Can dark matter drive electroweak symmetry breaking?

We consider the possibility of an oscillating scalar field accounting for dark matter and dynamically controlling the spontaneous breaking of the electroweak symmetry through a Higgs-portal coupling. This requires a late decay of the inflaton field, such that thermal effects do not restore the electroweak symmetry after reheating, and so inflation is followed by an inflaton matter-dominated epoch. During inflation, the dark scalar field acquires a large expectation value due to a negative non-minimal coupling to curvature, thus stabilizing the Higgs field by holding it at the origin. After inflation, the dark scalar oscillates in a quartic potential, behaving as dark radiation, and only when its amplitude drops below a critical value does the Higgs field acquire a non-zero vacuum expectation value. The dark scalar then becomes massive and starts behaving as cold dark matter until the present day. We further show that consistent scenarios require dark scalar masses in the few GeV range, which may be probed with future collider experiments.

In addition, the introduction of a dark scalar singlet may solve the Higgs vacuum stability problem. The Higgs vacuum is stable if its self-coupling, λ h , is positive for any scale of energy µ where the minimum of its potential is a global minimum. However, for the measured Higgs mass m h 125 GeV, λ h becomes negative for energy scales around µ ∼ 10 10 − 10 12 GeV [26,27], which are well below the GUT or the Planck scales. This could constitute a problem since it may lead to a possible instability in the Higgs potential (see, for e.g., Refs. [26,28,29] and references therein). The behaviour of λ h is mostly driven by the large contribution of the top Yukawa coupling at one-loop, i.e., strongly depends on the top quark mass. When the coupling constant becomes negative, the renormalization group-improved Higgs potential is V (h) = λ h h 4 4 < 0 and, therefore, the Higgs minimum could be only a local minimum, instead of a global minimum. However, if the time scale for quantum tunneling to this true minimum exceeds the age of the Universe, the Higgs vacuum is only metastable. In fact, Ref. [29] showed that the lifetime for quantum tunneling is extremely long: about the fourth power of the age of the Universe.
There have been several attempts to cure the (in)stability problem of the electroweak vacuum. For instance, Ref. [26] showed that a shift in the top quark mass of about δm t = −2 GeV would suffice to keep λ h > 0 at the Planck scale (this could also be a good reason to motivate more precise measurements of the top quark mass). Other ways include introducing physics beyond the Standard Model. In particular, coupling a scalar singlet with non-zero expectation value to the Higgs may stabilize the electroweak vacuum, provided that the contribution of the coupling between the Higgs and the singlet scalar maintains the Higgs self-coupling positive. This idea has been explored in the literature, and some of them promote this singlet scalar to a dark matter candidate, such as illustrated in Refs. [8,30,31]. In addition, one must consider the stability of the Higgs field during inflation, since de Sitter quantum fluctuations could drive the field to the true global minimum of the potential. This may potentially be avoided if the Higgs field is sufficiently heavy during inflation, which may be achieved by coupling it to other fields such as the inflaton itself [32] or a dark matter scalar as we propose in this work.
We consider a self-interacting dark scalar field, Φ, coupled to the Higgs field, H, through a standard biquadratic "Higgs-portal" coupling, and non-minimally coupled to gravity: where the Higgs potential V (H) takes the standard "mexican hat" shape. As in previous works [19,20], we assume an underlying scale invariance of the theory, spontaneously broken by some mechanism that generates the Planck and electroweak mass scales in the Lagrangian, but which forbids a bare mass term for the dark scalar. It is thus easy to see that, for a sufficiently large value of Φ, the minimum of the Higgs potential will lie at the origin, and it is natural to enquire whether the dark scalar can dynamically drive the spontaneous breaking of the electroweak symmetry 1 .
To prevent thermal effects from restoring the electroweak symmetry after inflation, we focus on scenarios with a late inflaton decay, such that the reheating temperature, T R , is below ∼ 100 GeV. Consequently, inflation is followed by a long matter-dominated epoch while the inflaton oscillates about the origin in an approximately quadratic potential. As we will see in more detail below, the negative sign of the non-minimal coupling to gravity leads to a large expectation value for the dark scalar during inflation, which makes the Higgs field heavy and stabilizes it at the origin during this period. After inflation the dark scalar starts oscillating about the origin in its quartic potential, and its amplitude decreases with expansion, such that at some point it falls below a critical value that allows the Higgs to develop a non-zero vacuum expectation value. The spontaneous breaking of the electroweak symmetry is thus dynamically controlled by the dark scalar, and once it occurs the latter gains a mass and starts behaving as cold (pressureless) dark matter.
This work is organized as follows. In the next section we discuss the dynamics of the dark scalar and the Higgs field during inflation. In section 3 we describe the postinflationary dynamics of both fields, discussing the possibilities of reheating occurring before or after the electroweak symmetry is spontaneously broken. We discuss the consistency of our analysis and parametric constraints in section 4 and present our results for the allowed values of the dark scalar mass and couplings in section 5. We summarize our discussion and main conclusions in section 6.

Inflation
During inflation, the relevant interaction Lagrangian for the dynamics of the Higgs and dark scalar field, assuming they have no significant interactions with the inflaton field, is given by: 2 and the Ricci scalar can be written in terms of the Hubble parameter, where H inf can be related to the tensor-to-scalar ratio of primordial curvature perturbations as: Since the interaction term between φ and R has a negative sign, the dark scalar acquires a vacuum expectation value (vev) during inflation, φ inf , with the minimum of the potential lying at: The dark scalar then provides a large mass to the Higgs field during inflation: We will see later that g/ λ φ ∼ 10 2 if the dark scalar accounts for all dark matter, such that m h H inf for ξ 10 −5 . This large Higgs mass has two related effects. First, it induces an additional quadratic term in the Higgs potential, thus shifting the field value at which the potential becomes unbounded (i.e. λ h < 0) towards values larger than H inf , i.e. above the 10 10 − 10 12 GeV scale at which it becomes unbounded in the Standard Model [27]. Second, it suppresses the Higgs de Sitter quantum fluctuations, which for a light Higgs (m h H inf ) would be ∼ H inf /2π ∼ 10 12 GeV unless the tensor-to-scalar ratio is very suppressed. For a massive Higgs field, the field variance during inflation on super-horizon scales is given by [34]: which, using Eq. (2.5), simplifies to corresponding to an average fluctuation amplitude h 2 10 11 GeV for r 10 −2 and ξ 0.1. Thus, the coupling between the Higgs and the dark scalar can prevent the former from falling into the putative large field true minimum during inflation.
We note that the dark scalar is also heavy during inflation, such that its de Sitter quantum fluctuations, with an amplitude δφ 2 0.05 ξ −1/4 H inf [19,20], have a negligible effect on its expectation value φ inf H inf , the latter setting the initial amplitude for field oscillations in the post-inflationary epoch.

Post-inflationary period
In this model we assume that, after inflation, the inflaton field, χ, does not decay immediately. Instead, the inflaton evolves as non-relativistic matter, while oscillating about the minimum of its potential, and an early matter-era follows inflation until reheating finally occurs. Therefore, there are some significant changes in the dynamics of the Universe with respect to the usual radiation-dominated epoch. The scale factor evolves in time as a ∼ t 2/3 and the Ricci scalar has a non-vanishing value, R = 3 H 2 , unlike its value during the radiation era (R = 0). The evolution of the inflaton energy density is thus given by: where the subscript "end" corresponds to the end of inflation. Note that H end depends on the particular inflationary model. Let us consider, for instance, the case where inflation is driven by a field with a quadratic potential, V (χ) = 1 2 m 2 χ χ 2 , where m χ is the inflaton's mass. The number of e-folds of inflation, after the observable CMB scales become superhorizon, is given by: where χ * is the value of the inflaton field when observable CMB scales become superhorizon during inflation, with χ * χ end . Inflation ends when = M 2 P (V /V ) 2 /2 ∼ 1, yielding χ end √ 2M P , from which we deduce that: Although the quadratic potential is already in some tension with Planck bounds on the tensor-to-scalar ratio [35], we will consider the above relation with N e = 60 henceforth in our discussion, bearing in mind that a different relation between H end and H inf may lead to somewhat different results. Note that this model dependence is nevertheless degenerate with the unknown value of the tensor-to-scalar ratio, which we take as a free parameter. At some stage, the inflaton decay reheats the Universe, establishing the beginning of the radiation-dominated epoch. This scenario resembles the so-called Polonyi problem found in many supergravity models, where the Polonyi field or other moduli decay at late times (see, for e.g., Refs. [36][37][38]). We assume that the inflaton transfers all its energy density into Standard Model degrees of freedom at a reheating temperature T R : where g * R is the number of relativistic degrees of freedom at reheating. The reheating temperature must be above ∼ 10 MeV, as the Universe must be radiation-dominated during Big Bang nucleosynthesis (BBN). As mentioned earlier, we will consider the case where reheating does not restore the electroweak symmetry, such that electroweak symmetry breaking is controlled by the dynamics of the dark matter scalar field, i.e, T R m W 80 GeV. It is important to note that before reheating there is no notion of temperature, since the inflaton has not yet decayed. Using Eqs. (3.1) and (3.4), the number of e-folds from inflation until reheating, N R , reads: where we used N e = 60. The interesting feature of this model is that the dark scalar will control a non-thermal EWSB. From Eq. (2.1), it is easy to see that the minimum of the Higgs potential occurs at EWSB then takes place when the amplitude of the field becomes smaller then the critical value: noting that, in a few e-folds, the Higgs field should attain its final vacuum expectation value |h| = v. In the following subsections, we will study the dynamics of the dark scalar when reheating occurs after or before EWSB. Note, however, that N R is determined solely by r and T R , being independent of when EWSB takes place. Hence, our model has five free parameters: r, ξ, g, λ φ and T R .

Reheating after EWSB
The first scenario we study is the one where reheating occurs after EWSB, as illustrated in figure 1. Before EWSB, the quartic term dominates the energy density of the dark scalar and it starts oscillating about the origin with initial amplitude φ inf . The amplitude decays as φ ∝ a −1 , such that the field behaves as dark radiation, ρ φ ∝ a −4 . Note that R ∝ H 2 ∝ a −3 , so that the effects of the non-minimal coupling to gravity decay faster than those of the quartic self-interactions and may thus be neglected. We will assume, for simplicity, that once the electroweak symmetry is spontaneously broken and the field becomes massive the associated quadratic term in the scalar potential becomes dominant, such that the field behaves as cold dark matter (CDM) from EWSB onwards. Therefore, the dark scalar exhibits two behaviors: where a c is the value of the scale factor at which EWSB takes place. At EWSB, we have: and, therefore, the number of e-folds from inflation until EWSB, N EW , is given by (3.10) Once reheating occurs, the Universe enters the usual radiation-dominated epoch. Thus, we can now consider a temperature and the number of dark matter particles in a comoving volume, n φ /s, becomes constant. The dark scalar amplitude at reheating is, then: Introducing the last equation into Eq. (3.11), the amplitude of the field at reheating becomes: (3.13) The number of particles in a comoving volume at T R is, then: where m φ stands for the dark scalar mass once the electroweak symmetry is spontaneously broken and is given by: The present dark matter abundance then reads: where g * 0 , T 0 and H 0 are the present values of the number of relativistic degrees of freedom, CMB temperature and Hubble parameter, respectively. Replacing Eq. (3.13) into the last expression and fixing Ω φ,0 = 0.26, we then obtain a relation between g and λ φ : (3.17)

Reheating before EWSB
The second putative scenario we should consider is the case where reheating occurs before EWSB, as illustrated in figure 2. Since the number of e-folds from inflation until reheating does not depend on when EWSB takes place, N R is given by Eq. (3.5) as in the previously discussed scenario. Similarly, N EW only depends on φ inf and φ c and, therefore, it is given by Eq. (3.10). The difference between this and the previous scenario is that N EW should now exceed N R . The dark scalar behaves like dark radiation from reheating until EWSB, after which n φ /s becomes constant. From reheating onwards, the Universe enters the usual radiationdominated epoch and R = 0.
The amplitude of the field at reheating is different from the previous scenario: and now we have a defined temperature and can write the amplitude of the field as a function of the temperature: This can be used to compute the temperature at which EWSB occurs, T c : At T c the dark scalar stops holding the Higgs at the origin. Notice, however, that T c must be smaller than the usual T EW ∼ 80 GeV, so that the dark scalar can control the EWSB and the latter is not restored by thermal effects. By proceeding as in the previous subsection, since n φ /s is constant as soon the field starts behaving as CDM, the present dark matter abundance is given by: Setting Ω φ,0 = 0.26 we then obtain for the temperature at which the field amplitude falls below the critical value: GeV . Hence, we conclude that, for reheating to occur before EWSB, T c must be well above T EW ∼ 80 GeV. This is not consistent with our reasoning given that, at that temperature, the Higgs thermal mass is still sufficiently large to hold the latter at the origin, such that EWSB does not occur at T c as assumed and, consequently, the dark scalar remains massless and behaves as dark radiation, as opposed to our starting assumption. In the remainder of this paper, we will thus focus only on the case where reheating occurs after EWSB, given that in this scenario the dark scalar, in addition to being a viable dark matter candidate, can also control a non-thermal EWSB.

Consistency analysis
In analyzing the dynamics of the dark scalar and of the Higgs field both during and after inflation we have made several technical assumptions. In this section, we discuss the parametric constraints imposed by these assumptions and also by the properties of the Higgs boson measured at the Large Hadron Collider (LHC). First, our scenario assumes that inflation is driven by a scalar field, χ, that is neither the dark scalar nor the Higgs field. Therefore, we have to ensure, in particular, that the dark scalar does not affect the dynamics of inflation. The dark scalar's contribution to the effective potential during inflation is given by: Requiring that this does not significantly reduce the inflationary energy density V (χ) 3H 2 inf M 2 P then implies the condition: which constrains the allowed values of the non-minimal coupling ξ and the self-coupling λ φ , depending on the tensor-to-scalar ratio, i.e. the scale of inflation: Second, we have assumed that the dark scalar field starts behaving as CDM as soon as the electroweak symmetry is spontaneously broken, i.e. when the field amplitude falls below the critical value φ c . This means that the quadratic term has to dominate over the quartic term at EWSB, that is, g 2 v 2 φ 2 c / λ φ φ 4 c > 1, which translates into the following condition: Finally, radiative corrections to the quartic coupling from the Higgs-portal coupling should be small, unless we accept some degree of fine-tuning: (4.5) From the experimental point of view, the Higgs may decay into dark scalar pairs with a decay width Γ h→φφ 1 8π leading to a Higgs branching ratio into invisible particles, assuming Γ h→inv = Γ h→φφ : Current limits from the LHC establish an upper bound for the branching ratio Br inv < 0.23 [39], and using Γ h = 4.07 × 10 −3 GeV [40], this yields an upper bound on the Higgs-portal coupling: g < 0.13 , (4.8) which translates into an upper bound m φ 22.6 GeV. From the dynamical perspective, we have also implicitly assumed that the dark scalar field remains in the form of an oscillating condensate, such that processes that may lead to its evaporation and subsequent thermalization (which would yield a WIMP-like dark matter candidate) must be inefficient, as we discuss in detail below.

Condensate evaporation
The dark scalar provides mass to the Higgs field during the period before EWSB. Since φ is oscillating, this could induce oscillations of the Higgs mass. This may pose a problem, since if the Higgs mass m h < 3 λ φ φ rad , Higgs production by the oscillating condensate is kinematically allowed and lead to the condensate's evaporation.
A solution to this problem is to provide initial conditions to the field such that its absolute value, and hence the Higgs mass, does not oscillate. This is possible if the dark scalar oscillates in the complex plane, as e.g. in the Affleck-Dine mechanism for baryogenesis [41,42], with no oscillations in the special cases where φ = A e ±iωt .
To generate the required angular momentum in field space, we need to introduce terms in the scalar potential that depend on the phase of the field and not only on its modulus, i.e. which violate the global U(1) symmetry of the original Lagrangian, Eq. (1.1). Since gravity is expected to violate such global symmetries, it is natural to envisage such terms in the gravitational sector of the model, in particular the non-minimal coupling to curvature as well as Planck-suppressed non-renormalizable operators: where n is a positive even integer, such that the Lagrangian is still invariant under a discrete Z 2 symmetry that ensures the stability of the dark scalar, and c is an O(1) dimensionless coupling. We note that both the non-minimal coupling to curvature and the non-renormalizable term decay faster than the quartic self-interaction term, such that they play a negligible role in the late time dynamics of the field. However, since the value of the Ricci scalar during inflation differs from its value at the end of inflation, the phase of φ at the minimum is different during and after inflation, thus making the dark scalar oscillate in the complex plane. This prevents |Φ| from oscillating significantly, such that the Higgs never becomes light enough to be produced, while ρ φ still redshifts as dark radiation before EWSB. Another way of solving the problem is to couple the Higgs field directly to the inflaton, χ, with an interaction term of the form [32] 1 2 g 2 χ χ 2 |H| 2 , (4.10) where g χ is the Higgs coupling to the inflaton, which yields an additional contribution to the Higgs mass, ∆m h = g χ χ/ √ 2 that is present until reheating occurs (after EWSB). Since φ ∝ a −1 and χ ∝ a −3/2 before EWSB, this contribution decays faster than the dark scalar's contribution to the Higgs mass, which we assume to be dominant. Nevertheless, this contribution may be sufficient to kinematically block the production of Higgs particles by the oscillating dark scalar, provided that it exceeds the latter's oscillation frequency before EWSB: which imposes a lower bound on g χ . The inflaton's amplitude at EWSB, χ c , is where χ end is the inflaton's amplitude at the end of inflation. Introducing the last expression into Eq. (4.11) and using the relation between g and λ φ , we obtain a lower bound on g χ : Notice that smaller values of r, corresponding to lower inflationary scales, allow for smaller couplings g χ . Also, note that both the inflaton and the dark scalar's contribution to the Higgs mass are oscillatory in nature. However, since they will not, in general, be in phase, the Higgs mass should not oscillate significantly, thus preventing its production.
Since reheating can only consistently occur after EWSB as we have shown above, and hence there is no cosmic thermal bath yet at this stage, the only other possible channel for the evaporation of the dark scalar field is the perturbative production of φ-particles by the oscillating background condensate. The dark scalar behaves like radiation until EWSB and the condensate decay width is given by [19,20]: 14) where, at EWSB, φ rad = φ c . Condensate evaporation is then avoided if this never exceeds the Hubble expansion rate until EWSB, noting that after EWSB this production channel is blocked since the dark scalar becomes massive (see e.g. [19,20]): Since the Universe is still in a matter-dominated era at EWSB, the Hubble parameter can be computed using the expression for the inflaton's energy density given in Eq. (3.1): Therefore, from Eq. (4.15) we find that g < 6 × 10 −13 r 0.01 (4.17) and using the relation between g and λ φ (Eq. (3.17)), the upper bound on g reads (for N e = 60): (4.18)

Results
In this section we summarize our results, taking into account all the different constraints analyzed earlier. We present the results for the regions in the (λ φ , g) plane where all model constraints are satisfied, namely Eqs. (4.2) -(4.5) and (4.17). We choose to represent the results for values of the tensor-to-scalar ratio r = 10 −2 and non-minimal coupling ξ = 0.1, 1, as illustrated in figure 3. In figure 3, we can see that there is a window where our model can explain all of the present dark matter abundance, for dark scalar masses larger than the ones we have obtained in previous Higgs-portal scenarios with an oscillating scalar field and an underlying scale invariance [18][19][20]. For instance, we can see that g ∼ 10 −2 is allowed, which corresponds to m φ ∼ 1 GeV. We may conclude that an early matter-era precludes sub-GeV dark scalar masses, mainly since these would lead to super-Planckian dark scalar values during inflation that could affect the latter'ss dynamics.
In fact, it is possible to get an analytic expression for the window of possible values for g and λ φ . Hence, since the dark scalar cannot affect inflation (Eq. (4.2)) and using the relation between the Higgs-portal coupling and the dark scalar's quartic coupling (Eq. (3.17)), the constraint on g becomes g > 10 3 T R 10 GeV In turn, requiring that the field behaves like CDM at EWSB (Eq. (4.4)), and using the relation between couplings (Eq. (3.17)), we find a lower bound on g: and, consequently, a lower bound on the mass The no fine-tuning constraint allows only Higgs portal couplings below the following threshold: g < 16 π 2 1/2 4 × 10 2 T R 10 GeV Taking into account all these restrictions, along with the bound coming from avoiding condensate evaporation, Eq. (4.18) and the LHC bound on the Higgs invisible partial decay width, Eq. (4.8), we may alternatively plot the allowed parametric regions in the (m φ , T R ) plane for different values of the non-minimal coupling to gravity and tensor-to-scalar ratio, as illustrated in figure 4.  . Allowed values for the dar scalar mass as a function of the reheating tempeature, for 10 MeV < T R < 80 GeV and considering different values for the non-minimal coupling to curvature ξ and tensor-to-scalar ratio r.
From figure 4 we may conclude that our model predicts masses for the dark scalar in the few GeV range, depending on the values of the tensor-to-scalar ratio and non-minimal coupling chosen. These may be within the reach of the LHC or its successors in the near future, since for instance Br inv 2 × 10 −3 for m φ 6 GeV, which is not too far from the current experimental limit (Eq. (4.7).) Notice, however, that large values of the nonminimal coupling to gravity, permitting heavier dark scalars, are only allowed for lower values of r, i.e. in scenarios with a low inflationary scale.

Conclusions
In this work, we have analyzed the possibility of an oscillating scalar field, accounting for all the dark matter in the Universe, driving a non-thermal spontaneous breaking of the electroweak symmetry. The dark scalar is coupled to the Higgs field through a standard "Higgs-portal" biquadratic term, has no bare mass terms due to an underlying scale invariance of the theory, and has a negative non-minimal coupling to curvature. The latter, in particular, allows the dark scalar to develop a large expectation value during inflation. This holds the Higgs field at the origin both during and after inflation, until the dark scalar's oscillation amplitude drops below a critical value at which EWSB takes place. This prevents, in particular, the Higgs field from falling into the putative global minimum at large field values during inflation, ensuring at least the metastability of the electroweak vacuum.
The proposed scenario assumes a late decay of the inflaton field, such that reheating does not restore the electroweak symmetry, while the reheating temperature is still large enough to ensure a successful primordial nucleosynthesis 2 . Therefore, the Universe is still dominated by the inflaton field for a parametrically long period after inflation, while it oscillates about the minimum of its potential and behaves as a pressureless fluid. In fact, we have shown that consistent scenarios require reheating to occur only after EWSB, such that the latter occurs in the inflaton matter-dominated epoch essentially in vacuum.
Compared to other scenarios of scalar field dark matter where the Higgs is the only source of mass for the dark scalar field [18][19][20], we have shown that this allows for larger Higgs-portal couplings and hence dark scalar masses, since there are no thermalized particles in the Universe that could lead to an efficient evaporation of the scalar condensate until EWSB takes place. The dark scalar's oscillations, while it behaves as dark radiation, could themselves lead to particle production, but this can either be kinematically blocked in the case of Higgs production or made less efficient by the faster expansion of the Universe in a matter-dominated regime, as compared to the standard radiation epoch.
Overall, we have concluded that consistent scenarios where the dark scalar (1) does not affect the inflationary dynamics, (2) has technically natural values for its self-coupling (i.e. requiring no fine tuning), and (3) starts behaving as cold dark matter after it breaks the electroweak symmetry, require dark scalar masses in the few GeV range, unless inflation occurs much below the grand unification scale. This looks promising from the experimental perspective, since it allows for Higgs invisible branching ratios 10 −3 , which may be within the reach of colliders in a hopefully not too distant future.
We thus reply "Yes, it can" to the fundamental question posed in this work and hope that testing this idea may shed a new light on the nature of dark matter and on its role in the cosmic history.