A possible interpretation of the Higgs mass by the cosmological attractive relaxion^*

Recently, a novel type of resolution for the electroweak (EW) hierarchy problem²⁾ has been proposed (relaxion mechanism) in Ref. [1], which is very different from the traditional approaches (either weak scale dynamics [2–6] or anthropics). In this relaxion mechanism, the relevant fields below the cutoff scale are just the standard model (SM) fields plus the axion field with an unspecified inflation sector being involved, and the Higgs mass is dependent on the axion field [1], which is motivated from Abbott's field-dependent idea to solve the cosmological constant problem [7]. Accordingly, the cosmological evolution of the Higgs mass and the specific axion potential choose the EW scale, which is smaller than the cutoff of the theory. The highest cutoff relaxed in Ref. [1] is about 10⁸ GeV. This relaxion mechanism can technically relax the EW hierarchy problem and has become a theoretical highlight in frontier studies of the hierarchy problem and exploring new physics beyond the SM [8–24]. Especially, this new mechanism opens a new window to understand some puzzles and key parameters in particle physics from the aspect of cosmological evolution.

Following this cosmological evolution idea, our toy model here takes advantage of the attractive properties of the "α-attractors" [25–37] to fix the Higgs mass at today's value rather than the increasing potential barriers of the axion potential in Ref. [1]. In recent years, Linde and Kallosh have proposed a broad class of supergravity inflationary models based on conformal symmetry in the Jordan frame, where a universal attractor behavior exists in the Einstein frame [25–37]. These classes of supergravity inflationary models are called "α-attractors", since their potentials involve a free parameter α [30–37]. These inflation models of "α-attractors" mainly have two classes: one is the so-called T-models with the potential and the other is the so-called E-models with the potential in the Einstein frame. In the limits of small α and large e-folding number N_e, these models have the same predictions [30–37] corresponding to the central area of the n_s − r plane favored by the data of Planck 2015 [38].

Our toy model here only tries to provide a possible cosmological interpretation of the light Higgs mass, and only the attractive inflation field is needed motivated from the above relaxion mechanism and the attractive inflation. Here, we do not consider the UV-completed theory for a fully natural theory. In addition, our models may be tested in particle physics experiments.

In Section 2, we first describe the cosmological scenario to explain the light mass of the Higgs boson by attractive inflation. In Section 3, we study the possible signals or constraints in future lepton colliders and the constraints from the muon anomalous magnetic moment. Then, in Section 4, we perform a detailed analysis of cosmological perturbations seeded by the inflation fields in a concrete model. Section 5 gives a brief summary.

In this section, we show a possible cosmological scenario to explain the light mass of the Higgs boson by attractive inflation. In our toy model, the field contents below the cutoff scale are just the SM fields and the inflaton field. The relevant potential can be written as

where the field h and ϕ represent the Higgs field and the inflaton field, respectively. The second term in Eq.(1) generally can be the form of (f(ϕ) − M²)h²/2. In this scenario, the Higgs mass in the early universe is field dependent, namely, . This is just the starting point of our discussion. We assume that the initial value of the inflation field ϕ starts at ϕ ≫ M/g, where M represents the cutoff scale in this toy model. Thus, the mass of the Higgs boson in the early universe is naturally set to be the order of the cutoff scale M. Here, V_att means the attractive potential, which can drive the cosmological inflation and fix the current Higgs mass. Interestingly, the potentials in a broad class of supergravity inflation models (the so-called "α-attractors") [30–37] can just satisfy the requirements. One class of potentials is given by [30–37]

where M_pl is the reduced Planck mass. This class of potential is just the potential in the T-models of "α-attractors". The potential V_T can be obtained from a canonical Kahler potential such as in the supergravity model. The non-minimal coupling case between the inflaton field and the gravitational field can also lead to the same predictions when compared to this type of attractive potential in some limits. The other type of attractive potential can be written as [30–37]

This potential can be motivated by considering a vector rather than a chiral multiplet for the inflation models in supergravity, and is just the potential in the so-called E-model. Here, the power exponents T and E in Eqs. (2) and (3) are integers and α is the free parameters. ϕ_c is a constant, which is related to the current Higgs boson mass.

We take the potential V_T as an example to illuminate the cosmological origin of the light Higgs boson mass. Firstly, under the slow-roll approximation, the spectral index n_s, the tensor-to-scalar ratio r, and the e-folding number N_e as functions of ϕ can be obtained:

For the E-models, the corresponding results are

These types of potential in the small α limit are favored by the data from Planck 2015 [38].

Firstly, in the very early universe ϕ ≫ M/g, the effective mass-squared of the Higgs boson is positive and the vacuum expectation value (vev) of the Higgs field is zero. The final solution is insensitive to the initial condition of ϕ as long as the initial mass-squared of the Higgs is positive, since it is slow-rolling due to Hubble friction. The inflaton field ϕ drives the slow-roll inflation by the attractive potential or as shown in Fig. 1. With the evolution of the universe, the inflaton field naturally crosses the critical point for the Higgs mass where , namely, a transition point occurs when ϕ = M/g. When the inflaton field across this critical point with ϕ < M/g, the mass-squared term transits from a positive value to a negative value, and then the Higgs field acquires a vev. Thereby, the cosmological inflation can scan the physical mass of the Higgs boson, as shown in Fig. 1.

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After reheating and dissipation¹⁾, the inflaton field ϕ is stuck in the vicinity of ϕ_c. Thus, the Higgs mass evolves from a large field-dependent mass to the currrent 125 GeV mass. In the early universe, the field dependent mass of the Higgs boson is , where ϕ(t) ≫ ϕ_e. During the cosmological evolution, the mass of the Higgs boson becomes much smaller. When the inflaton is stuck by the attractive potential of ϕ as shown in Fig. 1, the Higgs mass is fixed as . Here, μ² is the coefficient of the h² in the SM with . This provides a cosmological interpretation of the light Higgs mass, and . For M = 10⁶ GeV, g = 10⁻², ϕ_c ≈ 10⁸ GeV. We call the inflaton field ϕ the attractive relaxion, which has the above attractive inflation behavior.

In this section, we discuss the possible collider signals or constraints from particle physics experiments for two simple cases of Higgs portal interactions f(ϕ)h²/2.

3.1. Higgs invisible decay

For the case of f(ϕ) = g² ϕ² in Eq. (1), this toy model will contribute to the Higgs boson invisible decay. This scenario can be tested at a future lepton collider, such as the circular electro-positron collider (CEPC), by precisely measuring the width of the Higgs invisible decay. Here, the Higgs invisible decay channel is induced from the Higgs portal term g²h²ϕ²/2 in Eq. (1). We obtain the following interaction term

which leads to the Higgs invisible decay, and its decay width is

Figure 2 shows the relation between the Higgs portal coupling g and its decay width Γ_inv(h). The current Higgs portal coupling from LHC data is constrained as g < 0.088.

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For the future lepton collider, the expected accuracy for the branching ratio of the Higgs boson invisible decay BR(h → inv), normalized to 5 ab⁻¹ is about 0.14% combined [41]. If the signal is not observed at the future CEPC, it will provide an upper bound for the Higgs portal coupling of about g < 0.073 at the future CEPC [41].

3.2. Muon anomalous magnetic moment

For the case of f(ϕ) = g²ϕ, the mixing interaction g²υϕh between ϕ and h is induced, and ϑ is defined as the mixing angle between the inflaton and the Higgs boson. This will contribute to the muon anomalous magnetic moment. Through the effective interaction of the inflaton field with the SM particles by mixing effects, the inflaton ϕ can contribute to the muon anomalous magnetic moment. Up to now, there exists a 3.5σ deviation between SM predictions and experimental results [42]:

At the one-loop level, the contribution from the inflaton ϕ to the muon anomalous magnetic moment can be written as

where G_F is the Fermi constant with . Since it needs , the constraints on the model parameters can be obtained numerically as ϑ < 0.75.

In this section, we study a concrete attractive relaxion model, assuming a non-minimal coupling of the inflation field ϕ to gravity, namely (1/2)ξRϕ² [43]. Such a non-minimal coupling may arise from some quantum gravity effects, such as the Higgs field with asymptotically safe gravity [44, 45]. We begin the discussion with the following action [43] in the Jordan frame:

where

and κ²/8π = G, which G is Newton's gravitational constant.

Performing the Weyl conformal transformation with Ω² = 1 − ξκ²(ϕ − ϕ_c)² and defining a new field φ₁ to make the kinetic term of ϕ be canonical as

or

we obtain the following action in the Einstein frame:

with

We use the longitudinal gauge (Newton gauge),

and take Φ = Ψ here.

Firstly, we derive the background field evolution equations for the cosmic expansion rate H = /a and homogeneous parts of scalar fields by variation of the action in Eq.(12):

Secondly, following the standard perturbative methods in cosmological inflation [46], we obtain the Fourier-transformed, first-order Einstein equations for the metric and field fluctuations as

For , then e^2F = 1 − ξκ²(ϕ − ϕ_c)², e^−2F = 1/(1 − ξκ²(ϕ − ϕ_c)²), F_,φ1 = F_ϕ/(dφ₁/dϕ), and U_,φ1 = U_,φ/(dφ₁/dϕ). Note that we should be careful in dealing with the original field ϕ and the new field φ₁.

Write

then we have

Setting M_pl = 1, then the tensor power spectrum is obtained as

and the scalar power spectrum is

where the so-called Bardeen parameter is defined by

Here, we take the single field slow-roll approximation. Then, the detailed conditions of the cosmological inflation are described by the following slow-roll parameters:

where and η represent the first and second derivatives of the inflation potential in the Einstein frame, respectively. The number of e-foldings N_e is given by

Thus, the amplitude of density perturbations in k-space under the slow-roll approximation is defined by the power spectrum:

where A_S is the scalar amplitude at some "pivot point" k*, which is given by

which can be measured from cosmic microwave background radiation (CMB) experiments. At the leading level, the scalar spectral index n_s can be written as

and the corresponding tensor-to-scalar ratio r* is

Using the recent Planck 2015 data n_s = 0.9655 ± 0.0062 and ln(10¹⁰A_S) = 3.089 ± 0.036 [47], we can obtain the constraints on the model parameters, and fit the combined experimental results of Planck 2015 using this model in the n_s − r plane as shown in Fig. 3, where we choose the pivot scale k* = 0.002 Mpc⁻¹, and at 95% C.L. Figure 3 shows that our prediction is well within the joint 95% C.L. regions. Roughly speaking, during inflation, there may exist the effect of entropy perturbation from the Higgs field. After reheating, it can be converted to curvature perturbation if the Higgs field couples to the inflaton field. In particular, if the reheating process is realized by the so-called Higgs reheating, then this effect would be very manifest. But this only applies to the case when the coupling between the inflation field and the Higgs field is sufficiently strong. However, the coupling between them is weak in this attractive relaxion model, and thus the result for the curvature perturbation is almost unaffected¹⁾. This non-minimal coupling model just corresponds to a special case of the T-models, which can explain the light Higgs mass from the cosmological evolution.

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We have put forward a toy model, which aims at providing a possible interpretation of the Higgs mass by attractive inflation. Only the inflaton field is needed and a broad classes of inflation models with attractive potentials can satisfy the conditions. This proposal ties the puzzling light mass of the Higgs boson to an attractive inflaton field which plays an important role during cosmological evolution. The possible collider signals or constraints at the future lepton colliders, the possible constraints from the muon anomalous magnetic moment and the concrete models were also discussed in detail. The attractive relaxion idea here represents a new interplay between particle physics and cosmology, and these new ideas of cosmological evolution would open a new door to understand some key parameters of particle physics.

We thank Francis Duplessis for valuable discussions during the early work. FPH is pleased to recognize the hospitality of the Department of Astronomy when visiting at University of Science and Technology of China.