Advances in High Energy Physics

Volume 2017 (2017), Article ID 6295927, 8 pages

https://doi.org/10.1155/2017/6295927

## A Correlation between the Higgs Mass and Dark Matter

^{1}Center for Theoretical Physics and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA^{2}Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA

Correspondence should be addressed to Mark P. Hertzberg

Received 20 January 2017; Revised 16 May 2017; Accepted 7 June 2017; Published 27 July 2017

Academic Editor: Juan José Sanz-Cillero

Copyright © 2017 Mark P. Hertzberg. 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

Depending on the value of the Higgs mass, the Standard Model acquires an unstable region at large Higgs field values due to RG running of couplings, which we evaluate at 2-loop order. For currently favored values of the Higgs mass, this renders the electroweak vacuum only metastable with a long lifetime. We argue on statistical grounds that the Higgs field would be highly unlikely to begin in the small field metastable region in the early universe, and thus some new physics should enter in the energy range of order of, or lower than, the instability scale to remove the large field unstable region. We assume that Peccei-Quinn (PQ) dynamics enters to solve the strong CP problem and, for a PQ-scale in this energy range, may also remove the unstable region. We allow the PQ-scale to scan and argue, again on statistical grounds, that its value in our universe should be of order of the instability scale, rather than (significantly) lower. Since the Higgs mass determines the instability scale, which is argued to set the PQ-scale, and since the PQ-scale determines the axion properties, including its dark matter abundance, we are led to a correlation between the Higgs mass and the abundance of dark matter. We find the correlation to be in good agreement with current data.

#### 1. Introduction

Recent LHC results are consistent with the predictions of the Standard Model, including the presence of a new boson that appears to be the Higgs particle with a mass GeV [1, 2] (more recent measurements are summarized in [3]). With the Higgs at this mass, the Standard Model is well behaved up to very high energies if we evolve its couplings under the renormalization group (RG) equations. By no means does this imply that the Standard Model will be valid to these very high energies, and in fact there are good phenomenological reasons, such as dark matter, strong CP problem, baryogenesis, inflation, and hierarchy problem, to think it will be replaced by new physics at much lower energies, say (TeV). But it is logically possible, albeit unlikely, that the Standard Model, or at least the Higgs sector, will persist to these very high energies and the explanation of these phenomena will be connected to physics at these high, or even higher, energy scales.

So at what energy scale must the Standard Model breakdown? Obviously new physics must enter by the Planck scale where quantum gravity requires the introduction of new degrees of freedom. However, the RG running of the Higgs self-coupling can dictate the need for new physics at lower energies, depending on the starting value of . The Higgs mass is related to the self-coupling by , where the Higgs VEV is GeV. For moderate to high values of the Higgs mass, the initial value of , defined at energies of order of the electroweak scale, is large enough that it never passes through zero upon RG evolution. On the other hand, for small enough values of the Higgs mass, the self-coupling passes through zero at a sub-Planckian energy, which we denote by , primarily due to the negative contribution to the beta function from the top quark, acquiring an unstable region at large field values [4, 5]. The latter occurs for a light Higgs as has been observed. One finds that this renders the electroweak vacuum only metastable with a long lifetime. However, we will argue in this paper that it is highly unlikely for the Higgs field in the early universe to begin in the metastable region as that would require relatively small field values as initial conditions. Instead it would be much more likely to begin at larger field values, placing it in the unstable region. Hence, the energy scale sets the maximum energy scale for new physics beyond the Standard Model to enter.

There are many possible choices for the new physics. One appealing possibility is supersymmetry, which alters the running of the Higgs self-coupling due to the presence of many new degrees of freedom, likely entering at much lower energies, conceivably (TeV), or so. In addition to possibly stabilizing the Higgs potential, supersymmetry can also alleviate the hierarchy problem, improve unification of gauge couplings, and fit beautifully into fundamental physics such as string theory. So it is quite appealing from several perspectives. It is conceivable, however, that even if supersymmetry exists in nature, it is spontaneously broken at very high energies, and in such a scenario we would be forced to consider other possible means to stabilize the Higgs potential.

One intriguing possibility that we examine in this paper is to utilize dynamics associated with the solution of the strong CP problem; the problem that the CP violating term in the QCD Lagrangian is experimentally constrained to have coefficient , which is highly unnatural. The leading solution involves new Peccei-Quinn (PQ) dynamics [7], involving a new complex scalar field and a new global symmetry that is spontaneously broken at some energy scale . This leads to a new light scalar field known as the axion [8, 9]. Since it is bosonic, the field adds a positive contribution to the effective for the Higgs, potentially removing the unstable region, depending on the scale . This elegant mechanism to remove the unstable region was included in the very interesting reference [10], where this and other mechanisms were discussed, and was a source of motivation for the present work (also related is [11–13]).

In the present paper, we would like to take this elegant mechanism for vacuum stability and push it forward in several respects. Firstly, as already mentioned, we will argue on statistical grounds why the metastable vacuum requires stabilization. Secondly, we will allow the PQ-scale to scan and argue, again on statistical grounds, why it should be of order of the instability scale , rather than orders of magnitude lower. Finally, we will furnish a correlation between the Higgs mass and the axion dark matter abundance and use the latest LHC [1, 2] and cosmological data [6] to examine the validity of this proposal. The outcome of this series of arguments and computation is presented in Figure 1, which is the primary result of this work.