Phase structure of extra-dimensional gauge theories with fermions

We study the phase structure of five-dimensional Yang-Mills theories coupled to Dirac fermions. In order to tackle their nonperturbative character, we derive the flow equations for the gauge coupling and the effective potential for the Aharonov-Bohm phases employing the functional renormalization group. We analyze the infrared and ultraviolet fixed-point solutions in the flow of the gauge coupling as a function of the compactification radius of the fifth dimension. We discuss various types of trajectories which smoothly connect both dimensional limits. Last, we investigate the phase diagram and vacuum structure of the gauge potential for different fermion content.

Figure 1

Effective dimension for a trivial gauge background and as a function of the dimensionless compactification radius for different truncations of the KK-mode tower ($N_{\mathrm{tr}}$). Figure adapted from Ref. [39].

Figure 2

Examples of two-loop diagrams. The left-hand side diagram produces the two-loop effect proportional to $[J(\overline{R},0){]}^2$, where $J(\overline{R},0)$ is defined in Eq. (33), while the right-hand side one gives a new type of threshold function $I(\overline{R},0)$ different from $J(\overline{R},0)$.

Figure 3

In the top (bottom) row, the nontrivial UV and IR fixed-point solutions for an $SU(N_c)$ gauge coupling in the presence of $N_{\mathrm{f}}$ ($N_{\mathrm{ad}}$) fermion fields transforming in the fundamental (adjoint) representations and at different compactification radii; $\overline{R}=0.01$ (left column), $\overline{R}=0.1$ (center column), and $\overline{R}=1$ (right column). Brighter colors indicate a larger magnitude of the fixed point. In order to account for beyond one-loop fermionic contributions, we have employed three-loop $\overline{\mathrm{MS}}$ results in the anomalous dimension ${\eta }_g$ in Eq. (35).

Figure 4

$SU(3)$ gauge coupling beta functions as a function of the coupling for different fermion content in the fundamental (top row) and adjoint (bottom row) representations. As plain lines from red (dark) to yellow (bright), we show the evolution from $\overline{R}=0$ to $\overline{R}\to \infty$ employing the three-loop $\overline{\mathrm{MS}}$ results. In black we show the resumed beta functions at $\overline{R}=0$ (black dashed) and $\overline{R}\to \infty$ (black dotted). The resumed anomalous dimension does not show the CBZ fixed point in the $\overline{R}=0$ as lacks the complete two-loop contributions. Nevertheless, in the large and few $N_{\mathrm{f}},N_{\mathrm{ad}}$ limits, both beta functions qualitatively agree. Noninteger number of fermions are employed in some instances for best display of qualitative features.

Figure 5

Integrated renormalization group flows of the $SU(3)$ coupling for different fermion content. From left to right, we increase the number of fermions $N_{\mathrm{f}}=2$, 6, 8 transforming in the fundamental representation, while $N_{\mathrm{ad}}=0$. In each plot we show different curves for which the onset of the extra-dimension triggers at different scales. The vertical lines denote the onset of the extra dimension where $R/k=1$ is satisfied.

Figure 6

Configurations of Aharonov-Bohm phases and breaking patterns of $SU(3)$ gauge symmetry, namely $SU(3)\to {\mathcal{H}}_{\mathrm{sym}}$ via the Hosotani mechanism [86, 87]. Here, $\alpha \ne 0$, ${\beta }_1\ne {\beta }_2,-\frac{1}{2}{\beta }_2,-2{\beta }_2$. The confined configuration $X$ arises from the Haar-measure distribution which is uniformly and randomly distributed so as to be $P=0$ [87]. On the right-hand side, the Polyakov loop configurations [defined in Eq. (54)] for the phases summarized on the left-hand side Table are displayed.

Figure 7

On the left-hand side plot, the AB phase diagram in the presence of one adjoint fermion $(N_{\mathrm{f}},N_{\mathrm{ad}})=(0,1)$ as a function of the compactification radius and the adjoint fermion mass. Three phases are found to appear: deconfined [($a$), in blue], split [($b$), in green], and reconfined [($c$), in orange]. On the first set of plots on the right-hand side panel, we show the evolution of a slice of the effective potential along a fixed ${\overline{m}}_{\mathrm{ad}}=0.4$ for different finite compactification radii. In the top plot, we show the evolution from the deconfined (blue lines) to the split phase (green lines) and in the bottom plot, from the split (green lines) to the reconfined phase (yellow lines). We slightly vary the compactification radius in the interval $1/\overline{R}=[0.42,0.66]$. In the rightmost set of plots, we show the evolution of the gauge potential at two fixed radii $1/\overline{R}=\{0.1,{10}^2\}$. The adjoint fermion mass is slightly decreased showing the evolution of the potential from the deconfined (dark blue lines) to the split phase (dark yellow lines) in both dimensional limits. While in the $D=5$ limit (top plot), this phase transition displays a separating barrier, in the $D=4$ limit (bottom plot) the potential undergoes a flat shape.

Figure 8

AB phase diagram as a function of the compactification radius and ${\overline{m}}_{\mathrm{ad}}$. On the left-(right-)hand side plot, we show the phase diagram for $(N_{\mathrm{f}},N_{\mathrm{ad}})=(1,1)$ [$(N_{\mathrm{f}},N_{\mathrm{ad}})=(2,2)$] with periodic boundary conditions and $m_{\mathrm{f}}=0$. In both cases, three phases are found; deconfined [($a$), in blue], split [($b$), in green] and pseudoreconfined [($c^{*}$), in brown].