Advances in High Energy Physics

Volume 2018, Article ID 7168480, 8 pages

https://doi.org/10.1155/2018/7168480

## Composite Higgs Models after Run 2

Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, UK

Correspondence should be addressed to Veronica Sanz; ku.ca.xessus@znas.v

Received 30 October 2017; Revised 22 January 2018; Accepted 1 February 2018; Published 18 March 2018

Academic Editor: Alexey A. Petrov

Copyright © 2018 Veronica Sanz and Jack Setford. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We assess the status of models in which Higgs is a composite pseudo-Nambu Goldstone boson, in the light of the latest 13 TeV Run 2 Higgs data. Drawing from the extensive Composite Higgs literature, we collect together predictions for the modified couplings of Higgs, in particular examining the different predictions for and . Despite the variety and increasing complexity of models on the market, we point out that many independent models make identical predictions for these couplings. We then look into further corrections induced by tree-level effects such as mass-mixing and singlet VEVs. We then investigate the compatibility of different models with the data, combining Run 1 and recent Run 2 LHC data. We obtain a robust limit on the scale of 600 GeV, with stronger limits for different choices of fermion embeddings. We also discuss how a deficit in a Higgs channel could pinpoint the type of Composite Higgs model responsible for it.

#### 1. Introduction

Composite Higgs models [1–3] offer an elegant solution to the hierarchy problem of Higgs physics. They postulate the existence of a new strongly interacting sector which confines not far above the electroweak scale. In recent years there has been significant interest in a specific class of these models: models in which the Higgs emerges as a pseudo-Nambu Goldstone boson of the strong sector. This sector is taken to be endowed with a global symmetry which is spontaneously broken in the confining phase, protecting the Higgs mass from corrections above the compositeness scale. Although the idea is reasonably straightforward, there are, as with most theories Beyond the Standard Model, many possibilities for its realisation.

Although this plethora of models offers a variety of unique and interesting predictions, those that are most immediately testable are the modifications of the Higgs couplings to the rest of the Standard Model fields. Of particular interest are the values of the coupling modifiers and , as defined in [4].

In this paper we summarise the predictions for these couplings in Composite Higgs (CH) models. We make the case that, despite the diversity of models in the literature, these predictions have very generic structures, and we attempt to provide some intuition for this fact.

We then investigate some simple cases in which tree-level effects can modify these generic structures. These can occur, for instance, in models with extra singlets that get vacuum expectation values (VEVs) or models with an extra doublet that mixes with the Higgs. We point out that to leading order the modifications to and are precisely as one would expect in corresponding models where all the scalars are elementary, plus the usual CH corrections.

Taking the generic structures we have identified, we then perform a fit to the data, allowing for the possibility that different fermions couple in different ways. We place bounds on the compositeness scale and identify the classes of models that are most constrained.

#### 2. The Nonlinear Composite Higgs

In Composite Higgs models, the Higgs is realised as a pseudo-Nambu Goldstone boson (pNGB) of a broken global symmetry. This symmetry is a symmetry of a* new* strongly interacting sector, out of which Higgs emerges as a composite.

Let the global symmetry be denoted by and the subgroup to which it spontaneously breaks be denoted by . Then Higgs and the other pNGBs (denoted collectively by , one for each broken generator ) are parametrised via where is an energy scale associated with the spontaneous symmetry breaking. transforms nonlinearly under the global symmetry : where and . By nonlinear we mean that the transformation is field-dependent: .

In cases where the coset is symmetric (If and are the unbroken and broken generators respectively, then the Lie algebra of a symmetric coset obeys the schematic relations , , ) we are allowed to construct an object (which we will label as ) whose transformation under is* linear*. In all the models considered here [3, 5–22] and in the vast majority of models in the literature, will be symmetric. This reduces the task of writing down a low-energy effective theory for the pNGBs to a relatively trivial search for invariant combinations of and the other relevant fields.

We will assume that the Higgs boson is a doublet under , which, along with , must be embedded as an unbroken subgroup of . Although data strongly supports the doublet scenario (e.g., see LHC constraints on the ratio of couplings to and bosons [4]), nonlinear models have been studied in which the four scalar fields are actually a singlet and a triplet under [23–26] (note, though, that one could assume a custodially symmetric strong sector as in [27, 28]).

##### 2.1. Gauge Couplings

The couplings of Higgs to the gauge bosons come from the kinetic term for , which in the CCWZ prescription [29] iswhere , with for each gauged generator . We assume that Higgs is embedded in a bidoublet of a custodial ; this is necessary in order to protect the parameter from unwanted corrections [30]. Note that this imposes the nontrivial requirement that must contain an unbroken factor of .

Since we are interested in the couplings of the physical Higgs boson to SM fields, we will expand along the direction in which Higgs will get a VEV and set all other pNGB fields to zero. The term in (3) will generically (in unusual cases the coupling may be proportional instead to , but all this amounts to is a redefinition of and an effective rescaling of ) lead to a Higgs-gauge coupling of the form, which is valid as a series expansion around .

Expanding around Higgs VEV (where is the physical excitation of the Higgs field) we find the gauge boson masses and couplings: Identifying (here is defined as , as in the Standard Model) and defining , we find Thus, Since is defined as , we find

Since the structure of (3) is generic, so too is this result, at leading order, across all Composite Higgs models.

##### 2.2. Fermion Couplings

In Composite Higgs models the SM fermions usually couple to the strong sector via the* partial compositeness* mechanism [8, 31, 32]. As far as this mechanism pertains to the construction of the low-energy effective theory, it involves embedding the SM fermions in representations of the global symmetry and then constructing invariant operators out of these multiplets and . Such an embedding is sometimes called a spurion; the term spurion refers to the “missing” elements of the multiplet, since after all, the SM particles do not come in full multiplets of the new symmetry . The incompleteness of these spurious multiplets contributes to the explicit breaking of and allows Higgs to acquire a potential via loops of SM fermions.

The appropriate representation in which embedding the SM particles would, in principle, depend on the UV completion of the model. Some attempts towards UV completions of Composite Higgs models have been made (see, e.g., [5, 7, 9]); however, for the purposes of most model building the choice of representation is a “free parameter” of the model. There is, however, good cause to restrict the choice of representation into which the quark doublet is embedded. As shown in [33], embedding into a bidoublet of the custodial can prevent anomalous contributions to the coupling. This restriction forces one to choose representations that contain a bidoublet in their decomposition under the custodial subgroup of .

To treat the EFT in full generality, one should embed , , and into different multiplets , , and . The kind of representation that the three quarks are embedded into need not be the same. Thus, even for each coset , there are a bewildering number of possibilities. However, for the vast majority of models the form of is actually quite restricted. We tabulate a few examples in Table 1.