The KSVZ Axion Model with Quasi-Degenerate Minima: A Unified Model for Dark Matter and Dark Energy

We consider the possibility that dark matter and dark energy can be explained by the minimal KSVZ axion model. This is possible if the lowest energy metastable minimum of the scalar potential has zero energy density, which is possible in theoretical models of vacuum energy cancellation based on spacetime averaging and in models based on energy parity. Dark energy is then understood as being due to the energy density of the metastable electroweak vacuum relative to a second quasi-degenerate metastable minimum. The requirement of quasi-degenerate minima is a non-trivial condition which completely determines the form of the potential for a given value of the axion decay constant, $f_{a}$, and the PQ scalar self-coupling, $\lambda_{\phi}$. The existence of the second quasi-degenerate minimum imposes a new lower bound on the axion decay constant, $f_{a} \geq 2.39 \times 10^{10} \, \lambda_{\phi}^{-1/4}$ GeV. If the PQ symmetry is broken after inflation then the lower bound on $f_{a}$ implies a lower bound on the amount of axion dark matter, $\Omega_{a}/\Omega_{dm} \geq (0.28-0.46)\,\lambda_{\phi}^{-0.291}$, where the range is due to the uncertainty in the amount of axion dark matter produced by vacuum realignment, cosmic strings and domain walls, therefore at least 30$\%$ of dark matter must be due to axions if $\lambda_{\phi} \lesssim 1$. When axions constitute all of the dark matter and the PQ symmetry is broken after inflation, $f_{a}$, and so the form of the scalar potential, is completely fixed for a given value of $\lambda_{\phi}$, with only a weak dependence on $\lambda_{\phi}$. This will allow the inflation and post-inflation evolution of the model to be quantitatively studied for a given inflation model and dimensionally natural values of $\lambda_{\phi}$.


I. INTRODUCTION
The Standard Model (SM) has been successful in explaining the observed nature of strong and electroweak interactions. However, there are fundamental questions which are not addressed by the SM. Here we will consider an existing extension of the SM, the KSVZ version of the Pecci-Quinn (PQ) axion model [1,2], which is presently able to address two of these questions, the strong CP problem and the nature of dark matter. We will show that in the case where the energy density is zero at the lowest metastable minimum of the scalar potential, the KSVZ axion model can also account for dark energy in the form of the energy density of the electroweak vacuum state relative to a second metastable minimum. We will also show that the requirement that the scalar potential is very close to the limit of degenerate vacuum states is a non-trivial constraint. This has implications for the minimum amount of axion dark matter and for the form of the scalar potential for a given axion dark matter density.
One class of models which can account for the lowest energy metastable minimum being at zero energy density are those based on spacetime averaging, where the energy averaged over spacetime adds a negative constant energy density to the Lagrangian. If the universe is in the lowest energy metastable vacuum state over most of the volume of spacetime, then the energy of that vacuum state will be cancelled. The first model of this type was the universe multiplication model proposed in [3]. Other more recent approaches include vacuum sequestering [4] and a model based on non-local constraints [5]. A different approach, which can also result in the lowest energy metastable minimum having zero energy, is based on energy parity [6]. In this case the energy density of the lowest energy metastable minimum is zero if the mirror universe occupies the same vacuum state. In all of these models, all of which seek to explain the absence of a large vacuum energy density, the total energy of one of the vacuum states is effectively subtracted away. The KSVZ axion model adds a singlet complex scalar field, the PQ scalar, Φ, to the scalar sector of the SM. The axion is the angular mode of Φ. The axial anomaly couples the axion to the QCD topological term GG, which generates a potential for the axion after the QCD phase transition. The axion field cancels the CP-violating QCD θ-angle at the minimum of this potential, solving the strong-CP problem.
Extending the SM by the addition of Φ modifies the classical scalar potential of the model to a function of two scalar fields, h and φ, where h is the electroweak doublet Higgs scalar and φ is the radial mode of the PQ scalar. The minimum of the classical potential for a given value of h is then a curving trajectory in the (h, φ) plane. Therefore the question of the stability of the electroweak vacuum is quite different from the case of the SM. In addition, the observed SM Higgs boson is a mixture of φ and h. As a result, the electroweak doublet Higgs quartic coupling can be larger than the SM Higgs quartic coupling which is determined experimentally from the Higgs mass. The effect of increasing the electroweak Higgs doublet quartic coupling is to increase the stability of the electroweak vacuum with respect to quantum corrections. The combined effect of the classical modification of the scalar potential and the increase of the electroweak Higgs doublet quartic coupling was first discussed in [7], where it was called the scalar threshold effect.
In this paper we will show how the combination of the classical scalar potential and quantum corrections can allow the existence of a second vacuum, which is at a slightly lower energy than the electroweak vacuum. In the case where there is zero energy density at the lowest energy metastable minimum of the potential, the observed dark energy density can then be explained by the vacuum energy of the metastable electroweak minimum relative to the lowest energy minimum. (The possibility of metastable dark energy has previously been considered in [8][9][10][11][12][13].) Combined with axion dark matter, the KSVZ axion model can then serve as a minimal model for both dark matter and dark energy, as well as providing a solution to the strong CP problem.
In the case where the PQ symmetry is restored after inflation and subsequently broken, as may be necessary in order to satisfy axion isocurvature constraints, the axion dark matter density is determined for a given value of f a . We will show that the requirement of a quasi-degenerate second minimum imposes a lower bound on λ 1/4 φ f a , where λ φ is the PQ scalar self-coupling. This in turn imposes a lower bound on the density of axion dark matter.
In addition, we will show that the form of the potential is entirely determined by the value of λ 1/4 φ f a . Therefore, in the case of PQ symmetry breaking after inflation, the potential is determined by the axion dark matter density for a given value of λ φ , with only a weak dependence on this unknown coupling.
Our paper is organized as follows. In Section 2 we review the KSVZ axion model and explain our method for determining the quantum corrections to its potential. In Section 3 we discuss the conditions for a second quasi-degenerate vacuum as a function of the axion decay constant. In Section 4 we discuss the cosmology of the quasi-degenerate potential, in particular for the case where the PQ symmetry is broken after inflation. In Section 5 we present our conclusions.

A. The Classical Scalar Potential
We will consider the minimal KSVZ axion model in the following, which adds a singlet complex scalar field to the scalar sector of the SM 1 . The Lagrangian fermion term added to the SM is Here Q is a triplet of fermions under SU(3) c which is a singlet under the electroweak group, and Φ = φ/ √ 2e ia/ f a is the PQ scalar, where φ is the radial PQ scalar and a is the axion. These terms are invariant under an axial global U(1) PQ symmetry where Φ → e iα Φ and Q → e iαγ 5 Q. The axial anomaly of the U(1) PQ symmetry couples the axion to the QCD topological term, which generates the necessary potential to solve the strong CP problem 2 .
The scalar potential of the model is given by where H is the Higgs doublet. As a function of the electroweak doublet Higgs field h and radial PQ scalar φ, this becomes To determine the minima of this potential, we will consider the minimum in the φ direction as a function of h, φ(h), then follow the trajectory of this minimum in the (h, φ) plane. φ(h) is given by where This is true when h < h c , where h c is the value at which φ 2 (h) = 0 and so the U(1) PQ symmetry is restored, Since µ 2 Φ ≈ λ φ f 2 a to a good approximation for natural values of the couplings, we can also write this as This will be important in the following. The potential along the minimum trajectory as a function of h when h < h c is then At h > h c , φ(h) = 0 and so the potential along the minimum trajectory becomes where and The potential along the minimum trajectory depends purely on the combination λ 2 hφ /λ φ . The only explicit dependence on the PQ scalar couplings λ hφ and λ φ in our analysis is in the relation between h c and f a in Eq. (7), where there is a weak λ A key feature of the potential along the minimum as function of h is the change in behaviour of the h 4 term at h = h c , where the effective quartic coupling changes from λ h − λ 2 hφ /4λ φ at h < h c to λ h at h > h c . This jump in the value of the effective quartic coupling will play an important role in producing the degenerate minimum of the potential at h close to h c once quantum corrections are included.
In addition to creating a two-field potential, the addition of the PQ scalar also modifies the input value of the Higgs doublet quartic coupling λ h at the electroweak scale. This is due to mixing between h and φ. On expanding the field around the electroweak vacuum at (h, φ) = (v, f a ) as h =ĥ + v and φ =φ + f a , the mass matrix ofĥ andφ becomes Using f 2 a v f a v 2 , the eigenvalues of this matrix are and where h SM is the observed SM Higgs boson with m h SM = 125 GeV and φ is the orthogonal mass eigenstate. The mass eigenstates are related toĥ andφ by and where Although the SM Higgs h SM has only a very small admixture ofφ, the large value of f a relative to v causes a significant shift of the relation between the SM Higgs mass and the Higgs doublet quartic coupling, λ h . The SM Higgs quartic coupling, λ h SM , which is defined by Thus λ h is generally larger than λ h SM . The effect of the two-field potential and the increase of λ h relative to λ h SM is equivalent to the scalar threshold effect discussed in [7], where the heavy PQ scalar is integrated out to form an effective single field Higgs potential. In our analysis we will work with the full two-field scalar potential, without integrating out the heavy PQ scalar.

B. Quantum Corrections to the Scalar Potential
In order to calculate the quantum corrections to the potential, we need to calculate the effective action as a function of two scalar fields. For a single scalar field the effective action can be expressed as a functional Taylor series where ϕ is the classical field. The effective potential then corresponds to the sum of 1PI Green's functions with all external particles at zero momentum where Γ (n) (p i ; µ) are the 1PI Green's functions in momentum space, renormalized at a scale µ in a given renormalization scheme. On performing the sum over n, the effective potential to 1-loop order becomes the tree-level (i.e. classical) potential plus the 1-loop Coleman-Weinberg (CW) correction.
In the case with two scalar fields ϕ and χ, the effective action is a straightforward generalization of the functional Taylor expansion to the case with two scalar fields. The corresponding effective potential is therefore of the form The functions Γ (l,m) (p i ; µ) are 1PI Green's functions with l external ϕ fields and m external χ fields. However, there is then a problem when using perturbation theory to calculate the effective potential in the case where one of the fields has a large value compared to the renormalization scale of the theory whilst the other field remains small. For example, if ϕ is large compared to the initial renormalization scale, µ 0 , while χ ∼ µ 0 , then on summing over the terms with only the ϕ fields we will get a contribution to the effective potential which is just the conventional single field CW correction. We therefore need to run µ up to µ ∼ ϕ to keep this contribution to the effective potential perturbative. However, this will then cause the CW correction calculated from the sum over only the χ terms to have a large logarithm, ln(χ/µ). Therefore it is difficult to use perturbation theory to calculate the effective potential at points (ϕ, χ) where the magnitudes of ϕ and χ are very different.
Although there are multiscale renormalization group methods which attempt to address this problem [15][16][17][18] (and which are still being developed [19][20][21]), in this first analysis of the potential we will neglect the quantum effects of the λ hφ and λ φ by setting them to zero in the 1-loop Green's function calculations. This will produce the correct potential if the classical modification of the potential due to the φ field and its mixing with the electroweak doublet Higgs field dominates the effect of the quantum corrections due to λ hφ and λ φ . This will be true if these couplings are sufficiently small. In this case we can set the 1-loop contribution to Γ (l,m) in Eq. (21) to zero for m = 0 and for any internal φ lines, in the case where ϕ is identified with h and χ with φ. The quantum corrections to the effective potential then reduce to the sum over h in the effective potential Eq. (21), in which case the quantum corrections at the point (h, φ) are simply the SM CW corrections calculated for the field h. This will allow us to use the existing two-loop SM RG equations to calculate the effective potential in a well-defined limit. It is always possible choose both λ hφ and λ φ to be small enough that quantum corrections due to these couplings are negligible, whilst still obtaining a degenerate vacuum purely due to the classical effects of λ hφ and λ φ . This is because main effects of the axion sector scalars on the stability of the electroweak vacuum, namely the classical modification of the quartic electroweak doublet coupling λ h relative to the SM Higgs coupling λ h SM and the form of the potential along the minimum trajectory, V (h, φ(h)), both depend only on the combination λ 2 hφ /λ φ , which can be large even if λ hφ and λ φ are small. We define the tree-level classical potential using couplings evaluated at the renormalization point µ = m t in the MS scheme to be V 0 (h, φ; µ = m t ). There will also be a quantum correction to this potential at the point (h, φ), which we denote by ∆V q (h, φ). In order to calculate the quantum correction to the potential, we will start from the electroweak vacuum state, (h, φ) = (v, f a ). We first consider the potential in the direction corresponding to the electroweak doublet Higgs scalar h. Having calculated the correction at (h, f a ), we next fix the corresponding value of h and move in the φ direction to the point of interest on the minimum trajectory, (h, φ(h)). (The trajectory is schematically illustrated in Figure 1.) Since we are neglecting the quantum corrections due to λ hφ and λ φ , only the classical potential changes as we move from (h, f a ) to (h, φ(h)). The result of this is that the quantum corrected potential at the point (h, φ(h)) is given by To compute ∆V q (h, f a ), we run the 2-loop SM RG equations in the MS scheme from the electroweak vacuum (with µ = m t ) to µ = h to obtain the couplings at this renormalization point. We then add the 1-loop CW correction in the MS scheme, ∆V CW . The tree-level potential along (h, f a ) at h v is simply Thus Therefore, from Eq. (22) and Eq. (24), the quantum correction to the potential along the minimum trajectory (h, φ(h)) is given by simply replacing the λ h h 4 /4 term by λ h (µ = h)h 4 /4 and adding the 1-loop CW term. Thus, at h < h c , the quantum corrected potential is where all couplings are evaluated at µ = m t except for λ h (µ = h). At h > h c , where φ(h) = 0, the potential along the minimum trajectory becomes

III. QUASI-DEGENERATE MINIMA OF THE POTENTIAL
To calculate the potential, we will use the 2-loop SM RG equations given in Appendix A of [22] and the 1-loop SM CW potential in the MS scheme given in [23]. We use the initial values for the SM couplings in the MS scheme at µ = m t as given in [24]: λ h (m t ) = 0.12604, y t (m t ) = 0.93690, g 2 (m t ) = 0.64779, g Y (m t ) = 0.35830, g 3 (m t ) = 1.6666 and m t = 173.34 GeV.
To explain dark energy, we require an effectively degenerate second minimum of the scalar potential at V = 0 in addition to the electroweak vacuum. (The tiny increase of the electroweak vacuum relative to the second minimum is negligible on the scale of the potential.) The potential is a function of a single parameter, which can be taken to be the combination λ hφ / λ φ . To obtain the degenerate minimum, our procedure is to first consider the potential with h c >> h 1 , where h 1 is the value of h at which V = 0 for a given value of λ hφ / λ φ . We next set h c , the value of h at which φ(h) = 0, to a value close to h 1 . The effect is to change the value of the quartic coupling from λ h (µ) − λ 2 hφ /4λ φ at h < h c to λ h (µ) at h > h c (where µ = h). The jump in the quartic coupling causes the potential, which had been approaching zero at h < h c to increase at h > h c , hence generating a minimum. In practice we have to adjust h c to a value which is a little below h 1 in order to obtain a degenerate minimum with V = 0 once the RG running of λ h (µ = h) is included. The value of h c then determines the value of λ 1/4 φ f a via Eq. (7).
In Table 1  max for the case of degenerate minima as a function of λ hφ / λ φ . Here h min is the value of h at the degenerate minimum, h max is the value of h at the maximum of the potential between the electroweak vacuum and the second metastable vacuum along the minimum trajectory, and V 1/4 max gives the value of potential at the maximum. We find that there is a finite range λ 1/4 φ f a for which a degenerate minimum is possible, between 0.10 and 0.297. The upper bound corresponds to values of h at the degenerate minimum which are greater than the Planck scale, which we assume is a natural cutoff for the effective theory. The lower bound corresponds to values of λ 2 hφ /λ φ which are too small to sufficiently modify the potential to produce a second metastable minimum. The lower bound on λ hφ / λ φ corresponds to a lower bound on λ 1/4 φ f a equal to 2.39 × 10 10 GeV. This lower bound is an important prediction of metastable dark energy in the KSVZ axion model.
In Table 2 we give a more detailed set of parameters for the range of λ 1/4 φ f a which is relevant to axion dark matter in the case where PQ symmetry is broken after inflation (discussed in the next section).
In Figure 2 we show the potential along the minimum trajectory with a degenerate minimum at the lower limit of the λ 1/4 φ f a range, corresponding to λ hφ / λ φ = 0.1. In Figure 3 we show a degenerate minimum with h min close to the Planck scale, corresponding to λ hφ / λ φ = 0.297. In Figure 4 we show the potential for a value of λ 1/4 φ f a which is typical of the case where dark matter is entirely due to axions in the case of PQ symmetry breaking after inflation. In Figure 5 we show that the second minimum is metastable, with an unbounded negative potential at large h.
In Figure 6 we show that it is also possible to obtain inflection point behaviour of the potential, by reducing λ 1/4 φ f a for a given λ hφ / λ φ below the case of a degenerate potential. This may have applications to inflection-point inflation in the context of the KSVZ axion model, although we will not explore this possibility further here.

IV. AXION COSMOLOGY WITH A QUASI-DEGENERATE POTENTIAL
If the PQ symmetry is broken during inflation such that the universe is in the electroweak vacuum (h, φ) = (v, f a ) when observable scales exit the horizon (N 60), and if axions account for dark matter, then there is a strong constraint from axion isocurvature perturbations. The isocurvature contribution to the total curvature plus entropy power spectrum, α, due to axions is FIG. 2. The scalar potential with quasi-degenerate minima along the minimum trajectory when λ hφ / λ φ is at the lower bound for quasidegenerate minima to exist, λ hφ / λ φ = 0.1.
where M Pl = (8πG) −1/2 = 2.4 × 10 18 GeV, θ a is the initial displacement of the axion field angle (θ a = a/ f a ) from the minimum of its potential, and Ω a and Ω dm are the density in axions and the observed dark matter density respectively. The present bound on isocurvature perturbations is α < α lim = 0.03 [26]. Using r = 16ε for the tensor-to-scalar ratio r, we obtain The axion dark matter density due to the misalignment mechanism is given by Eq.(2.8) of [27] Ω mis a = 3.8 × 10 −4 θ 2 a f a 10 12 GeV 1.165 .
This assumes that the axion mass has the form given in Eq.(22) of [28] with Λ QCD = 400 MeV and n = 8.16 [29,30]. Combining Eq. (28) and Eq. (29) gives an upper bound on r from isocurvature perturbations, Inflation models commonly have H in f larger than 10 10 GeV (i.e. r larger than 10 −8 ). In such models, in the case where there is a significant amount of axion dark matter (in particular, where axions account for all of the dark matter) and f a M Pl , the PQ symmetry must be restored and subsequently broken after inflation in order to satisfy the isocurvature constraint. In addition, and independently of the isocurvature constraint, if the fields are at the quasi-degenerate minimum during inflation, then the electroweak and PQ symmetry must be restored after inflation to allow the fields to subsequently evolve to the electroweak vacuum. Therefore, if the fields are at either of the vacuum states during inflation (and if axions account for a significant amount of dark matter in the case of the electroweak vacuum), the PQ symmetry must be restored and subsequently broken after inflation.
In the case of PQ symmetry breaking after inflation 3 , the axion dark matter density is due to vacuum realignment and the decay of the cosmic strings and domain walls which form after PQ symmetry breaking and the QCD transition. The resulting dark matter density is related to f a by Eq.
Since λ 1/4 φ f a ≥ 2.39 × 10 10 GeV is necessary in order for a degenerate minimum to exist, we find a lower bound on the amount of axion dark matter in the case where PQ symmetry is broken after inflation where Ω dm h 2 = 0.12 is the observed dark matter density. Thus if λ φ 1 then at least 30% of the dark matter must be due to axions 4 . This lower bound is a direct consequence of the assumption that dark energy is due to the energy of the electroweak vacuum relative to a second metastable minimum at zero energy 5 .
If all of the dark matter is due to axions, then the value of f a is fixed by Eq. (33) to be in the range 4.7 × 10 10 GeV to 7.2 × 10 10 GeV. The potential is determined by the value of λ hφ / λ φ , which is fixed (via the degenerate minimum constraint) by the value of λ 1/4 φ f a via Table 1. Therefore, up to a weak dependence on λ φ (and uncertainties in the axion dark matter density from cosmic string and domain wall decay), the form of the potential is fixed. This will allow the cosmology of the model to be quantitatively studied. For λ φ in the dimensionally natural range 0.01-1, the uncertainty in λ 1/4 φ f a due to the unknown value of λ φ corresponds to a factor between 0.32 and 1.

V. CONCLUSIONS AND DISCUSSION
We have shown that dark energy in the KSVZ axion model can be understood as being due the energy density of a metastable electroweak vacuum relative to a second metastable minimum of the potential which is at zero energy density. The resulting model is therefore able to account for both the dark matter and dark energy densities, as well as providing a solution to the strong CP problem. The requirement of a second almost degenerate minimum of the scalar potential imposes a non-trivial constraint on the parameters of the model. In particular, it implies a lower bound on the axion decay constant for a given PQ scalar selfcoupling via a lower bound on λ 1/4 φ f a . In the case where PQ symmetry is broken after inflation, this implies a lower bound on the amount of axion dark matter of about 30% of the total dark matter density if λ φ 1. In addition, if axions constitute all of the observed dark matter, then the value of f a is fixed (up to uncertainties in the production of axions from cosmic string and domain φ . This will allow the cosmology of the KSVZ model with axion dark matter and metastable dark energy to be quantitatively studied.
The energy density of the lowest energy metastable minimum being equal to zero can arise, for example, in theoretical approaches which explain the cancellation of the vacuum energy density via spacetime averaging, such as vacuum sequestering, and in models with an energy parity.
Both of these approaches (more generally, any approach that effectively subtracts off the total energy of one of the vacuum states) will ensure that there is no cosmological constant term, as any constant term in the potential is cancelled. Importantly, this ensures that the large quantum contribution to the vacuum energy density is also cancelled. Dark energy must then be explained either by adding a new quintessence component to the energy density, or by the vacuum energy density due to the difference in energy between a pair of vacuum states. We have shown that the KSVZ axion model can provide the necessary pair of vacuum states. The KSVZ axion model with quasi-degenerate minima can therefore serve as a minimal model for dark energy, in the sense that it does not require any additional component to account for dark energy. In this context, the question of the smallness of the observed dark energy density becomes the question of the underlying physics responsible for the near-degeneracy of the vacuum states.
In this first study we have not included the quantum corrections to the scalar potential due to the PQ scalar couplings λ hφ and λ φ . This will be a good approximation if the classical modification of the potential due to the φ field and its mixing with the electroweak doublet Higgs field are the dominant effects on electroweak stability. A complete analysis of the effective potential requires a multiscale RG analysis of the two-field scalar potential in order to keep the CW correction to the potential small for all values of h and φ. We will return to the full analysis of the KSVZ effective potential via multiscale RG methods in future work.
It has recently been suggested that, in generalizations of the SM Higgs potential which have a second minimum due to the scalar threshold effect, primordial black holes can form via quantum fluctuations [33]. The KSVZ axion model with a degenerate potential, PQ symmetry breaking after inflation and axion dark matter, would therefore provide a well-defined potential of the required form. We will consider this in a future analysis.
More generally, it would be interesting to explore the possible cosmological evolution of the model in different inflation scenarios. In particular, the non-minimally coupled PQ scalar inflation model of [34] would be interesting in this context. Combined with the quasi-degenerate vacuum explanation of the dark energy density, this would provide a minimal model for FIG. 6. An example of an inflection point in the potential at λ hφ / λ φ = 0.1. Although we do not consider inflection points in this article, it is interesting to note that they can easily be found by reducing the value λ 1/4 f a from the case of the degenerate potential. dark matter, dark energy and inflation, as well as a solution to the strong CP problem 6 .