{'fraction_non_alphanumeric': 0.045125666412795126, 'fraction_numerical': 0.013994668697638995, 'mean_word_length': 4.20307082714215, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Brans-Dicke gravity: from Higgs physics to (dynamical) dark energy

The Higgs mechanism is one of the central pieces of the Standard Model of electroweak interactions and thanks to it we can generate the masses of the elementary particles. Its fundamental origin is nonetheless unknown. Furthermore, in order to preserve renormalizability we have to break the gauge symmetry spontaneously, what leads to a huge induced cosmological constant incompatible with observations. It turns out that in the context of generalized Brans-Dicke theories of gravity the Higgs potential structure can be motivated from solutions of the field equations which carry harmless cosmological vacuum energy. In addition, the late time cosmic evolution effectively appears like an universe filled with mildly evolving dynamical dark energy mimicking quintessence or phantom dark energy.

Thus, paradoxically, while the famous Englert-Brout-Higgs mechanism [5,6] for spontaneous symmetry breaking (SSB) of the gauge symmetry is very much welcome in pure particle physics, it nevertheless carries a profound and unsolved enigma in the context of gravity and cosmology that seems to clash violently with observations. The problem might be suggesting that the vacuum energy in quantum field theory (QFT) is not the same as the cosmological vacuum energy. The two kind of vacua might be conceptually different and in such case there would be no reason to mix them up. While a solution does not seem possible in General Relativity (GR), it may be feasible in Brans-Dicke (BD) type gravity.

II. GENERALIZED BRANS-DICKE ACTION AND FIELD EQUATIONS
Consider the standard BD-action [7] with BD-field, ψ, coupled to the curvature R: where ω is the standard BD-parameter [7]. Recall that in the limit ω → ∞ this action is supposed to reproduce Einstein's GR. In this theory, the effective gravitational coupling at any time of the cosmic expansion is slowly varying with the cosmic time t, and reads Equivalently, the effective value of the (reduced) Planck mass squared at any time is just given by the BD-field: ψ(t) = M 2 P (t). At t = t 0 (our time) we have M P (t 0 ) ≡ M P = 1/ √ 8πG ≃ 2.43 × 10 18 GeV, hence G eff (t 0 ) = G is the current value of Newton's coupling.
Let us extend non-trivially the above action by invoking a new scalar field, φ, which is non-minimally coupled to gravity (both derivatively and non-derivatively) and also coupled with the BD-field. The suggested new piece of the action is Here G µν = R µν − (1/2)g µν R is the Einstein tensor and ξ, ς, η are dimensionless coefficients.
As we shall see, they are crucial for our purposes. While ψ(t) varies very slowly with the This epoch is characterized by the scalar field dominance and we can neglect S m . Within the (flat) FLRW metric (ds 2 = −dt 2 + a 2 (t)dx 2 ), the variation of the total action, S tot , with respect to the BD-field ψ yields where H =ȧ/a is the Hubble rate. Similarly, the variation with respect to the metric gives the following 00-component and the variation with respect to φ renders Let us search for power-law solutions of the above equations, namely The first expression is equivalent to a ∝ t n . Here n, α, γ are dimensionless parameters and φ 1 and ψ 1 are the values of the scalar fields at some early time t = t 1 in the past when the power-law solutions hold good, e.g. during the EW phase transition. From its interpretation we must have ψ 1 > 0, and |γ| ≪ 1. The above scaling solutions are associated to asymptotic states of the different phases of the cosmic evolution, e.g. the radiation-dominated (n = 1/2) and the matter-dominated (n = 2/3) epochs, inflation (n ≫ 1) [10,11].

III. COSMOLOGICALLY HARMLESS HIGGS
One of the above field equations contains the potential V and the other the derivative dV /dφ. The final expression for V can be derived self-consistently by requiring that the field equations determine the same form when we substitute in them the power-law solutions (7).
Let us consider arbitrary n values 1 . From Eq. (4), which does not depend on V , we find It determines the unique value α = −1 for the power of φ in Eq. (7). Using this value, Eq. (5) leads to and the field equation for φ leads to an alternative expression for V : In both of them the two powers t γ−2 and t −4 are consistently present. When we revert back to φ through t/t 1 = φ 1 /φ (using α = −1 ), these powers become associated to φ 2−γ and φ 4 , respectively.
Therefore it is possible to match the coefficients of each power. We obtain the following unique form for the effective potential (up to an additive constant): In the second equality we used the fact that |γ| ≪ 1 and thus the coefficient of φ 2 remains essentially constant. The obtained potential is essentially a Higgs-like potential for φ. The coefficient λ of the quartic coupling is given by For instance, for ξ > 0 and ς < 0 the necessary condition λ > 0 for vacuum stability is fulfilled both for n = 1/2 and n = 2/3. The form (10) of the potential tells us that there will be SSB of the EW symmetry provided η < 0, so this crucial sign is uniquely determined. Note that |η| ≪ 1 because ψ 1 ∼ M 2 P and the physical Higgs mass squared is of order M 2 H ∼ |η|ψ 1 ∼ 10 4 GeV 4 . The fact that |η| must be a very small parameter has nothing to do with fine-tuning [10]. As pointed out long ago by Bjorken [12,13], it is instead a well-known feature expected of any theory claiming a possible connection between the gravity scale and the EW scale.
It is also interesting to mention that the consistency conditions leading to the unique expression of the Higgs potential V indicated above imply a noticeable relation between the power γ that governs the cosmic time evolution of ψ and the BD-parameter ω. For ω ≫ n it takes the simple form It follows that γ is naturally small for large ω. A counterpart to this relation for the present universe will be elucidated in the next section, which will provide also a connection between ω and the (different) power-law evolution of the BD-field in the current epoch.

IV. LATE TIME UNIVERSE AND EFFECTIVE DYNAMICAL DARK ENERGY
As we have seen, in the present BD-gravity context the standard solutions H 2 ∼ a −2/n (n = 1/2, 2/3) for matter and radiation can perfectly support SSB with large EW vacuum energy but essentially zero cosmological vacuum energy. At late time the scalar field φ becomes suppressed: Correspondingly, for ψ = 1/(8πG) =const. it boils down to the standard pressure equation in GR.
The product ωǫ becomes determined by the remaining two BD-equations (4) and (14) upon using the ansatz (15). After a detailed calculation and evaluation at a = 1 (redshift z = 0) one Thus, ωǫ 2 ∼ ǫ and the two terms defining ν = ǫ + ωǫ 2 /6 are of the same order. The ∼ ωǫ 2 part of (16) is easily seen to originate from the non-canonical kinetic term of the BD-field, cf. Eq. (1), interpreted roughly in the manner of an effective potential contribution to the vacuum dynamics: where we have used (15). The precise and rigorous contribution, however, appears at the level of the field equations and stems from the third term on the l.h.s of Eq. (13). Together with the second term in that equation (which is linear in ǫ within the context of the power-law solution we are considering) it leads to the running vacuum form (13) [4,14]. On comparing (12) with (17) we can see that in both cases the power-law evolution of the BD-field becomes progressively smaller (hence closer to a constant) for larger and larger values of ω, but at different rates: for the current epoch ǫ ∼ 1/ω, whereas in the early universe γ ∼ 1/ √ ω.
The above cosmological solution is obviously different from GR and has distinct properties.
The pressure equation (14) can also be put in approximate ΛCDM fashion using (15). Neglecting the small parameter corrections except for those in the dynamical terms, such as ∼ a −3+ǫ and ∼ νM 2 P H 2 , we find the leading expression for the effective acceleration equation: It follows that the equation of state (EoS) for the effective DDE reads Thus, for ǫ < 0 (ǫ > 0) we have ν < 0 (ν > 0) and the effective DDE behaves quintessence (phantom)-like. For ǫ = 0 (i.e. ν = 0) we have w = −1 (ΛCDM) and only then the BD-parameter is forced |ω| → ∞ from (17).
Needless to say, matter is locally and covariantly conserved in BD-theory, as there is actually no interaction of matter with the BD-field ψ. However, when we try to encapsulate the observational behavior of BD-gravity in a strict GR-form, we find that the former may effectively appear as a model of interactive quintessence or phantom DE. The eventual detection of this kind of "anomalies" in the behavior of the ΛCDM could be signaling the possible presence of BD-gravity at a more fundamental level. This does not preclude, of course, that other sources of fundamental DE dynamics associated to new fields might also be concomitant with the same physical phenomenon, see e.g. [20]. However, BD-gravity alone does indeed have the capacity to mimic dynamical DE.
Let us finally note that if the tiny time dependence in the coefficient of φ 2 in the first expression of the potential in Eq. (10) had not been neglected, the expectation value of the Higgs field derived from BD-gravity would carry a mild time-evolution, φ ∼ t γ/2 (|γ| ≪ 1), which would be transferred to all the particle masses in the universe, a conclusion that has also been reached from alternative considerations in [21][22][23].

V. CONCLUSIONS
Generalized BD-gravity theories admit cosmologically viable solutions which lead to the precise structure of the Higgs potential. The inherent QFT vacuum energy of the potential is as large as usual but the cosmological vacuum energy density remains very small. The latter emerges only at late times through the usual constant term ρ Λ , but accompanied with an additional (albeit dynamical) contribution induced by the slow time evolution of the BD-field ψ. Effectively it leads to Λ(t)CDM, with a moderate time-evolving Λ(t) = Λ(H(t)) and EoS very close to −1. More specifically, it mimics the running vacuum model, which has recently been shown to provide a fit to the overall cosmological data better than the ΛCDM (with rigid cosmological constant Λ) .
Therefore, finding mild vacuum dynamics, accompanied with unsuspected time-evolving particle masses, could be the "smoking gun" signaling that the underlying gravity theory is not GR but BD. It could provide the missing link between gravity and particle physics.