A light dilaton in a metastable vacuum

We identify a parametrically light dilaton by studying the perturbations of metastable vacua along a branch of regular supergravity backgrounds that are dual to four-dimensional confining field theories. The branch includes also stable and unstable solutions. The former encompass, as a special case, the geometry proposed by Witten as a holographic model of confinement. The latter approach a supersymmetric solution, by enhancing a condensate in the dual field theory. A phase transition separates the space of stable backgrounds from the metastable ones. In proximity of the phase transition, one of the lightest scalar states inherits some of the properties of the dilaton, despite not being particularly light.

We pursue an alternative approach to the study of the dilaton, along the programme announced in Ref. [55], which is inspired by Refs. [56][57][58][59], but is implemented within the rigorous framework of supergravity. We generalise the notion of proximity to the BF unitarity bound [60]-central to the arguments in Ref. [56]-in order to explore non-AdS backgrounds dual to confining theories, in regions of parameter space near tachyonic instabilities. We aim at ascertaining whether the spectrum of bound states includes a light dilaton.
We focus on three branches of solutions: i) regular solutions that include the Witten model and are interpreted as duals of four-dimensional confining theories, ii) a class of supersymmetric solutions, and iii) a branch of non-supersymmetric solutions, that (locally) preserve six-dimensional Poincaré invariance, but are badly singular-they do not even meet Gubser's criteria [72]. We compute the spectrum of fluctuations of the relevant scalar and spin-2 tensor states, using the gaugeinvariant formalism of Refs. [73][74][75][76][77], hence extending the study of the spectra performed in Ref. [78] and Ref. [70]. We compare to the result of applying the probe approx-imation [79], in order to ascertain whether any of the scalar states have significant overlap with the trace of the stress-energy tensor, and can hence be identified with an approximate dilaton.
In a region of parameter space the spectrum contains a parametrically light dilaton. We study the energetics along the three branches of solutions, by computing the free energy using holographic renormalisation [80][81][82], and employing a simple scale-setting procedure to compare different backgrounds [83]. We present firm evidence of the existence of a phase transition in the gravity theory (see also Ref. [84]). The parametrically light dilaton emerges along the portion of the regular branch of solutions which contains metastable solutions, the lifetime of which is not known (but see Ref. [85]).

II. THE GRAVITY MODEL
We denote with hatted symbols quantities characterising the theory in D = 7 dimensions. The action, truncated to retain the scalar φ coupled to gravity, is the following [79] (see also Refs. [62,63,70]): where the potential (see Fig. 1) is This potential admits two critical points. The one with φ = φ U V = 0 will play a central role in this paper, as it corresponds to a UV fixed point in the dual field theory. It yields V 7 (φ U V ) = − 15 8 . Another critical point of V 7 has φ IR = − 1 √ 5 log(2), for which V 7 (φ IR ) = − 5 2 7/5 . Following the notation in Refs. [70,79], we reduce to D = 5 dimensions by adopting the following ansatz: and the background profiles φ(r), χ(r), ω(r), and A(r) depend only on the the radial coordinate r. The angles 0 ≤ η, ζ < 2π parametrise a torus. We apply the change of variables dρ = e −χ dr. The domain-wall (DW) ansatz in D = 7 dimensions is recovered by imposing the constraints ω = 0 and A = A − χ = 3 5 A = 3 2 χ, and hence the AdS 7 solution has ∂ ρ A = 1 2 , ∂ ρ χ = 1 3 , and ∂ ρ A = 5 6 . The bulk action in D = 5 dimensions is the following: , ω, χ}, and the sigma-model metric is G ab = diag 1 2 , 1, 15 4 . We verified that

III. CLASSES OF SOLUTIONS
All the solutions of interest approach φ = φ U V = 0 at large ρ. We write them as a power series of the small coordinate z ≡ e −ρ/2 , as follows They are characterised by seven integration constants: φ 2 , φ 4 , ω U , ω 6 , χ U , χ 6 , and A U . The DW solutions have ω U = ω 6 = χ 6 = 0 and χ U = 2 5 A U , leaving A U , φ 2 and φ 4 as independent non-trivial free parameters. What we will call confining solutions have χ 6 = 0.
We find it convenient to define a scale Λ as follows [83]: with ρ o the end of space. While other choices might be admissible, this has the advantage of being applicable to all the solutions of interest.

A. SUSY solutions
The supersymmetric DW solutions satisfy the following first-order differential equations: The superpotential with the warp factor given by where A o and τ o are real integration constants. The IR expansion of these solutions in terms of the radial coordinate ρ and the new constants ρ o and A I can be written explicitly in the following form and Their holographic interpretation involves an operator of dimension ∆ = 4 developing a vacuum expectation value (VEV) in the dual field theory. The conjugate superpotential entering the calculation of the free energy is known as a perturbative expansion: where κ is scheme dependent.

B. Singular DW solutions
A class of singular DW solutions is characterised by the harmless A I , the end of space ρ o , and the non-trivial φ 5 . As anticipated, these solutions are badly singular: their Ricci scalar tensor R 7 diverges, and the potential is not bounded from above, violating the requirement from Ref. [72]. The IR expansion of solutions of this class reads as follows:

C. Confining solutions
The regular solutions of this class obey the constraint A = 5 2 χ + ω. They depend on two harmless constants χ I and ω I , besides ρ o and φ I . The IR expansion of these solutions reads as follows: We restrict attention to solutions flowing from the UV critical point, which requires φ I > φ IR . The invariants R 7 , R 7MN RMN 7 , and RP 7MNQ RMNQ 7P are finite. We impose the constraint ω I = 3 2 χ I in order to avoid a conical singularity.

IV. GLUEBALL MASSES
We compute the spectrum of fluctuations of the fivedimensional theory, by employing the gauge-invariant formalism developed in Refs. [73][74][75][76][77]. We introduce the IR regulator ρ 1 with ρ o < ρ 1 , and the UV regulator ρ 2 . The physical results are recovered in the limits ρ 1 → ρ o and ρ 2 → +∞ (see Refs. [70,71,77]). The scalar fluctuations are written as the gauge invariant combinations where M is the mass in the dual theory, ϕ a are fluctuations of the scalars Φ a and h of the trace of the fourdimensional portion of the metric. They obey the following linearised equations and boundary conditions: where in all these expressions the quantities A, Φ a , and V are evaluated on the background, and The gauge invariant spin-2 tensor fluctuations obey the linearised equation and Neumann boundary conditions ∂ ρ e µ ν | ρi = 0. The probe approximation for the scalars is defined by ignoring the term proportional to h in Eq. (24). According to the dictionary of gauge-gravity dualities, h is the bulk field associated with the trace of the stress-energy tensor, which is the field theory operator associated with dilatation, and sourcing the dilaton. Hence, this approximation holds for scalar bound states that decouple from the dilatation operator, and cannot be interpreted as a dilaton. The equations for the scalar fluctuations greatly simplify, as only the first term in Eq. (26) survives, and the boundary conditions reduce to Dirichlet.
In Fig. 2, we show the spectra of tensors and scalars, compared to the probe approximation, normalised to the lightest spin-2 fluctuation. For φ I < 0 the scalars agree with Ref. [70]. The new results for φ I > 0 show that one of the scalars becomes parametrically light, and eventually tachyonic, for positive φ I . When this state is light, or tachyonic, the probe approximation does not capture it correctly, indicating that the state has a non-trivial component along h, and hence is sourced by the trace of the stress-energy tensor, as expected by a dilaton. We also notice that several of the heavy scalar states are not well captured by the probe approximation, showing that mixing effects with the dilaton are not restricted to the lightest states.

V. FREE ENERGY
To compute the free energy, we write explicitly the boundary terms of the theory in D = 7 dimensions: whereg denotes the determinant of the pullback of the induced metric, K is the Gibbons-Hawking-York (GHY) term and λ i are localised boundary potentials. The potential terms are chosen according to the same prescription as in Ref. [55]: in the UV we replace λ 2 = W 2 , which allows one to cancel all the divergences and perform the programme of holographic renormalisation [80][81][82], while in the IR we impose in such a way that the variational problem be well defined in the presence of the IR boundary at ρ = ρ 1 . The free energy density F is defined in terms of the complete on-shell action to be By making use of the equations of motion we arrive at which is identical to Eq. (5.22) of Ref. [55]. We make use of the UV expansions of the background solutions of interest. By replacing the UV expansions in Eqs. (7)-(10) into the form of the free energy density in Eq. (30), supplemented by the specific form of the superpotential W 2 in Eq. (17), we arrive at the expression:  The divergence of the contribution to the free energy proportional to φ 2 2 is cancelled by W 2 . This implies that, as for the circle reduction of the Romans supergravity [55], the concavity theorems do not apply to F. This expression still contains a residual scheme-dependence, in the logarithmic term. We set κ = e −4/3 φ 2 2 , and hence our final expression for the free energy density is We also remind the reader that χ 6 = 0 in the background solutions of interest. In Fig. 3 we show the free energy of the three classes of solutions, as a function of the deforming parameterφ 2 ≡ φ 2 Λ −2 , and setting A U = 0 = χ U . The SUSY solutions have F = 0. We verified explicitly that regulating the free energy with κ = Λ 2 yields results that are almost identical to those in Fig. 3.
The figure shows evidence of the existence of a firstorder phase transition. The confining solutions minimisê nite free energy densityF, so that the solutions along the confining branch are at best metastable when φ I > φ c I , and eventually become unstable, with one of their fluctuations becoming tachyonic when φ I > ∼ 0.447. Most interestingly, along the metastable branch, the lightest state becomes parametrically light, before becoming tachyonic (see Fig. 2). The probe approximation fails to capture correctly its mass squared when it is either small or negative. This eigenstate of the system is hence an admixture containing a significant contribution from the trace of the fluctuation of the metric-we interpret this finding as evidence that the state is approximately a dilaton.

VI. OUTLOOK
We presented evidence of the emergence of a parametrically light dilatonic state along the metastable portion of a branch of regular backgrounds of the supergravity system in D = 7 dimensions that yields also the Witten model, the first known holographic description of a four-dimensional confining theory [66]. Furthermore, the results of our analysis confirm, in the rigorous context of top-down holography, the expectations from Ref. [59] that along the stable portion of the regular branch a dilatonic state persists, but it is not parametrically light.
The metastable vacua, and the accompanying parametrically light dilatonic state, are new findings. Comparison with Ref. [55] indicates that this is a generic feature, which emerges in a broad class of theories. It would be interesting to discover examples in which the phase transition is weaker, and the spectrum along the stable branch exhibits a light approximate dilaton. It would also be useful to identify the requirements a supergravity theory must fulfil for such features to emerge.