Higgs as a holographic pseudo-Goldstone boson

Abstract

The AdS/CFT correspondence allows one to relate 4D strongly coupled theories to weakly coupled theories in 5D AdS. We use this correspondence to study a scenario in which the Higgs appears as a composite pseudo-Goldstone boson (PGB) of a strongly coupled theory. We show how a non-linearly realized global symmetry protects the Higgs mass and guarantees the absence of quadratic divergences at any loop order. The gauge and Yukawa interactions for the PGB Higgs are simple to introduce in the 5D AdS theory, and their one-loop contributions to the Higgs potential are calculated using perturbation theory. These contributions are finite, giving a squared-mass to the Higgs which is one-loop smaller than the mass of the first Kaluza–Klein state. We also show that if the symmetry breaking is caused by boundary conditions in the extra dimension, the PGB Higgs corresponds to the fifth component of the bulk gauge boson. To make the model fully realistic, a tree-level Higgs quartic coupling must be induced. We present a possible mechanism to generate it and discuss the conditions under which an unwanted large Higgs mass term is avoided.

Introduction

Despite its tremendous phenomenological success, the standard model is almost certainly not a fundamental theory of nature. Quantum instabilities of the Higgs potential strongly suggest that it must be replaced by some other theory at energies not much higher than the electroweak scale. Such a theory, for example, can be a supersymmetric theory [1], a strongly coupled gauge theory [2], or a theory of quantum gravity [3]. In these theories the quadratic divergence of the Higgs squared-mass parameter is cutoff either by embedding the Higgs boson into some larger multiplet or by giving it an internal structure. A physical scale then exists, Λ_NP, at which many new particles appear revealing the underlying symmetry or dynamics.

In this paper we consider a class of theories where the standard-model Higgs boson arises as a composite particle of some strongly coupled dynamics. The dynamical scale Λ_NP then must be parametrically larger than the scale of the Higgs mass in order to avoid strong constraints coming from direct and indirect searches of new particles at colliders. This suggests that the Higgs mass must be protected by some (approximate) symmetry even below the scale Λ_NP. A natural candidate for such a symmetry is an internal global symmetry under which the Higgs boson transforms non-linearly: the Higgs is a pseudo-Goldstone boson (PGB) of the broken global symmetry. This situation is somewhat similar to that of the pion in QCD, although for the Higgs there are additional requirements. It must have a sizable quartic self-coupling and appropriate Yukawa couplings to the quarks and leptons. In this paper we aim to build theories of this kind, which are well under control as effective field theories, and in which some quantities are even calculable despite the strongly interacting dynamics.

The basic observation is the following. Suppose we have a strongly coupled gauge theory that produces the Higgs boson as a composite state. In general, it is quite difficult to obtain quantitative low-energy predictions in such a theory because of non-perturbative effects; one can at best derive estimates by using certain scaling arguments. This is indeed the case if the gauge interaction is asymptotically free, as in QCD. However, it is not necessarily true if the theory remains strongly coupled in the UV and approaches a non-trivial conformal fixed point. In this case it is possible that the theory, in the limit of large number of “colors”, has an equivalent description in terms of a weakly coupled 5D theory defined on the truncated anti-de Sitter (AdS) space [4], [5]. This allows us to construct theories where the Higgs boson is interpreted as a composite state of a strong dynamics and yet some physical quantities such as the Higgs potential can be computed using perturbation theory.

The actual implementation of the above idea is quite simple, as far as gauge and Yukawa interactions are concerned. These interactions explicitly break the global symmetry, but, as we will see, they do not induce quadratic divergences for the Higgs mass at any loop order. The global symmetry protects the Higgs mass parameter at high energies and predicts it to be a one-loop factor smaller than the mass of the lightest resonance ∼Λ_NP. This property makes this class of theories quite interesting.

To make the model fully realistic, however, we must generate a sizable Higgs quartic coupling. This is crucial not only to obtain a large enough physical Higgs mass, but also to obtain the needed mas gap between the electroweak scale and Λ_NP (as suggested by experiments). We will propose a mechanism that allows to generate a Higgs quartic coupling at tree level. This mechanism needs specific assumptions on the form of the explicit breaking of the global symmetry. However, these are assumptions on the underlying physics around the Planck scale, and not on the TeV-scale physics which yields the Higgs boson as a composite particle. Therefore, once the particular pattern of breaking is assumed (which is not quite unnatural from the viewpoint of the 4D picture), we can compute the Higgs potential generated at loop level through the explicit symmetry-breaking effects.

Since the cutoff of our theory is around the Planck scale, there is no obstacle in extending the theory beyond Λ_NP, up to very high energies. In this respect, our framework may be viewed as a way to provide a UV completion for “little Higgs” theories [6] in which realistic models of the PGB Higgs have been constructed. It might be interesting to construct a UV completion of the existing little Higgs models [7] using a warped spacetime as outlined in the present paper.

In the next section we start by defining the framework in more detail. We describe the basic structure of our theories in terms of both 4D and 5D pictures. An explicit model is given in Section 3, in which the Higgs boson is identified with a PGB arising from a scalar field located on the infrared brane. We present a possible mechanism to obtain a sizable Higgs quartic coupling and discuss the size of quantum corrections to the Higgs potential, which trigger electroweak symmetry breaking. In Section 4 we consider theories where the Higgs boson arises from the extra-dimensional component of a gauge field in a warped 5D spacetime. We point out that also in this case the Higgs is interpreted as a composite PGB in the 4D picture, and we present several realistic models. Conclusions are given in Section 6.