Besides the Higgs particle discovered in 2012, with mass 125 GeV, recent LHC data
show tentative signals for new resonances in diboson as well as diphoton searches at
high center-of-mass energies (2 TeV and 750 GeV, respectively). If these signals
are confirmed (or other new resonances are discovered at the TeV scale),
the large hierarchies between masses of new bosons require a dynamical
explanation. Motivated by these tentative signals of new physics, we investigate
the theoretical possibility that large hierarchies in the masses of glueballs
could arise dynamically in new strongly-coupled gauge theories extending the
standard model of particle physics. We study lattice data on non-Abelian gauge
theories in the (near-)conformal regime as well as a simple toy model in
the context of gauge/gravity dualities. We focus our attention on the ratio
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\Lambda_{{\rm QCD}} . \end{equation} In this limit, the spectrum of an asymptotically-free gauge theory can be extracted with perturbation theory. In this section, we restrict our discussion to the case of a (3+1)-dimensional hypercubic volume $L^{3} \times T$, with $L$ satisfying the corresponding condition in eq.~(\ref{eq:femto_qcd}). Perturbation theory in such small cubic boxes can be complicated by the global toroidal structure (``torons'') of the periodic lattice~\cite{GonzalezArroyo-ml-1981vw,Coste-ml-1985mn}. This complication can be removed by employing the colour-twisted boundary conditions (TBC)~\cite{bib-ml-tHooft-ml-1979uj,Coste-ml-1986cb}. To keep our discussion simple, we concentrate on results obtained using TBC in refs.~\cite{GonzalezArroyo-ml-1988dz,Daniel-ml-1989kj,Daniel-ml-1990iz}. The glueball spectrum of the pure Yang-Mills theories in the femto-universe with TBC takes the generic form \begin{equation} \label{eq:femto_glueball_generic} M_{G} = \frac{X_{0}}{L} + \frac{g^{2} X_{1}}{L} + \ldots , \end{equation} \looseness=-1 where $g$ is the coupling, and $X_{0}$ is a constant that is completely determined by the geometry and the boundary condition of the finite volume, as well as the cubic-group representation of the glueball state. Eq.~(\ref{eq:femto_glueball_generic}) is the result of perturbation theory. With a specific choice of the twist that preserves the cubic symmetry, $X_{0}$ takes the same value for all the states associated with the irreducible representations of the cubic group~\cite{Daniel-ml-1989kj}. In particular, this means that the scalar and the tensor glueballs are degenerate, $R=M_{T}/M_{0} = 1$, in the $g^{2} \rightarrow 0$ limit. We stress that this value of the mass ratio, $R$, is the consequence of the box geometry, the boundary condition and group theory. It does not result from the underlying dynamics. \looseness=-1 The gauge-field dynamics begins to set in at the first non-trivial order in the expansion of eq.~(\ref{eq:femto_glueball_generic}). In general, the coefficient $X_{1}$ depends on the spin of the glueball. For the case of TBC considered in ref.~\cite{Daniel-ml-1989kj}, it increases $R$.\footnote{This statement is also found to be valid when lattice artefacts are accounted for in the perturbative calculations~\cite{Daniel-ml-1990iz}.} This coefficient depends on the boundary conditions as well. In fact, in the computation employing periodic boundary conditions, the authors of refs.~\cite{Luscher-ml-1982ma,F} find that the tensor glueball is lighter than the scalar state at the first non-trivial order of perturbation theory, although the mass ratio, $R$, is also close to unity. In the femto-universe, the light-fermion masses are well below the scale $1/L$. Therefore, practically they can be considered as massless. Since the fermions couple to the pure gauge degrees of freedom only perturbatively, they will not have any significant (non-perturbative) effects on the glueball spectrum discussed above. The authors of ref.~\cite{Daniel-ml-1990iz} computed the fermionic contribution in continuum perturbation theory to $O(g^{2})$, and found it to be small. ]]>
1.4$. At the smallest available values of $L M_0$, $R$ is systematically below the pure Yang-Mills value, although still compatible with $R=1.44(4)$ within the large errors. \looseness=-1 We conclude this subsection by noting that for the smallest volumes at $a m = -1.15$ studied in~\cite{DelDebbio-ml-2015byq} $R$ is compatible with the expected femto-universe results. The explicit observation of the deviation from the femto-universe regime in the results discussed here is a good indication of absence of severe small volume effects for the data presented in figure~\ref{fig:fitrvsm0PP_2}. \begin{figure} \centering \includegraphics[width=0.95\columnwidth]{FIGS/figure3} ]]>
2 M_0$ deserves further comments. In fact, one should expect that if the resonance in the tensor channel has a mass that is much larger than that of the scalar, the tensor correlator will be asymptotically dominated by scattering states~\cite{Maiani-ml-1990ca}. Hence, one might wonder whether we have identified the wanted resonance or we are observing some spurious object. In order to provide a convincing answer to this question, calculations with extended statistics and purpose designed analysis methods~\cite{Luscher-ml-1985dn,Luscher-ml-1985dn.m001} are needed, which would go well beyond our current aims. Our preliminary investigation, based on the scaling of the mass with the lattice volume and on the expected degeneracy of the continuum tensor state in two representations of the rotational group of the cube in the case of a single particle, suggests that the extracted mass identifies a resonance. The apparent absence of scattering state contributions in the correlator might be due to the particular construction of the trial variational operators, which are optimised for single particles. } ]]>
5$). This is encouraging, in the light of the recently observed LHC diphoton excess, which would require $R\sim 6$. The model in which these indications arise is the mass deformation of a gauge theory with $\SU(2)$ gauge group and $N_f=1$ generations of Dirac fermions in the adjoint representation. In the massless limit, this theory is believed to have an IR fixed point with large anomalous dimensions. As a consistency check, we compared the numerical results to the lattice results for the case $N_f=2$, in which the IR fixed point has small anomalous dimensions. We found $R\sim {\cal O}(1)$, which is compatible with expectations. We also compared to a toy model built in the context of gauge/gravity dualities. We drew a parallel between the cutoff effects on the lattice and in gravity models, finding qualitative agreement. In the physically relevant region of parameter space, we found remarkably good numerical agreement with the lattice for the relation between $R$ and the dimension $\Delta$ of the deforming parameter. These are preliminary results, and require further investigation with dedicated studies. We summarise some of the directions this research program could encompass. On the lattice side, a dedicated program for the study of glueball spectra, both in $\SU(2)$ with adjoint matter, as well as in other candidate theories with IR conformal dynamics, is needed. Particular attention should be devoted to the $\SU(2)$ theory with $N_f=1$, the only known case to date for which $R$ is found to be large. New studies with larger lattices might ascertain whether large values of $R$ are truly physical or affected by lattice artefacts. On the gravity side, a more general exploration of models that describe the dynamics of deformations of IR-conformal theories is needed. The results presented here are based on a simplified model built within the bottom-up approach to holography. It would be useful to identify and study a full model derived from string theory in the top-down context. Yet, it should be possible to carry out similar studies in more general contexts, within the bottom-up approach. In the field theory context, it would be interesting to study the ratios of masses, aided by statistical field theory arguments. The claim of universality that underpins the comparison we make of lattice and gravity results might be taken literally (in the sense that $R$ is only a function of $\Delta$), or just in a broad sense: the behaviours emerging in different models share qualitative features. In this paper, we took the latter view, in the absence of a rigorous proof of a more robust relation, though our numerical results show such a good level of agreement with gravity calculations that it might be an indication of a more fundamental physical principle. On the phenomenological side, several additional questions arise. Under the assumption that new strong dynamics can explain a large mass hierarchy $R\gsim 5-6$, the construction of a realistic model of electroweak symmetry breaking requires to introduce many additional ingredients. It is worth bringing to the attention of the reader for example the papers in~\cite{VS,VS.m001,VS.m002}, where some relevant considerations pertaining to model-building and LHC phenomenology are discussed. This paper is a first step towards a future proof of principle that comparatively heavy spin-2 resonances can emerge as composite states of new strong dynamics, leaving all model-building and phenomenological issues aside. { We close by repeating the main results of this study. We collected a significant body of empirical evidence, both from lattice studies of $\SU(2)$ theories with adjoint matter and from simplified gravity dual models, that points in two intertwined directions. First of all, we find agreement (within errors) in the results for $R$ computed in completely different theories that share the same dimension $\Delta$ for the relevant deformation, suggesting that the quantity $R$ might be a manifestation of some form of universality. Secondly, when the deforming operator has large anomalous dimension, we find a parametric enhancement of $R$, the ratio of masses of the lightest tensor and scalar glueballs. Both results need to be further tested with more extensive, specific future studies. } ]]>