We study a class of UV-complete, strongly coupled, confining three-dimensional field theories, that exhibit a novel stabilization mechanism for the mass of the lightest scalar composite state, relying on the existence of a critical point. The theories admit a holographic dual description in terms of regular backgrounds in 11-dimensional supergravity, which retains its rigorous microscopic interpretation in field theory. The phase diagram includes a line of first-order phase transitions ending at the critical point, where the transition becomes of second order. We calculate the mass spectrum of bound states of the field theory, by considering fluctuations around the background solutions, and find that, near the critical point, a hierarchy of scales develops, such that one state becomes parametrically light. We identify this state as the dilaton, the pseudo-Nambu-Goldstone boson associated with the spontaneous breaking of approximate scale invariance, demonstrating the emergence of this composite state in an ab initio calculation that has a field theory origin. A stabilization mechanism of this type might be exploited to address hierarchy problems in particle and astroparticle physics.

Portugal

Daniel Elander

Spain

Antón F. Faedo

United Kingdom

Maurizio Piai

Sweden

Ronnie Rodgers

Netherlands

Javier G. Subils

Light dilaton near critical points in top-down holography

We develop a bottom-up holographic model that provides the dual description of a strongly coupled field theory, in which the spontaneous breaking of an approximate global symmetry yields the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo>/</mo><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> coset relevant to minimal composite-Higgs models. The gravity background is completely regular and smooth, and has an end of space that mimics confinement on the field theory side. We add to the gravity description a set of localized boundary terms, that introduce additional symmetry-breaking effects, capturing those that would result from coupling the dual strongly coupled field theory to an external, weakly coupled sector. Such terms encapsulate the gauging of a subgroup of the global <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></math> symmetry of the dual field theory, as well as additional explicit symmetry-breaking effects. We show how to combine spurions and gauge fixing and how to take the appropriate limits, so as to respect gauge principles and avoid violations of unitarity. The interplay of bulk and boundary-localized couplings leads to the breaking of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></math> symmetry to either its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></math> subgroup, via vacuum misalignment. In field theory terms, the model describes the spontaneous breaking of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> gauge symmetry to its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></math> subgroup. We expose the implications of the Higgsing phenomenon by computing the spectrum of fluctuations of the model, which we interpret in four-dimensional field-theory terms, for a few interesting choices of parameters. We conclude by commenting on the additional steps needed to build a realistic composite Higgs model.

Ali Fatemiabhari

Holographic vacuum misalignment

Within the bottom-up approach to holography, we construct a class of six-dimensional gravity models and discuss solutions that can be interpreted, asymptotically in the far UV, in terms of dual five-dimensional conformal field theories deformed by a single scalar operator. We treat the scaling dimension of this operator, related to the mass of the one scalar field in the gravity theory, as a free parameter. One dimension in the regular geometry is compactified on a shrinking circle, hence mimicking confinement in the resulting dual four-dimensional theories. We study the mass spectrum of bosonic states. The lightest state in this spectrum is a scalar particle. Along the regular (confining) branch of solutions, we find the presence of a tachyonic instability in part of the parameter space, reached by a smooth deformation of the mass spectrum, as a function of the boundary value of the background scalar field in the gravity theory. In a region of parameter space nearby the tachyonic one, the lightest scalar particle can be interpreted as an approximate dilaton, sourced by the trace of the stress-energy tensor, and its mass is parametrically suppressed. We also compute the free energy, along several branches of gravity solutions. We find that both the dilatonic and tachyonic regions of parameter space, identified along the branch of confining solutions, are hidden behind a first-order phase transition, so that they are not realized as stable solutions, irrespectively of the scaling dimension of the deforming field-theory operator. The (approximate) dilaton, in particular, appears in metastable solutions. Yet, the mass of the lightest state, computed close to the phase transition, is (mildly) suppressed. This feature is amplified when the (free) parameter controlling the scaling dimension of the deformation is <math><mn>5</mn><mo>/</mo><mn>2</mn></math>, half the dimension of space-time in the field theory.

France

Phase transitions and light scalars in bottom-up holography

We study a bottom-up, holographic description of a field theory yielding the spontaneous breaking of an approximate <math><mi>S</mi><mi>O</mi><mo>(</mo><mn>5</mn><mo>)</mo></math> global symmetry to its <math><mi>S</mi><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> subgroup. The weakly coupled, six-dimensional gravity dual has regular geometry. One of the dimensions is compactified on a circle that shrinks smoothly to zero size at a finite value of the holographic direction, hence introducing a physical scale in a way that mimics the effect of confinement in the dual four-dimensional field theory. We study the spectrum of small fluctuations of the bulk fields carrying <math><mi>S</mi><mi>O</mi><mo>(</mo><mn>5</mn><mo>)</mo></math> quantum numbers, which can be interpreted as spin-0 and spin-1 bound states in the dual field theory. This work supplements an earlier publication focused only on the <math><mi>S</mi><mi>O</mi><mo>(</mo><mn>5</mn><mo>)</mo></math> singlet states. We explore the parameter space of the theory, paying particular attention to composite states that have the right quantum numbers to be identified as pseudo-Nambu-Goldstone bosons (PNGBs). We find that in this model the PNGBs are generally heavy, with masses of the same order as other bound states, indicating the presence of a sizeable amount of explicit symmetry breaking in the field-theory side. Nevertheless, we also find a qualitatively new, unexpected result. When the dimension of the field-theory operator inducing <math><mi>S</mi><mi>O</mi><mo>(</mo><mn>5</mn><mo>)</mo></math> breaking is close to half the space-time dimensionality, there exists a region of parameter space in which the PNGBs and the lightest scalar are both parametrically light in comparison to all other bound states of the field theory. Although this region is known to yield metastable classical backgrounds, this finding might be relevant to model building in the composite Higgs context.

Toward minimal composite Higgs models from regular geometries in bottom-up holography

We continue our study of strongly-coupled, approximately scale-invariant gauge theories with a large number of flavours, which provide a suitable ultraviolet completion of the composite-Higgs scenario. We identify the requisite operators to realise partial compositeness of the Standard-Model fermions. In order to compute the spectrum of composite fermionic states, we extend the bottom-up holographic models, which we previously introduced to capture the main features of the non-perturbative dynamics in the Veneziano limit, by adding fermion fields in the bulk. We identify regions in parameter space where some fermionic bound states become light, depending in particular on the number of flavours, the operator scaling dimensions, and the bulk Yukawa couplings. We also observe a dense spectrum of states, when multi-scale dynamics is induced by a large backreaction of bulk scalars on the geometry. Adapting the formalism of the holographic Wilsonian renormalisation group, we study the linear coupling between the composite and elementary fermions, as a function of energy scale. We find that, in some circumstances, the associated operators are dangerously irrelevant: the renormalisation-group flow gives rise to a large linear coupling in the infrared, even when it is irrelevant from the point of view of the ultraviolet fixed point. We finally compute the partially composite spectrum, correlate it with the analysis of the flow, and assess the potential phenomenological implications, e.g. for the top-quark partners.

Michele Frigerio

Marc Knecht

Jean-Loïc Kneur

Holographic models of composite Higgs in the Veneziano limit. Part II. Fermionic sector

Within the context of top-down holography, we study a one-parameter family of regular background solutions of maximal gauged supergravity in seven dimensions, dimensionally reduced on a 2-torus. The dual, four-dimensional confining field theory realises the global (spontaneous as well as explicit) symmetry breaking pattern SO(5) → SO(4). We compute the complete mass spectrum for the fluctuations of the 128 bosonic degrees of freedom of the five-dimensional gravity theory, which correspond to scalar, pseudoscalar, vector, axial-vector, and tensor bound states of the dual field theory, and includes particles with exotic SO(4) quantum numbers. We confirm the existence of tachyonic instabilities near the boundaries of the parameter space.

Towards top-down holographic composite Higgs: minimal coset from maximal supergravity

We find a parametrically light dilaton in special confining theories in three dimensions. Their duals form what we call a scion of solutions to the supergravity associated with the large-<math><mi>N</mi></math> limit of the Coulomb branch of the <math><mi>N</mi><mo>=</mo><mn>4</mn></math> Super-Yang-Mills theory. The supergravity description contains one scalar with bulk mass that saturates the Breitenlohner-Freedman unitarity bound. The new solutions are defined within supergravity, they break supersymmetry and scale invariance, and one dimension is compactified on a shrinking circle, yet they are completely regular. An approximate dilaton appears in the spectrum of background fluctuations (or composite states in the confining theory), and becomes parametrically light along a metastable portion of the scion of new supergravity solutions, in close proximity of a tachyonic instability. A first-order phase transition separates stable backgrounds, for which the approximate dilaton is not parametrically light, from metastable and unstable backgrounds, for which the dilaton becomes parametrically light, and eventually tachyonic.

John Roughley

Coulomb branch of <math><mrow><mi>N</mi><mo>=</mo><mn>4</mn></mrow></math> SYM and dilatonic scions in supergravity

The spectrum of bound states of special strongly coupled confining field theories might include a parametrically light dilaton, associated with the formation of enhanced condensates that break (approximate) scale invariance spontaneously. It has been suggested in the literature that such a state may arise in connection with the theory being close to the unitarity bound in holographic models. We extend these ideas to cases where the background geometry is non-anti-de Sitter, and the gravity description of the dual confining field theory has a top-down origin in supergravity. We exemplify this program by studying the circle compactification of Romans six-dimensional half-maximal supergravity. We uncover a rich space of solutions, many of which were previously unknown in the literature. We compute the bosonic spectrum of excitations and identify a tachyonic instability in a region of parameter space for a class of regular background solutions. A tachyon only exists along an energetically disfavored (unphysical) branch of solutions of the gravity theory; we find evidence of a first-order phase transition that separates this region of parameter space from the physical one. Along the physical branch of regular solutions, one of the lightest scalar particles is approximately a dilaton, and it is associated with a condensate in the underlying theory. Yet, because of the location of the phase transition, its mass is not parametrically small, and it is, coincidentally, the next-to-lightest scalar bound state, rather than the lightest one.

Dilatonic states near holographic phase transitions

We study strongly-coupled, approximately scale-invariant gauge theories, which develop a mass gap in the infrared. We argue that a large number of fermion flavours is most suitable to provide an ultraviolet completion for the composite Higgs scenario. The holographic approach allows to describe the qualitative features of the non-perturbative dynamics in the Veneziano limit. We introduce new bottom-up holographic models, which incorporate the backreaction of flavour on the geometry, and show that this can correlate the mass gap to the scale of flavour-symmetry breaking. We compute the mass spectrum for the various composite bosonic states, and study its dependence on the scaling dimension of the symmetry-breaking operators, as well as on the number of flavours. The different regions with a light dilaton are critically surveyed. We carefully assess the domain of validity of the holographic approach, and compare it with lattice simulations and the Nambu-Jona-Lasinio model.

Holographic models of composite Higgs in the Veneziano limit. Part I. Bosonic sector

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.