Running coupling from gluon and ghost propagators in the Landau gauge: Yang-Mills theories with adjoint fermions

Non-Abelian gauge theories with fermions transforming in the adjoint representation of the gauge group (AdjQCD) are a fundamental ingredient of many models that describe the physics beyond the Standard Model. Two relevant examples are $\mathcal{N}=1$ supersymmetric Yang-Mills (SYM) theory and minimal walking technicolor, which are gauge theories coupled to one adjoint Majorana and two adjoint Dirac fermions, respectively. While confinement is a property of $\mathcal{N}=1$ SYM, minimal walking technicolor is expected to be infrared conformal. We study the propagators of ghost and gluon fields in the Landau gauge to compute the running coupling in the MiniMom scheme. We analyze several different ensembles of lattice Monte Carlo simulations for the SU(2) adjoint QCD with $N_f=1/2,1,3/2$, and 2 Dirac fermions. We show how the running of the coupling changes as the number of interacting fermions is increased towards the conformal window.

Figure 1

(a, b) Bare gluon and ghost dressing functions for $\mathcal{N}=1$ SYM as a function of the lattice momentum $ap$ at ${\beta }_{\mathrm{lat}}=1.9$. (c, d) Renormalized gluon and ghost dressing functions for $\mathcal{N}=1$ SYM on a logarithmic scale of the physical momentum $p$. The gluon dressing function tends to zero for $p\to 0$.

Figure 2

(a, b) The running of the coupling $\alpha (\mu )$ for the $\mathcal{N}=1$ SYM theory. There is only a very weak dependence of $\alpha (\mu )$ on the gluino mass.

Figure 3

(a) The running of the coupling $\alpha (\mu )$ for $\mathcal{N}=1$ SYM. The running can be related to four-loop perturbation theory at high energy. (b) The ratio of the lattice results over the perturbative prediction, including the lattice correction of Eq. (22), is constant (blue points). Consequently there is a fairly large window to where perturbation theory can be trusted. The fit is performed in the window delimited by the green vertical lines. The orange points show the ratio of the lattice strong coupling to the perturbative prediction from the same fit, but excluding from the ratio the fitted correction of Eq. (22). (c) Lattice corrections to the perturbative running $K_{\mathrm{lat}}$ including (blue) and excluding (orange) the term $\sum _{\mu }{p}_{\mu }^6$ from the fit.

Figure 4

Bare gluon and ghost dressing functions for $N_f=2$ AdjQCD as a function of the lattice momentum $ap$ at ${\beta }_{\mathrm{lat}}=1.5$ and ${\beta }_{\mathrm{lat}}=1.7$.

Figure 5

Strong coupling constant for $N_f=2$ AdjQCD as a function of the renormalization scale $a\mu$ at different ${\beta }_{\mathrm{lat}}$. The results show a clear fermion mass dependence and the strong coupling effectively freezes toward the massless limit.

Figure 6

Mass dependence of the running coupling for $N_f=2$ AdjQCD as a function of the bare $a{m}_{\mathrm{PCAC}}$ at different $a\mu$. The data shown come from the ensembles at ${\beta }_{\mathrm{lat}}=1.5$ and ${\beta }_{\mathrm{lat}}=1.7$, (there is a tiny horizontal offset for each $\mu$ to make the individual points visible). All the couplings collapse to unique point at $a{m}_{\mathrm{PCAC}}\simeq 0.03$, corresponding to $\kappa =0.1350$ [green points of Fig. 5]. At the smallest $\kappa =0.1325$, the red and cyan points are still compatible within the errors (${\alpha }^{\kappa =0.1325}(1.333)=0.305(13)$ and ${\alpha }^{\kappa =0.1325}(1.447)=0.313(12)$), meaning that the (near-)zero of the ${\beta }_{\mathrm{lat}}$-function can survive even for nonzero fermion masses.

Figure 7

“Hyperscaling” of the running coupling for $N_f=2$ AdjQCD at ${\beta }_{\mathrm{lat}}=1.5$ and ${\beta }_{\mathrm{lat}}=1.7$ as a function of $\mu {(m_{\mathrm{PCAC}})}^{-1/(1+{\gamma }^{*})}\approx \mu {(m_{\mathrm{PCAC}})}^{-0.77}$. The strong coupling at different fermion masses appears to collapse toward a universal function. Some deviation is observed for small $\mu$ at the smallest fermion mass, probably due to finite volume effects.

Figure 8

Bare gluon and ghost dressing functions for $N_f=3/2$ AdjQCD as a function of the lattice momentum $ap$ at ${\beta }_{\mathrm{lat}}=1.5$.

Figure 9

The running of the running coupling $\alpha (\mu )$ on a logarithmic scale $\mu$ for $N_f=3/2$ AdjQCD. Many ensembles show that a rising tendency of $\alpha (\mu )$ that peaks in the infrared region.

Figure 10

Fit of $\alpha (\mu )$ to perturbation theory at various ${\beta }_{\mathrm{lat}}$ with the largest $\kappa$ available plotted on a linear scale $\mu$. The fitting intervals are (a) (1.3,2.05), (b) (1.3,2.05), (c) (0.9,2.15).

Figure 11

Bare gluon and ghost dressing functions for $N_f=1$ AdjQCD as a function of the lattice momentum $ap$ at ${\beta }_{\mathrm{lat}}=1.75$.

Figure 12

(a) The running of the running coupling $\alpha (\mu )$ on a logarithmic scale $\mu$ for the $N_f=1$ AdjQCD theory. (b) Comparison of finite volume effects at $\kappa =0.1650$.

Figure 13

Comparison of the fit of $\alpha (\mu )$ to two and four loops perturbation theory plotted on a linear scale $\mu$. The fitting interval is (0.47,2.2). The fit to two-loop perturbation theory might appear better, but the agreement at low $a\mu$ is most likely accidental, given that for $a\mu <0.3$ the effects of the fermion mass on $\alpha (\mu )$ are not negligible, see Fig. 12.