Higgs chameleon

The existing constraints from particle colliders reveal a suspicious but nonlethal meta-stability for our current electroweak vacuum of Higgs potential in the standard model of particle physics, which is, however, disfavored in the early Universe if the inflationary Hubble scale is larger than the instability scale when Higgs quartic self-coupling runs into negative value. Alternative to previous trials of acquiring a positive effective mass-squared from Higgs quadratic couplings to Ricci scalar or inflaton field, we propose a third approach to stabilize the Higgs potential in the early Universe by regarding Higgs as chameleon coupled to inflaton alone without conflicting to the present constraints on either Higgs or chameleon.


I. INTRODUCTION
The state-of-art measurements [1] on Higgs mass M h = 125.10 ± 0.14 GeV and top quark mass M t = 172.9 ± 0.4 GeV continue to reenforce the longstanding conspiracy of Higgs near-criticality [2][3][4][5][6] (see also [7,8] for recent reviews and references therein). The running of Higgs quartic self-coupling starts becoming negative around the dubbed instability scale Λ I = 9.92 × 10 9 GeV [9] (see also [10][11][12][13] for its gauge dependence), where the Higgs potential develops a shallow barrier unstable against quantum fluctuations of order H inf /(2π) during inflation if the inflationary Hubble scale H inf is larger than Λ I . Therefore, the survival of our current electroweak (EW) vacuum throughout a high scale inflation seems highly unnatural and undesirable, even though we are temporarily safe in the EW vacuum for a lifetime of order 10 161 yrs [14] against Coleman-de Luccia (CdL) instanton with decay rate estimated around 10 −554 Gyr −1 Gpc −3 [15,16] (see also [17] for lattice simulation result and [18] for most recent results with thermal corrections). This is known as Higgs meta-stability, a special case of Higgs near-criticality, since the running of Higgs quartic selfcoupling could otherwise be fairly stable all the way to Planck scale within the current uncertainties mainly from top quark mass and strong coupling.
Inspired by the chameleon mechanism [81][82][83][84][85] by coupling the chameleon to ambient matter where the effective potential of chameleon becomes heavier in the denser environment, we propose in Sec. III to stabilize the Higgs field in the early Universe by recognizing Higgs as chameleon coupled to inflaton after we first generalizing the chameleon coupling for arbitrary background in Sec. II. The idea is simple enough but has never been explored before, which is also free from all the current constraints on Higgs from particle colliders and on chameleon from local gravity experiments if we restrict ourselves to couple Higgs chameleon to inflaton alone.

II. HIGGS AS CHAMELEON
Choosing the scalar field h as the chameleon field introduces extra interactions between h and other matter arXiv:2005.12885v1 [hep-ph] 26 May 2020 fields ψ i with action in the Einstein frame of form where the reduced Planck mass M 2 Pl = (8πG) −1 and the chameleon couplings to the metric g µν induce new metricsg (i) µν = Ω 2 i (h)g µν for each fields ψ i that are assumed to be independent for simplicity. The corresponding action variation (the variations δψ i are not shown here) reads with the Einstein tensor G µν ≡ R µν − 1 2 g µν R and the energy-momentum tensors defined by where the last contribution (4) could be rewritten with respect to the Einstein-frame metric as with traceT i ≡T On the other hand, δS m could also be expressed in terms of chain rule as which, after compared with (7), leads to identificatioñ ThusT µν (i) The energy-momentum tensorT µν (i) is conserved bỹ ∇ (i) µT µν (i) = 0 in Jordan frame where ψ i is minimally coupled to the Jordan-frame metricg (i) µν . However, the energy-momentum tensor is not conserved as ∇ µ T µν (i) = 0 in Einstein frame. In fact, note thatΓ For a perfect fluid ansatz for T µ if EoS parameter w i is treated as a constant. This defines a covariantly conserved density in Einstein frame bŷ which is also h-independent from 0 = ∇ tρi =ρ h (h)∇ t h. Now requiring vanishing variation for the sum of (2), (3) and (7) gives rise to the equation-of-motions (EoMs) for the metric field g µν and scalar field h as where the scalar EoM (14) could be rewritten as Note that for radiation domination,ρ is covariantly constant in time and hence h-independent. Hereafter, we will choose the scalar field h as Higgs field specifically.

III. HIGGS CHAMELEON IN THE EARLY UNIVERSE
For the sake of simplicity, Higgs field is assumed to have no chameleon coupling to all the other fields except inflaton field, then the Higgs effective potential V eff only receives its contribution of U i from inflaton field alone as The SM Higgs potential at zero temperature with higher loop-order quantum corrections could be approximated as [45] V where the Higgs quartic coupling turns negative at a critical value h c 5 × 10 10 GeV and b ≈ 0.16/(4π) 2 . To save Higgs from the instability developed around h c , there are infinitely many choices for the conformal factor Ω 1−3w φ φ (h) as long as it exhibits a higher power than h 4 .

A. Dilatonic chameleon coupling
As an illustrative example, the conformal factor is parameterized as where the dimensionless conformal factor α is regarded as a constant parameter for simplicity. Now the Higgs effective potential could be normalized with respect to where the second term is characterized by two effective parameters defined by This effective potential is shown in the upper left panel of Fig. 1, where the SM Higgs potential (red line) corrected by the chameleon contribution from coupling to inflaton could be easily stabilized with appearance of a second minimum (blue lines) until its disappearance at an inflection point (green line) with increasing ξ or c. The second minimum h min is one of the roots of the extreme points h 0 from V eff (h 0 ) = 0 by with Lambert function W (z) defined by z = W (z)e W (z) . On the one hand, for the second minimum being the de- which, after combing with (21), could solve for ξ deg from given c as shown in red line in the right panel of Fig. 1.
On the other hand, for the second minimum being the inflection point with V eff (h 0 ) = 0, it admits which, after combing with (21), could solve for ξ inf from given c as shown in blue line in the upper right panel of Fig. 1. The difference between ξ deg and ξ inf is asymptotically vanishing at large c limit, both of which are decreasing with power-law at large c limit, approaching to the green dashed line, ξ ∞ = 4c −1/4 , determined by first solving log(h deg /h c ) as a whole from (22) and then plugging into (21) with asymptotic expansion of Lambert function W (z → 0) ∼ z + O(z 2 ). The corresponding h deg /h c in the c → ∞ limit approaches c 1/4 .

B. Absolutely stable region
Without the appearance of the second minimum when ξ > ξ inf , the Higgs field is absolutely stable against any quantum fluctuations. For large enough c, the absolutely stable region could be approximately estimated by To further transform above constraints on (c, ξ) into more physical constraints on the inflationary Hubble scale H inf and the dimensionless conformal factor α, we could first set the EoS parameter w φ = −1 during inflation without loss of generality, then α = ξ/4 and c is related to H inf by To ensure that the Higgs effective potential energy V eff (0)/V c ≡ c at the desirable stable vacuum h = 0 is sub-dominated to the background Hubble expansion, namely c 3M 2 Pl H 2 inf /V c , the amplitude of conformal factor should be small, Ω φ (0) 1 . Now the absolute stability condition ξ ξ ∞ reads This suggests an absolute stability bound by the product α · Ω φ (0) in power law with respect to the inflationary Hubble scale shown as the green region in the lower left panel of Fig. 1, which, without adopting the asymptotic form ξ ∞ = 4c −1/4 , is precisely computed by ξ > ξ inf with respect to the inflection case (blue lines) for Ω φ (0) = 10 −1 , 10 −2 , 10 −3 , 10 −4 from top to below. Nevertheless, for given Ω φ (0), the corresponding red shaded region below ξ = ξ inf is NOT everywhere unstable as specified below.

C. Presence of a second minimum
The second minimum appears when ξ < ξ inf , which is higher or lower than the h = 0 vacuum if ξ deg < ξ < ξ inf The region for an absolutely stable Higgs effective potential without presence of a second minimum (green shaded) is shown above the blue lines computed from ξ > ξ inf for some illustrative values of the amplitude of Higgs chameleon coupling Ω φ (0) = 10 −1 , 10 −2 , 10 −3 , 10 −4 from top to below. The gray shaded region is ruled out by current constraint on the tensor-to-scalar ratio r < 0.06. The stability analysis in the red shaded region below the blue lines with presence of a second minimum is presented in the next panel. Lower right : For given amplitude of Higgs chameleon coupling Ω φ (0) (black numbers), the directions of arrows point to larger position, higher height, and broader width of Higgs potential barrier with respect to Higgs quantum fluctuation scale, hmax/H inf (red), V 1/4 bar /H inf (blue), and |V bar |/(4H 2 inf ) (green) as well as larger position of Higgs potential barrier at finite temperature with respect to the position of the second minimum at zero temperature h T max /hmin (purple).
or ξ < ξ deg , respectively. The degeneracy cases ξ = ξ deg are shown as red lines in the lower left panel of Fig. 1 for Ω φ (0) = 10 −1 , 10 −2 , 10 −3 , 10 −4 from top to below. In the presence of a second minimum, the Higgs stability against quantum fluctuations is guaranteed in all e 3N0 Hubble patches in our past lightcone if [36,45] h max where h max is the other root of (21), N 0 ≈ 60 is the efolding number of our current Hubble scale leaving the Hubble horizon before the end of inflation, and m eff is given by For given Ω φ (0) = 10 −2 , 10 −3 , 10 −4 (black numbers) in the lower right panel of Fig. 1, we have tested the condition (27) as red curves with red arrows pointing to a larger value than n stab , which automatically guarantees a much higher potential barrier V bar ≡ V eff (h max ) − V eff (0) > H 4 inf (blue curves) than the inflationary Hubble scale for the same Ω φ (0). This largely suppresses the decay processes via either CdL instanton or Hawking-Moss (HM) instanton depending on the broadness of poten-tial barrier estimated by |V eff (h max )|/(4H 2 inf ) [44] (green curves), to the upper-left/lower-right of which are dominated by CdL/HM instantons (if ever happened via decay channel), respectively. Therefore, the Higgs stability region against the quantum fluctuations could be extended from the absolutely stable region (green shaded) into the red shaded region in the lower left panel of Fig.  1 bounded by the red curves in the lower right panel of Fig. 1 for given Ω φ (0). However, this is not the whole story. Even for the parameter region to the lower-right direction of red curve with given Ω φ (0) where the second minimum is accidentally achieved during inflation either by the rare decay instantons or random walks over the potential barrier in some of the Hubble patches, there is still hope for them to be saved by the thermal corrections to the Higgs potential during radiation dominated era as elaborated below.

D. Thermal rescue
For an instantaneous reheating history, the reheating temperature at the onset of radiation domination approximately reads from the inflationary energy, with the number of degrees of freedom g reh = 106.75 for SM. The Higgs effective potential simply reads V eff (h) = V 0 (h) + V T (h) +ρ r withρ r independent of h (ρ r could be chosen as zero since the trace of energy-momentum tensor in (14) is vanished for radiation dominance), and the thermal corrections could be conveniently approximated which pushes the potential barrier to larger position, The thermal rescue [45] occurs when the local maximum h T max at finite temperature T reh is large enough for the Higgs field in the second minimum h min achieved during inflation could subsequently roll back to h = 0 vacuum during radiation era, which is shown as purple curves in the lower right panel of Fig. 1 with the direction of arrows pointing to the larger ratio of h T max /h min than unity value. After the thermal rescue, the thermal fluctuations of order temperature T have been checked to be much smaller than the thermal potential barrier, h T max T .
For non-instantaneous reheating, U i (h) in (15) during pre/reheating is smaller than that from inflationary era due to smaller power 1 − 3w i < 4 with −1/3 < w i < 1/3 and smallerρ i that dissipates into radiations, which could push the second minimum (if ever reached during inflation) to larger and deeper values until gradually connecting to the thermal Higgs potential in radiation era, thus invalidating the thermal rescue mechanism. Furthermore, one still has to avoid the broad resonance even though the positive effective mass-squared at either h = 0 vacuum or the second minimum could evade the tachyonic resonant production of Higgs during preheating. Therefore, a conservative safe zone is that V eff (h) never develops a second minimum to be ever reached during inflation and relaxed during pre/reheating, namely (26). We hope to revisit this issue in more details in a separate paper in future.

IV. CONCLUSION AND DISCUSSIONS
We propose a new mechanism to stabilize the Higgs potential in the early Universe by regarding Higgs as chameleon coupled to inflaton, which simply adds positive contribution to the original Higgs potential as shown in (16). We have tested this proposal in an illuminating example with conformal factor of form exponential to Higgs field as shown in (19). Other forms of this conformal factor should also work as long as it contributes positively to the effective potential. The absolutely stability bound (24), or expressed in terms of inflationary Hubble scale as (26), is analytically derived from the disappearance of inflection point in the effective potential. We also preliminarily extend the stability regime beyond the absolutely stable region into the case with the presence of a second minimum. Several comments are in order below.
Firstly, our solution for the Higgs stability problem in the early Universe only requires a chameleon coupling of Higgs to inflaton alone, while the chameleon couplings of Higgs to other fields are not necessarily demanded, which buys us extra benefit of evading all the current constraints on Higgs from either particle colliers or local gravity experiments.
Secondly, we neglect the effects on the running of SM Higgs couplings from Higgs-inflaton chameleon-like coupling, which, after expanding the conformal factor in power of h, only contributes to SM Higgs couplings with terms proportional to the same power of product αΩ φ (0), which is quite small (δm 2 ∼ 10 −14 , δλ ∼ 10 −28 ) according to the typical value of absolute stability bound (26).
Thirdly, three possible traces of Higgs ever as chameleon in the early Universe could be the isocurvature perturbations and non-Gaussianity due to its chameleon coupling to inflaton, as well as the productions of domain walls [86][87][88] when the second minimum is accidentally achieved during inflation in some Hubble patches, which merits further studies in future.