Naturally light dilatons from nearly marginal deformations

TL;DR

The paper shows that in CFTs deformed by a nearly marginal operator, confinement to a soft-wall IR together with a small IR beta function yields a naturally light dilaton, identifiable with fluctuations of the condensate $\langle \mathcal{O} \rangle$. A central result is a mass formula for the dilaton, $m_{\rm dil}^2\simeq\big[\int_{0}^{z_s}dz\,{1\over a^{d-1}\beta^2}\int_{z}^{z_s}dz'\,a^{d-1}\beta^2\big]^{-1}$, which shows suppression when the beta function is walking and the rise to confinement is rapid. The work studies two holographic models (Model A with $\langle \mathcal{O} \rangle\neq0$ and Model B with $\langle \mathcal{O} \rangle=0$) and finds that in both cases a dilaton remains light, with $m_{\rm dil}^2\sim\Delta_-\Lambda_{\rm IR}^2$ for Model A and $m_{\rm dil}^2\sim\Delta_-^2\Lambda_{\rm IR}^2$ for Model B, aligning with Contino–Pomarol–Rattazzi. The paper also presents a holographic method to compute $\langle \mathcal{O} \rangle$ via IR regularity of the beta function, discussing implications for naturalness and potential connections to Higgs-like dilaton scenarios and the cosmological constant problem.

Abstract

We discuss the presence of a light dilaton in CFTs deformed by a nearly-marginal operator O, in the holographic realizations consisting of confining RG flows that end on a soft wall. Generically, the deformations induce a condensate <O>, and the dilaton mode can be identified as the fluctuation of <O>. We obtain a mass formula for the dilaton as a certain average along the RG flow. The dilaton is naturally light whenever i) confinement is reached fast enough (such as via the condensation of O) and ii) the beta function is small (walking) at the condensation scale. These conditions are satisfied for a class of models with a bulk pseudo-Goldstone boson whose potential is nearly flat at small field and exponential at large field values. Thus, the recent observation by Contino, Pomarol and Rattazzi holds in CFTs with a single nearly-marginal operator. We also discuss the holographic method to compute the condensate <O>, based on solving the first-order nonlinear differential equation that the beta function satisfies.

Figures (7)

• Figure 1: Form of ${d-1\over2}\,{V'\over V}$ (blue dotted) and $-\beta(\phi)$ (continuous red) for the model 'A'. Three regimes (deformation, condensation and confinement) are clearly distinguishable in $\beta(\phi)$. The black dashed lines are the behaviour of the beta function in the three regions. In the deformation region, $-\beta(\phi)\simeq \Delta_- \phi$. In the condensate-dominated region it is given by the $\Delta_+-$type form $\beta\simeq -\Delta_+ (\phi-\phi_{cond})$, or even better, Eq. \ref{['tanh']}; in the confinement region $-\beta(\phi)\simeq \nu \sqrt{d(d-1)}$. $-\beta(\phi)$ tracks the function ${(d-1)\over2}\,{V'\over V}$ everywhere except on the condensate-dominated region. The shaded area corresponds to regular and confining flows. The plot is for $\Delta_-=0.05$, $\nu=0.8$, $\phi_{conf}=2.6$, $d=4$ and a smoothing of the jump in $V'/V$ with $N=20$.
• Figure 2: Plot of ${d-1\over2}\,{V'\over V}$ (blue dotted line) and of $-\beta(\phi)$ (continuous red line) for the model 'B', with $\Delta_-=0.2$, $\nu=0.8$, $\phi_{conf}=2.6$, $d=4$ and a smoothing of the jump in $\beta$ with $N=15$. The blue dotted line is ${(d-1)\over2}\,{V'\over V}$. The black dashed lines are the behaviour of the beta function in the deformation region and confinement regions. There is still something like a 'condensate-dominated' region where $\beta$ does not track $V'/V$, but the condensate ($\langle {\cal O} \, \rangle\propto s_c$) in this flow vanishes.
• Figure 3: Spectrum for the Type-A model as a function of $\Delta_-$ -- the scaling dimension of the nearly-marginal coupling $\lambda$. The first 6 modes are depicted. The lightest mode -- the dilaton -- scales like $m_{\rm \hbox{\scriptsize dil}}^2 \sim \Delta_-$.
• Figure 4: Spectrum for the Type-B model as a function of $\Delta_-$. The first 6 modes are depicted. The lightest mode -- the dilaton -- scales like $m_{\rm \hbox{\scriptsize dil}} \sim \Delta_-$.
• Figure 5: The dilaton wave function $\xi(\phi)$ as a function of the $\phi-$coordinate (continuous blue line), obtained by numerically solving Eq. \ref{['xieigen']}. The UV- and IR- approximations to $\xi(\phi)$ (Eqs. (\ref{['UVapprox']}) and (\ref{['IRexp']}) respectively), and the background beta function are also plotted. The dilaton wavefunction switches on at the condensation threshold $\phi_{cond}$. The pink bullets indicate $\xi_{\rm \hbox{\scriptsize dil}}^{{\rm \hbox{\tiny UV}}}$ computed up to the $m^2 \,{\cal I}_{\rm \hbox{\tiny UV}}$ term in Eq. (\ref{['UVexp']}) at some sample points. This already approximates quite well to the numerical $\xi(\phi)$, indicating a rather fast convergence of the perturbative expansion. The left (right) panel refers to model A (B).