Quenched SU(2) adjoint scalar propagator in minimal Landau gauge

It is a longstanding question whether the confinement of matter fields in QCD has an imprint in the (gauge-dependent) correlation functions, especially the propagators. In particular, in the quenched case, a fundamental difference could be expected between adjoint and fundamental matter. In a preceding investigation, the propagator of a fundamental scalar was studied, showing no obvious sign of confinement. Here, as a complement, the adjoint scalar propagator is investigated over a wide range of parameters in the minimal Landau gauge using lattice gauge theory. This study is performed in two, three, and four dimensions in quenched SU(2) Yang-Mills theory, both in momentum space and position space. No conclusive difference between the two cases is found.

Figure 1

Unrenormalized propagator (left panels) and renormalized propagator (right panels). The top panels are for two dimensions, the middle panels for three dimensions, and the bottom panels for four dimensions. The values are $m_r=m=1\mathrm{GeV}$ and $\mu =1.5\mathrm{GeV}$. If the (statistical $1\sigma$) error bars here and hereafter are not visible then they are smaller than the symbol size.

Figure 2

Scale dependence at $m_r=m=1\mathrm{GeV}$ (left panels) and scheme dependence at $\mu =1.5\mathrm{GeV}$ (right panels) of the renormalized propagator. The top panels are two dimensions, the middle panels three dimensions, and the bottom panels four dimensions. The tree-level propagator is shown for comparison as a full line.

Figure 3

The wave-function renormalization constant as a function of the lattice cutoff and the lattice volume in two dimensions for $\mu =1.5\mathrm{GeV}$. The top-left panel shows the case of $m=m_r=0\mathrm{GeV}$, the top-right panel of $m=m_r=0.1\mathrm{GeV}$, the bottom-left panel of $m=m_r=1\mathrm{GeV}$, and the bottom-right panel of $m=m_r=10\mathrm{GeV}$. The hatched band is the fit (5) with the parameters given in Table 1.

Figure 4

The wave-function renormalization constant as a function of the lattice cutoff and the lattice volume in three dimensions for $\mu =1.5\mathrm{GeV}$. The top-left panel shows the case of $m=m_r=0\mathrm{GeV}$, the top-right panel of $m=m_r=0.1\mathrm{GeV}$, the bottom-left panel of $m=m_r=1\mathrm{GeV}$, and the bottom-right panel of $m=m_r=10\mathrm{GeV}$. The hatched band is the fit (5) with the parameters given in Table 1.

Figure 5

The wave-function renormalization constant as a function of the lattice cutoff and the lattice volume in four dimensions for $\mu =1.5\mathrm{GeV}$. The top-left panel shows the case of $m=m_r=0\mathrm{GeV}$, the top-right panel of $m=m_r=0.1\mathrm{GeV}$, the bottom-left panel of $m=m_r=1\mathrm{GeV}$, and the bottom-right panel of $m=m_r=10\mathrm{GeV}$. The hatched band is the fit (5) with the parameters given in Table 1.

Figure 6

The mass renormalization constant as a function of the lattice cutoff and the lattice volume in two dimensions for $\mu =1.5\mathrm{GeV}$. The top-left panel shows the case of $m=m_r=0\mathrm{GeV}$, the top-right panel of $m=m_r=0.1\mathrm{GeV}$, the bottom-left panel of $m=m_r=1\mathrm{GeV}$, and the bottom-right panel of $m=m_r=10\mathrm{GeV}$. The hatched band is the fit (6) with the parameters given in Table 2.

Figure 7

The mass renormalization constant as a function of the lattice cutoff and the lattice volume in three dimensions for $\mu =1.5\mathrm{GeV}$. The top-left panel shows the case of $m=m_r=0\mathrm{GeV}$, the top-right panel of $m=m_r=0.1\mathrm{GeV}$, the bottom-left panel of $m=m_r=1\mathrm{GeV}$, and the bottom-right panel of $m=m_r=10\mathrm{GeV}$. The hatched band is the fit (6) with the parameters given in Table 2. Note that the hatched band can be as narrow as the lines, and, therefore, not be visible.

Figure 8

The mass renormalization constant as a function of the lattice cutoff and the lattice volume in four dimensions for $\mu =1.5\mathrm{GeV}$. The top-left panel shows the case of $m=m_r=0\mathrm{GeV}$, the top-right panel of $m=m_r=0.1\mathrm{GeV}$, the bottom-left panel of $m=m_r=1\mathrm{GeV}$, and the bottom-right panel of $m=m_r=10\mathrm{GeV}$. The hatched band is the fit (6) with the parameters given in Table 2. Note that the hatched band can be as narrow as the lines, and, therefore, not be visible.

Figure 9

The propagator (left panel) and the dressing function (7) (right panel) in two dimensions for $m=m_r=0\mathrm{GeV}$ and $\mu =1.5\mathrm{GeV}$.

Figure 10

The propagator (left panel) and the dressing function (7) (right panel) in two dimensions for $m=m_r=0.1\mathrm{GeV}$ and $\mu =1.5\mathrm{GeV}$.

Figure 11

The propagator (left panel) and the dressing function (7) (right panel) in two dimensions for $m=m_r=1\mathrm{GeV}$ and $\mu =1.5\mathrm{GeV}$. Note the different scale in the right-hand panel compared to Figs. 9 and 10.

Figure 12

The propagator (left panel) and the dressing function (7) (right panel) in two dimensions for $m=m_r=10\mathrm{GeV}$ and $\mu =1.5\mathrm{GeV}$. Note the different scale in the right-hand panel compared to Figs. 9 and 10.

Figure 13

The propagator (left panel) and the dressing function (7) (right panel) in three dimensions for $m=m_r=0\mathrm{GeV}$ and $\mu =1.5\mathrm{GeV}$.

Figure 14

The propagator (left panel) and the dressing function (7) (right panel) in three dimensions for $m=m_r=0.1\mathrm{GeV}$ and $\mu =1.5\mathrm{GeV}$.

Figure 15

The propagator (left panel) and the dressing function (7) (right panel) in three dimensions for $m=m_r=1\mathrm{GeV}$ and $\mu =1.5\mathrm{GeV}$. Note the different scale in the right-hand panel compared to Figs. 13 and 14.

Figure 16

The propagator (left panel) and the dressing function (7) (right panel) in three dimensions for $m=m_r=10\mathrm{GeV}$ and $\mu =1.5\mathrm{GeV}$. Note the different scale in the right-hand panel compared to Figs. 13 and 14.

Figure 17

The propagator (left panel) and the dressing function (7) (right panel) in four dimensions for $m=m_r=0\mathrm{GeV}$ and $\mu =1.5\mathrm{GeV}$.

Figure 18

The propagator (left panel) and the dressing function (7) (right panel) in four dimensions for $m=m_r=0.1\mathrm{GeV}$ and $\mu =1.5\mathrm{GeV}$.

Figure 19

The propagator (left panel) and the dressing function (7) (right panel) in four dimensions for $m=m_r=1\mathrm{GeV}$ and $\mu =1.5\mathrm{GeV}$. Note the different scale in the right-hand panel compared to Figs. 17 and 18.

Figure 20

The propagator (left panel) and the dressing function (7) (right panel) in four dimensions for $m=m_r=10\mathrm{GeV}$ and $\mu =1.5\mathrm{GeV}$. Note the different scale in the right-hand panel compared to Figs. 17 and 18.

Figure 21

The value of the screening mass $D(0{)}^{-1/2}$ as a function of lattice extent in two (top panel), three (middle panel), and four dimensions (bottom panel). The various dashed lines show the dependence at roughly fixed $a$. The right-hand-side is at $m_r=m=0$.

Figure 22

The effective mass (8) in two dimensions at lattice spacing $a^{-1}\approx 1.5\mathrm{GeV}$. The top-left panel shows the case of $m=m_r=0\mathrm{GeV}$, the top-right panel of $m=m_r=0.1\mathrm{GeV}$, the bottom-left panel of $m=m_r=1\mathrm{GeV}$, and the bottom-right panel of $m=m_r=10\mathrm{GeV}$. Points with more than 100% relative error are suppressed.

Figure 23

The effective mass (8) in three dimensions at lattice spacing $a^{-1}\approx 1.5\mathrm{GeV}$. The top-left panel shows the case of $m=m_r=0\mathrm{GeV}$, the top-right panel of $m=m_r=0.1\mathrm{GeV}$, the bottom-left panel of $m=m_r=1\mathrm{GeV}$, and the bottom-right panel of $m=m_r=10\mathrm{GeV}$. Points with more than 100% relative error are suppressed.

Figure 24

The effective mass (8) in four dimensions at lattice spacing $a^{-1}\approx 1.5\mathrm{GeV}$. The top-left panel shows the case of $m=m_r=0\mathrm{GeV}$, the top-right panel of $m=m_r=0.1\mathrm{GeV}$, the bottom-left panel of $m=m_r=1\mathrm{GeV}$, and the bottom-right panel of $m=m_r=10\mathrm{GeV}$. Points with more than 100% relative error are suppressed.

Figure 25

The maximum of the effective mass (8) in two dimensions (top-left panel), three dimensions (top-right panel), and four dimensions (bottom panel) as a function of lattice size and lattice spacing.

Figure 26

The effective mass (8) in three dimensions for different renormalization schemes. The top-left panel shows the case of $m=m_r=0\mathrm{GeV}$, the top-right panel of $m=m_r=0.1\mathrm{GeV}$, the bottom-left panel of $m=m_r=1\mathrm{GeV}$, and the bottom-right panel of $m=m_r=10\mathrm{GeV}$. Points with more than 100% relative error are suppressed.