_{1}

^{*}

In spite of tremendous progress in experimental high-energy physics such as the apparent discovery of the Higgs boson at CERN, there exist a number of inconsistencies in theoretical physics which continue to go either unnoticed or unstated. These include the Higgs mechanism itself as well as recent discussions of problems with inflationary cosmology. The subject will be addressed in the context of this author’s recent paper [1] on the requirement for compatible asymptotic states in the study of the cosmological constant problem (CCP). Inconsistency in the Higgs mechanism is eliminated by using scalar-tensor gravity where the scalar field is a gravitational field with zero spin that represents the spontaneous symmetry breaking potential.

In recent comments regarding inflation and misinterpretations of BICEP2 [

Inconsistencies arise when authors fail to state what they are assuming or do not understand. These arise throughout theoretical physics and go far beyond inflation. In particular, those in particle physics likewise go unnoticed.

One involves the prominent Higgs mechanism [

In the afore-mentioned paper [

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 [

Only this current paper and [

The Abbott-Deser (AD) method [

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

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 _{ }

Einstein discovered VED in 1917 when he added the cosmological term to his theory of gravitation [

This section, culled from [

The Schwarzschild-de-Sitter metric (SdS) [

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

A canonical formulation of EG as a Hamiltonian system for the simple Schwarzschild case

For the case of (1) and (2) with

Circumstances change significantly, however, when

The ADM approach used above was extended by Abbott and Deser (AD) [

where the term

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

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 [

where

whose Gaussian curvature

The global energy of any cosmology, in particular the FLRW case (4), is determined by the ADT charges for APdS spacetime with

SSB per se is not due to Higgs et al. [

The Einstein-Hilbert action2 is

equations

with

^{2}R is the scalar curvature,

recognizing that

To the right-hand-side must be added the source term for matter

There are many examples of symmetry breaking potentials

where

Treated as a quantum field,

where

Another example is the more general self-interacting quartic case

investigated by [

A variation of (11) was used in [

with

The task now is to complete the scalar-tensor picture beginning with (6). The total Lagrangian

where

The field Equations (6) are now transformed, with the energy momentum tensor

(16) resolves the mass dimensionality of

Recalling that the Lagrangian for the FLW hadron model _{,} we have

The

with

^{3}To recover the pure Higgs mechanism with g_{μν}, set

The

The terms in (17) appearing in (18) involve quarks

, (21)

with counter terms not shown. _{3} Gell-Mann matrices and structure factors are

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).

Now we can turn to the energy-momentum tensor

^{4}In (26),

and is independent of the gravitational

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 [

The derivation of

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

where the coefficients A, B, C, D, and E are to be found. Taking the covariant divergence of (27), recalling the ansatz

with

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

while putting (29) into the trace

where

From the SSB potential

along with (31), the

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

After moving the

The interpretation of the scalar field arising from the well-known quartic Higgs potential for the Higgs complex doublet

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 the discussion here, the cosmological de Sitter background with a cosmological constant

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 [

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

The point of this analysis has been to demonstrate the procedure for introducing SSB mechanisms for scalar Spin-0 fields into scalar-tensor theories of gravity in a consistent fashion. This procedure has been careful to treat particle physics on an asymptotic FLRW cosmology representing an accelerating Universe [

The Higgs et al. mechanism [

Based upon the arguments presented here, the Higgs mechanism at best is incomplete. Its popularity has become folklore, but folklore is scientifically meaningless. Much in physics today is actually metaphysics5, examples of which are principles and assumptions such as the principle of relativity, the Pauli exclusion principle, or multiverses. These cannot be measured or proven experimentally. The first two are articles of faith that always seem to work. They are beyond physics yet they are used every day. The third is not observable.

On the other hand, inconsistencies that persist often become folklore and are also scientific meaningless. These are an artifact of misunderstanding some portion of physics, or they are based upon commonplace human error.

As long as particle physics has little or no respect for the asymptotic structure of curved spacetime discussed in Section 3, the inconsistency problem addressed here will go unresolved as will the CCP. A consistent treatment of VED in both cosmological gravity and particle physics is necessary. The scalar-tensor theory presented here may certainly be incomplete, but it is not inconsistent.