In the afore-mentioned paper [1] on the cosmological constant problem [CCP], this author demonstrated that the cosmological term in Einstein gravity (EG) as a scalar is therefore a potential term―a characteristic of EG that has been noted elsewhere [7] . It was also pointed out in [1] that Brans-Dicke theory is known to have been eliminated experimentally as a scalar-tensor alternative to EG. It was then demonstrated how to construct a consistent scalar-tensor theory in hadron physics to account for the two known values of the vacuum energy density (VED), one inside the hadron as a bag in quantum chromodynamics (QCD), for particle and nuclear physics, and one for the de Sitter phase gravitational background recently observed as an accelerating Universe in Friedman-Lemaitre-Robertson-Walker (FLRW) cosmology [8] -[10] . The two VED states are a consequence of SSB using a Higgs-type mechanism for a hadron potential in a de Sitter space background where the cosmological constant λ is not zero. Such de Sitter spacetimes where are referred to as cosmological gravity (CG).

It is true that the Higgs has been introduced as the scalar in scalar-tensor gravity in the literature, but not for reasons addressed here. These exceptions are therefore quite by accident. Examples include studies of the Higgs particle in the very early Universe and what role it may have played in inflationary models [11] .

Only this current paper and [1] address the inconsistency in particle physics of comparing energy calculations between incompatible asymptotic spacetimes. It is also argued that de Sitter space or CG is the mandatory background in order to solve the CCP in the observed accelerating Universe where [8] -[10] .

The Abbott-Deser (AD) method [12] for identifying mass and energy and their Killing-charge successors was shown to be the only consistent means for identifying them as the unique quantities associated with the asymptotic geometry at spatial infinity of de Sitter spacetime [1] . Hence a consistent definition and usage of global energy (Killing charge) in asymptotic spacetime must be adopted in theoretical (particle) physics. Currently, it is ignored; hence, there exists the CCP and other curved background problems.

When one attempts to compare and draw conclusions by cross-comparison of incompatible asymptotic states (with differing Killing charges), infinities arise and the results are an exercise in futility. They also disregard and contradict the known results of the AD method. Such comparisons for typical metrics will be addressed in Section 3 below, illustrating how this process is carried out.

Stated differently, there has been a great deal of theoretical work on the unification of gravity with QFT on curved backgrounds, quantum gravity (QG), zero-point energy fluctuations, and our understanding of VED in particle physics as well as cosmology. Furthermore, the difficulties of quantum gravity or performing QFT on curved backgrounds are well-known.

Yet the new requirement in Section 2.2 above follows using an obvious example commonly done in particle physics. Relativistic QFT has pursued VED physics in flat Minkowski space, resulting in the remarkable SSB mechanism used by Higgs et al. Even though EG is nonrenormalizable, its gravitational field couples minimally and universally to all of the fields of QFT’s renormalizable standard model [13] . To turn on gravity one simply introduces EG along with covariant derivatives in QFT that represent the transition from flat to curved background metrics. This ties everything nicely together except for the gravitational versus flat-space VED problem seen in the CCP. It is inconsistent to compare flat Minkowski space terms, whose metric is not even a solution of EG, with results based upon the metric in CG where. This is now discussed in Section 3.

The Schwarzschild-de-Sitter metric (SdS) [18] is

where

This curved background represents important global properties that relate to the definition of energy and energy conservation in Einstein gravity. In (1) and (2), we have with a unit 2-sphere metric, and M the Schwarzschild mass1. The SdS metric (1) becomes Schwarzschild for and de Sitter for.

A canonical formulation of EG as a Hamiltonian system for the simple Schwarzschild case in (2) was first derived by Arnowitt, Deser, and Misner (ADM) [19] . They determined the ADM energy, momentum, and mass defined by the asymptotic symmetries of (1) and (2) at spatial infinity. Conserved charge (mass, energy, etc.) is associated with a conserved Noether current which is determined by reducing the stress tensor density conservation law in EG to a conserved vector current law using Killing vectors. The ADM mass results and is equivalent to the Schwarzschild mass M, in (2).

For the case of (1) and (2) with, one obtains the Schwarzschild metric which is asymptotically flat as. Assuming further that and results in flat Minkowski space. Accordingly, the energy of Minkowski space is zero as expected.

Circumstances change significantly, however, when is re-instated in (2). The full SdS metric (1) is not asymptotically flat and becomes an asymptotic de Sitter space. It is forever distinguished from Minkowski space as. With, flat Minkowski space is no longer a relevant background for particle physics because it is not a solution of the Einstein equations [12] , and it is not an asymptotically flat de Sitter space.

The ADM approach used above was extended by Abbott and Deser (AD) [12] [20] who proceeded from the ADM results used in the Schwarzschild case and defined the AD Killing charges for the full SdS metric when it asymptotically becomes de Sitter space (dS), in contrast to the asymptotic flat case above. Because of their relevance to CG and the CCP, these AD charges have become very important. That work was later extended by Deser and Tekin (ADT) [21] -[24] who added Weyl and Gauss-Bonnet quadratic curvature terms (scaled by and respectively) to the Einstein-Hilbert Lagrangian [21] [22] , and found the generalized AD mass to be

where the term is the volume of the dS spacetime and has been added to account for the asymptotically pure de Sitter (APdS) case with in (2) and (3). Creating an energy density by dividing (3) by, this same term has been found by Padmanabhan [25] using different methods.

The total gravitational energy E of spacetime (3) is well-defined using ADM and ADT methods, provided it is being compared with a metric that has the same asymptotic structure. However, comparison of energies between asymptotically flat Minkowski and asymptotically de Sitter metrics is a misguided exercise. The concepts of global energy and energy conservation become ill-defined when compared to a non-existent solution (Minkowski space) in EG. There is no Einstein gravitational metric for a Minkowski metric because the latter has no gravity. One must use asymptotic de Sitter spaces to obtain any nonzero E in (3) at all [23] .

At this point, one can see from (3) that flat Minkowski space has no asymptotic structure.

FLRW cosmology is the accepted model for current observations of an accelerating Universe [8] -[10] . Its metric is

where is the scale factor and

whose Gaussian curvature [26] -[28] . Metric (4) is asymptotically an accelerating de Sitter space in its late stages, determined by the cosmological parameter as derived from the Einstein-Friedmann equations [28] .

The global energy of any cosmology, in particular the FLRW case (4), is determined by the ADT charges for APdS spacetime with in (1), (2), and (3) (no ADM or Schwarzschild mass).

There are many examples of symmetry breaking potentials. These include the well-known quartic Higgs potential for the Higgs complex doublet

where and. (9) has minimum potential energy for with.

Treated as a quantum field, has the vacuum expectation value. Following SSB, one finds

, indicating the appearance of the Higgs particle. In order to determine the mass of

one expands (9) about the minimum and obtains

where is negative definite and acquires the Higgs mass.

Another example is the more general self-interacting quartic case

investigated by [31] [32] to examine the ground states of nonminimally coupled, fundamental quantized scalar fields in curved spacetimes (1) or (4), that will be pursued in Section 4.2 below. (11) is based upon the earlier work of T.D. Lee et al. [33] -[36] and Wilets [37] [38] for modelling the quantum behavior of hadrons.

is arbitrary and represents a cosmological term in all cases, and all are unrelated except that they represent the VED or VEDs of the associated scalar field. The terms in have a mass-dimension of four as required for renormalizability. In the case of (9)-(10), it is the addition of the Higgs scalar that makes the standard electroweak theory a renormalizable gauge theory. Also, the electroweak bosons acquire the mass as a result of their interaction with the Higgs field if it is present in the vacuum.

A variation of (11) was used in [1] to address hadrons, which exhibit two VEDs (Figure 1 in [1] ), by establishing the direct relationship between Einstein’s λ and the hadron theory of Friedburg, Lee, and Wilets (FLW) where (11) becomes

with representing a self-interacting hadron scalar -field whose scaling coefficients are to be determined.

The task now is to complete the scalar-tensor picture beginning with (6). The total Lagrangian for the action involved must merge gravity with matter as being the sum of gravity, matter, and their interaction term. At this point, the Einstein-Hilbert action giving rise to (6) is extended to include the scalar field used in (12) with and λ encompassed into as where B is the well-known VED in QCD, Yang-Mills, and FLW hadron theory. This is accomplished by introducing the scalar-tensor action of Jordan-Fierz-Brans-Dicke (JFBD) [39] -[41]

where is a function of the -field, and in (6) is likewise as. For purposes here, the original JFBD ansatz is adopted although there are others. Note that special care must be given throughout to which has five degrees of freedom (DOFs) with Spin-0, Spin-1, and Spin-2 states of spin, discussed in [1] App. A-4, in addition to the scalar Spin-0 -field.

The field Equations (6) are now transformed, with the energy momentum tensor begin represented by a matter and a potential contribution, where contributes to the -field tensor. The matter tensor is the original tensor in (6), and their sum is conserved by the Bianchi identities:

(16) resolves the mass dimensionality of and B in that both sides of the equation have mass dimension two.

Recalling that the Lagrangian for the FLW hadron model is that for QCD (quarks pluscolor) supplemented by the nonlinear -field and a quark- mixing term_, we have

The term will become the -field interaction term with scalar-tensor gravity in the total Lagrangian that includes a nonminimally coupled Einstein-Hilbert term in (13) as

with

³To recover the pure Higgs mechanism with g_μν, set and use (9) rather than (11) or (12). This is for pedagogical purposes only because λ actually has two VED states, one for cosmology and one for hadrons and QCD. Hence, the procedure has no physical basis. cannot be ignored in particle physics.

The -field, again, is the scalar field of the scalar-tensor gravity presented here, giving a scalar field that couples to QCD in the FLW hadron model. Lee [42] has noted that QCD has no scalar field except for gluon and color condensates arising from nonlinear interactions of the color fields. Regardless of its origin and composition, this scalar is the basis for the scalar-tensor model under discussion3.

The terms in (17) appearing in (18) involve quarks, scalar, and colored gluons, which are defined as

, (21)

with counter terms not shown. is the -quark coupling constant, the strong coupling, the non- Abelian gauge field tensor, the quark flavor mass matrix, the gauge-covariant derivative, and the gravitation-covariant derivative (also in) with the spin connection derivable upon solution of (14) above, defining the geodesics. is the phenomenological dielectric function introduced by Lee et al. [34] , where and in order to guarantee color confinement. The SU₃ Gell-Mann matrices and structure factors are and.

Using (18)-(21), variation of (17) which neglects gravity in (18), gives the FLW equations of motion for σ and,

when one neglects the gluonic contribution (21). is the curved-space Laplace-Beltrami operator, and is

Now we can turn to the energy-momentum tensor in (15). It is comprised of two terms. The first is the usual matter contribution which includes all matter fields in the Universe except gravitation,

⁴In (26),.

and is independent of the gravitational -field4. The second term in (15) is new and must include the effects of in (19). Introducing a superscript “R” for renormalizable and consolidating all of the terms, we have in short-hand derivative notation

Based upon (26) and (27), variation of (13) will now give the final equations of motion. In order not to sacrifice the success of the principle of equivalence in Einstein’s theory [13] , a Brans-Dicke assumption must also be made. Only and not enters the equations of motion for matter (consisting of particles and photons). The interchange of energy between matter and gravitation thus must follow geodesics as assumed by Einstein [43] . The energy-momentum tensor for matter is hence assumed to be conserved in the standard fashion,.

The derivation of is a textbook problem [43] with specific details given in [1] where since that is the case here. Also, the potential is present with its renormalization restrictions in (12) being unique to this more thorough case.

The most general symmetric tensor of the form (27) which can be built up from terms each of which involves two derivatives of one or two scalar -fields, and itself, is

where the coefficients A, B, C, D, and E are to be found. Taking the covariant divergence of (27), recalling the ansatz, taking the trace of (14) and (15) for, modifying (22) to include the gravitational coupling with (still assuming) to produce the trace for, and obtaining the remaining trace from (38), the desired energy-momentum tensor for the -field follows as

with and.

Inserting (29) into (14) and (15) gives the full field equations

while putting (29) into the trace yields the scalar wave equation (with) for the -field

where is the source of -coupling to the traditional trace in JFBD theory. There is now coupling to the trace in (31) compared to (23). If, (30) is a conformally mapped set of Einstein field equations.

From the SSB potential in (12), we see that a has mass-dimension two or. By taking the derivative.

along with (31), the -field has acquired mass

Therefore it is a short-range field with only short-range interaction. (31) can be re-written

After moving the term to the left-hand side, is the remainder of (32) and a wave equation results. Hence a static solution for the -field must have a Yukawa cutoff where.

The interpretation of the scalar field arising from the well-known quartic Higgs potential for the Higgs complex doublet in (9) is legendary, having made the standard model of particle physics renormalizable. Its possible discovery at CERN is significant for particle physics without gravity.

The discussion here, however, has shown such a treatment to be inconsistent and certainly incomplete in spite of years of speculation in the literature about “Higgs gravity”. Nevertheless, one feature of discussions regarding the Higgs boson addresses its quality of giving some particles their mass (not all of them, just those in the standard electroweak model). Figuratively speaking, these particles acquire their mass by interacting with the universal background Higgs field in (9).

In the discussion here, the cosmological de Sitter background with a cosmological constant acts to give particles and photons an effective “mass” in the sense that they must follow geodesics in the curved spacetime of de Sitter space rather than flat trajectories in Minkowski space. The interaction is with the curved CG background, and the metaphors are related.

Furthermore, there exist two Spin-0 degrees of freedom in a scalar-tensor theory of gravity. As mentioned in Section 4.2, special care must address these DOFs in order to guarantee that the combined Spin-0, Spin-1, and Spin-2 states of spin do not create negative energy modes and instabilities, as discussed in [1] App. A-4.

There is an additional problem, involving the fact that the SSB mechanisms addressed in the scalar (Spin-0) potentials (9)-(12) are different mechanisms. Future work is necessary to explain why there would be two different SSB events in the vacuum such as (9) and (12). That subject lies far beyond the point of the present discussion. One naïve resolution to this quandary is simply to set the -field mass in (34) to the value measured at CERN, [44] -[47] .