the sole use of ingredients known from the standard perturbative treatment of the theory, with no additional assumptions. However, the subtle interplay of terms operating at the level of the exact equation gets distorted by the approximations usually employed when evaluating the aforementioned kernel. This fact is reflected in the form of the obtained solutions, for which the deviations from the correct behavior are best quantified by resorting to appropriately defined renormalization-group invariant quantities. This analysis, in turn, provides a solid guiding principle for improving the form of the kernel, and furnishes a well-defined criterion for discriminating between various possibilities. Certain renormalization-group inspired Ansätze for the kernel are then proposed, and their numerical implications are explored in detail. One of the solutions obtained fulfills the theoretical expectations to a high degree of accuracy, yielding a gluon mass that is positive definite throughout the entire range of physical momenta, and displays in the ultraviolet the so-called “power-law” running, in agreement with standard arguments based on the operator product expansion. Some of the technical difficulties thwarting a more rigorous determination of the kernel are discussed, and possible future directions are briefly mentioned.

Figure 1

The vertex $U_{\alpha \mu \nu }$ is composed of three main ingredients: the transition amplitude, $I_{\alpha }(q)$, which mixes the gluon with a massless excitation; the propagator of the massless excitation $i/q^2$; and the massless excitation gluon vertex $B_{\mu \nu }$. The omitted terms are not relevant for this analysis; they can be found in Refs. [65, 66].

Figure 2

Diagrammatic representation of the gluon mass equation.

Figure 3

The SDE for the three gluon vertex, in the conventional (first row) and Bethe–Salpeter version (second row). Note that the Bose symmetry of ${\mathrm{\Gamma }}_{\mu \alpha \beta }^{abc}(q,r,p)$ implies that $(a_4{)}_{\mu \alpha \beta }^{abc}(q,r,p)=(a_5{)}_{\mu \beta \alpha }^{acb}(q,p,r)$; when the color has been factored out, as in Eq. (3.5), we have instead $(a_4{)}_{\mu \alpha \beta }(q,r,p)=-(a_5{)}_{\mu \beta \alpha }(q,p,r)$.

Figure 4

Schematic representation of the conversion of a typical diagram into its RGI equivalent.

Figure 5

The quenched lattice data and the corresponding fits for the $SU(3)$ gluon propagator (left panel) and ghost dressing function (right panel) renormalized at three different scales ${\mu }_i$. Lattice data are taken from Ref. [30].

Figure 6

Left panel: The RGI combination $d(q^2)/4\pi$ obtained using Eq. (2.29) for the three renormalization points ${\mu }_i$ chosen. Right panel: The running coupling in the MOM scheme, ${\alpha }_{\text{MOM}}(q)$ [73] for ${\mathrm{\Lambda }}_{\text{QCD}}=350\mathrm{GeV}$ and $N_f=0$. Each point represents the values used for ${\alpha }_s({\mu }_i)$ in our calculations.

Figure 7

Left panel: The numerical solution for the dynamical gluon mass, $m^2(q^2,{\mu }^2)$, for the three values of ${\mu }_i$. Right panel: The corresponding RGI mass $4\pi {\overline{m}}^2(q^2)$ obtained from Eq. (2.30) for the three values of ${\mu }_i$.

Figure 8

The gluon dynamical mass $m^2(q^2,{\mu }^2)$ (left panel) and the corresponding RGI mass $4\pi {\overline{m}}^2(q^2)$ (right panel) obtained using the model of Eq. (6.6) for the three values of ${\mu }_i$.

Figure 9

The numerical solution for $m^2(q^2,{\mu }^2)$ (left panel) using the model of Eq. (6.7) for the three renormalization scales ${\mu }_i$; the corresponding RGI mass $4\pi {\overline{m}}^2(q^2)$ given by Eq. (2.30) is shown on the right panel.

Figure 10

Comparison of the numerical results obtained for $m^2(q^2,{\mu }_1^2)$ using the original gluon mass equation (black continuous); the improved versions with $W_1$ of Eq. (6.6) (blue dashed) and $W_2$ of Eq. (6.7) (red dotted). The inset shows a zoom in the UV tail of the same quantities.

Figure 11

The numerical solution for $m^2(q^2)$ obtained using the RG-improved Ansatz $W_2$ of Eq. (6.7) (black circles). The (black) continuous curve represents the fit of Eq. (6.9) while the (blue) dashed curve is the asymptotic fit for the ultraviolet tail given by Eq. (6.10).

Figure 12

Schematic representation of a gluonic BSE.

Figure 13

Comparison of the lattice data for the gluon propagator for ${\mu }_1=4.3\mathrm{GeV}$ (black continuous) with the simple massive gluon propagator with $m_0^2=0.14{\mathrm{GeV}}^2$ (red dashed). Lattice data are taken from Ref. [30].