^{1}

^{2}

^{1}

^{2}

^{3}.

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.

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

So at what energy scale must the Standard Model breakdown? Obviously new physics must enter by the Planck scale

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

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 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

Dark matter density

The outline of our paper is as follows. In Section

We begin with a reminder of the structure of the Higgs sector of the Standard Model. The Higgs field is a complex doublet

Performing the RG evolution leads to the energy dependent renormalized coupling

Higgs self-coupling

If we think of the field value

We could plot

Schematic of the effective potential

In this situation, the electroweak vacuum is only metastable. Its quantum mechanical tunneling rate can be estimated by Euclideanizing the action and computing the associated bounce action

It is conceivable that it is an acceptable situation for the electroweak vacuum to be metastable. However, here we would like to present an argument that such a situation is statistically disfavorable. We imagine that, in the very early universe, the Higgs field was randomly distributed in space. For instance, during cosmological inflation the Higgs field could have been frozen at some value as the universe rapidly expands (if there is high scale inflation) until after inflation when the field will oscillate and its initial value could plausibly have been random and uniformly distributed. If this is the case, then what is the probability that the Higgs field began in the metastable region

So, for instance, for

One of the phenomenological reasons for new physics beyond the Standard Model is the fine tuning of the CP violating term in the QCD Lagrangian. The following dimension 4 operator is gauge invariant and Lorentz invariant and should be included in the QCD Lagrangian with a dimensionless coefficient

Since

In the most common case then, this leads to a reduction or removal of the unstable region depending on the scale

We obviously require ^{11} GeV. In some landscape, we can imagine

The light scalar axion particle is neutral, is very stable, and acts as a form of dark matter. The computation of its abundance is nontrivial and has been studied in many papers, including [

The quantity

Let us summarize our argument: holding other parameters fixed, the Higgs mass

Improved accuracy in testing this scenario comes in several experimental directions. This includes measuring the Higgs mass

Related to this uncertainty is the particular prior distribution for ^{−8})^{4} = 10^{−32} or so, an alteration in prior probabilities would need to be quite drastic to change the conclusions.

An important test of this scenario involves unravelling the nature of dark matter directly. The QCD-axion is actively being searched for in a range of experiments, including ADMX [

The discovery of the Higgs boson at the LHC is a final confirmation of the Standard Model. This leaves the scale at which the theory breaks down unclear. Here we have investigated the possibility that the theory, or at least the Higgs sector, remains intact until the scale at which the Higgs potential runs negative which would lead to a runaway instability at large field values. By introducing Peccei-Quinn dynamics, we can potentially solve the strong CP problem, remove the unstable region, and obtain roughly the correct amount of dark matter due to a collection of statistical arguments that sets

In this Appendix, we list the RG equations for the couplings

For the Higgs quartic coupling, we have

The wave function renormalization of the Higgs field is

The author declares that there are no conflicts of interest regarding the publication of this paper.

The author would like to thank Alan Guth and Frank Wilczek for helpful discussions and would also like to acknowledge support from the Center for Theoretical Physics at MIT and the Tufts Institute of Cosmology. This work is supported by the U.S. Department of Energy under cooperative research agreement Contract no. DE-FG02-05ER41360.