Light dilaton from top-down holographic confinement with magnetic fluxes

Abstract

A two-parameter class of higher-dimensional, strongly coupled, confining field theories in the presence of magnetic fluxes for two Abelian gauge groups admits a top-down, holographic dual description. The corresponding two-parameter family of regular background solutions of the classical equations of maximal supergravity in seven dimensions descends from maximal supergravity in eleven dimensions. We study the global and local stability properties of these solutions. We identify lines of zero-temperature first-order phase transitions, describing a polygon (a square) in the space of parameters, identified with the two fluxes. In the spectrum of fluctuations of the supergravity equations, interpreted as bound states of the dual, confining field theories, we find no evidence of local instabilities (tachyons). Over a significant portion of parameter space, that extends far away from the proximity to the transition, we identify an approximate dilaton, the mass of which is one order of magnitude smaller than the scale set by confinement. Our findings complement those emerging in other holographic models discussed in the literature, in which either the dilaton mass is only mildly lower than the confinement scale (when approaching a first-order transitions), or parametrically suppressed (when reaching the proximity to a second-order one).

Figures (5)

• Figure 1: Parameters and functions appearing in the soliton (confining) solutions, as a function of $\varrho_0$, that sets the end of space in the geometry. In each plot the four colours (blue, orange, green, red), with long to short dashing, correspond to branches of solutions with ${\cal A}_{7, U}^{(2)}/{\cal A}_{7, U}^{(1)}=\left(0,\tan\left(\frac{\pi}{9}\right),\tan\left(\frac{\pi}{6}\right),\tan\left(\frac{\pi}{4}\right)\right)$, respectively. We shall use the same color-coding also in Fig.\ref{['Fig:PhaseDiagram']}. In the first three plots we mark a region of instability in the solutions as a dark shaded area, this is given when $\varrho_0^2\leq Q_i$, or equivalently $\sin(2\theta)\geq \varrho_0^2/(2-\varrho_0^2)$. In these regions, at least one of the $H_i$ functions changes sign in the region of the geometry corresponding to the deep IR, at which point the metric would become singular and would result in the background values of $\phi_1$ or $\phi_2$ to become imaginary. These cases are hence unphysical, and excluded from the rest of the analysis in the paper. In the fifth and sixth panels we plot the functions $\phi_1(\varrho), \phi_2(\varrho)$, evaluated at the end of space, $\varrho=\varrho_0$, where we see that $\phi_2$ diverges at the boundary of parameter space.
• Figure 2: Left panel: phase diagram of the model, in the plane defined by the sources ${\cal A}_{7, U}^{(1)}$ and ${\cal A}_{7, U}^{(2)}$. Inside the shaded area, the vacuum is given by regular soliton solutions, corresponding to confining solutions with magnetic flux in the dual theory. Outside the shaded region, the vacuum is given by domain wall solutions leading to AdS$_7$ geometry. Along the sides of the square one has first-order phase transitions. The dashed line correspond to lines of constant ratio ${\cal A}_{7, U}^{(2)}/{\cal A}_{7, U}^{(1)}=\left(0,\tan\left(\frac{\pi}{9}\right),\tan\left(\frac{\pi}{6}\right),\tan\left(\frac{\pi}{4}\right)\right)$ (long dashing to short dashing). Right panel: the free energy density, $\hat{\cal F}$ of the dual, strongly coupled field theory living in six dimensions, expressed in units of the scale $\Lambda$, as a function of the parameter ${\cal A}_{7, U}^{(1)}$, and for values of ${\cal A}_{7, U}^{(2)}$ chosen along the four straight lines in the phase diagram in the left panel, with constant ratio ${\cal A}_{7, U}^{(2)}/{\cal A}_{7, U}^{(1)}=\left(0,\tan\left(\frac{\pi}{9}\right),\tan\left(\frac{\pi}{6}\right),\tan\left(\frac{\pi}{4}\right)\right)$---long dashed to short dashed lines in blue, orange, green, and red, respectively.
• Figure 3: Top four panels: mass spectra, normalised to the mass of the lightest spin-2 fluctuation, $M_2$, as a function of $\varrho_0$, for four examples of one-parameter subclasses of soliton (confining) solutions, obtained by fixing the ratio of the two magnetic fluxes---see the free energy of the same solutions in Fig. \ref{['Fig:PhaseDiagram']}. Spin-2 states are shown as (red) triangles and spin-0 as (blue) disks. The far left of each plot corresponds to points along the line of first order phase transitions in Fig. \ref{['Fig:PhaseDiagram']}. Fifth panel: comparison of the spin-0 states with the result of using the probe approximation, shown with the orange rectangles, for solutions with $\theta=0$. The probe approximation entirely misses the lightest state, suggesting this state contains a substantial contribution from the dilaton. The numerical results are obtained for $\varrho_{IR}=10^{-6} \varrho_0$. Sixth panel: ratio of the mass of the lightest spin-2 states, $M_2$, to the universal energy scale, $\Lambda$, for the solutions in which $\theta=0$; the ratio is roughly constant across the parameter space, suggesting either of the two observables could be used to set the physical scale, and compare between theories of this class.
• Figure 4: Gravitational invariants for a selection of soliton (confining) solutions, as a function of the holographic direction, $\varrho$. Panels \ref{['Fig:Ricci']}, \ref{['Fig:R2']}, and \ref{['Fig:R4']}, show three gravitational invariants for two particular choices of regular confining solutions, parametrized by $\{{\cal A}_{7,U}^{(1)}, {\cal A}_{7,U}^{(2)}\}=\{0.396,0.216\}$ and $\{0.796,0.565\}$, in blue and orange respectively. The grey dashed vertical lines correspond to the value of $\varrho_0$, where the space ends for each solution. All invariants are found to be smooth and regular over the entire space. The bottom right panel, \ref{['Fig:RicciBoundary']}, shows the Ricci scalar calculated for a selection of solutions along the line of phase transitions, along which $\mu=0$ and the free energy vanishes (the sides of the square in Fig. \ref{['Fig:PhaseDiagram']}). The red, green, orange, purple and black lines correspond to soliton solutions for which the value of the angle $\theta={\frac{\pi}{4}, \frac{\pi}{6}, \frac{\pi}{9}, \frac{\pi}{20}}$ and $\frac{\pi}{25}$ respectively. The dashed lines show the corresponding end of space for each solution, at which point the Ricci scalar diverges.
• Figure 5: Examples of mass spectra, normalised to the mass of the lightest spin-2 fluctuation, $M_2$, as a function of $\varrho_0$, for four examples of one-parameter subclasses of soliton (confining) solutions, obtained by fixing the ratio of the two magnetic fluxes--- see the free energy of the same solutions in Fig. \ref{['Fig:PhaseDiagram']}---and choosing a representative value of $\varrho_0$. Spin-2 states shown as (red) triangles and Spin-0 as (blue) disks. To demonstrate the suppressed IR cutoff dependence of the spectra, we evaluated them at values of $\varrho_0$ close to, but not at, the phase transition. The black dashed vertical line shows the value of the cutoff used in the calculations presented in the main body of the paper, with $\varrho_{IR}=10^{-6}\varrho_0$