_{1}

^{*}

We proceed to obtain a polynomial based iterative solution for early universe creation of the Higgs boson mass, using a derived polynomial of the form with the coefficients for derived through a series of specific integral formulations with the mass of a Higgs boson, and the construction of the coefficients derived as of the using a potential system, for the Higgs, largely similar to the usual Peskin and Schroeder quantum field theoretic treatment for a Higgs potential, which subsequently is modified, i.e. this is for the regime of space-time as up to the Electro-weak regime of cosmology, in terms of a spatial regime. The linkage to Dark matter is in terms of the Scalar Singlet Dark matter model proposed by Silvera and Zee, i.e. what we do is to use a procedure similar to the usual Standard model Higgs, but to find ways to iterate to isolate key inputs into electro weak symmetry breaking procedure for the creation of Dark Matter. Afterwards, we will use specific inputs into the Scalar Singlet Dark matter model which would isolate out input parameters which we think are amendable to experimental testing. We conclude with a discussion of entropy so generated, along the lines of a modification of the usual branching ratios used in Higgs physics, with spin offs we think are relevant to the Dark Matter problem. We also use it to critique some linkage between Dark Matter, and gravity, which may explain some of the findings of LIGO, which were reviewed in 2016 in one of our listed references.

We begin first with an accounting of the traditional Higgs mass calculation, much of which is going to be taken from Peskin and Schroeder (1995) [

A ˜ 1 m h 4 + A ˜ 2 m h 3 + A ˜ 3 m h 2 + A ˜ 4 m h 1 + A ˜ 5 = 0 . (1)

Note that what we are doing is to eventually walk this to Equation (31) of our text, in terms of what we think the probable outcomes are, with the outcomes summarized in Equation (33). The entire plan of the first part of the paper is to ascertain criteria as to possibly make implementation of Equation (33) of probable outcomes obtainable.

In particular, the quote from [

Quote:

In summary, although debate may continue over whether or not quantum wave functions themselves represent anything real, there is clear evidence that quantum effects such as coherence and quantum collapse can at least occur over many kilometers and at superluminal speeds.

Links into our supposition given in [

Essentially what we will do here is to discuss the usual procedures done to isolate out the Higgs mass. In doing so, we do not do it to refute the existence of the Higgs, itself, but more to the point to help in the elucidation of background which may give more evidence as to [

Quote:

As in the SM, the electroweak vacuum is metastable, it is important to explore if an extended scalar has an answer in its reserve. As the scalar weakly interacting massive scalar particles protected by Z2 symmetry can serve as viable dark matter candidates, it is interesting to explore if they help prolong the lifetime of the Universe. The effective Higgs potential gets modified in the presence of these new extra scalars, improving the stability of electroweak vacuum.

End of quote.

As given in [

ϕ = U ( x ) 2 ( 0 v + h ( x ) ) . (2)

Peskin and Schroeder [

ℑ = | D u ϕ | 2 + μ 2 ϕ † ϕ − λ ( ϕ † ϕ ) 2 . (3)

This after a unitary transformation will lead to the following representation of an explicit Higgs Boson potential

ℑ V = − μ 2 h 2 ( x ) − λ μ h 3 ( x ) − h 4 ( x ) 4 . (4)

This can be written as:

ℑ V = − 1 2 m h 2 h 2 ( x ) − λ 2 m h h 3 ( x ) − λ h 4 ( x ) 4 . (5)

The method in all of this is to isolate out the quadratic term and to read off, in doing so a mass value as from going from Equation (3) and Equation (4) and then go to Equation (5). We will use a different approach, i.e. one which relies upon an expression for mass times acceleration equal to spatial integration of derivatives of Equation (5) above, with an approximation of found in [

Use the following from [

m h = 4π ( d 2 z d t 2 ) acceleration ⋅ ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ r 2 ⋅ { } { } = ( 1 r 2 ⋅ ∂ ∂ r ⋅ r 2 ⋅ ∂ ∂ r ⋅ ℑ V ( r ) ) ⋅ ( 1 r 2 ⋅ ∂ ∂ r ⋅ r 2 ⋅ ℑ V ( r ) ) ℑ V ( r ) = − 1 2 m h 2 h 2 ( r ) − λ 2 m h h 3 ( r ) − λ h 4 ( r ) 4 h ( r ) ∼ exp ( − α ˜ r )

m h = 4 π ( d 2 z d t 2 ) acceleration ⋅ [ ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ r ⋅ [ A 1 ⋅ h 4 ( r ) ] ) ⋅ λ 0 ⋅ m h 4 + ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ r ⋅ [ A 2 ⋅ h 5 ( r ) ] ) ⋅ λ 1 / 2 ⋅ m h 3 + ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ r ⋅ [ A 3 ⋅ h 6 ( r ) ] ) ⋅ λ 1 ⋅ m h 2 + ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ r ⋅ [ A 4 ⋅ h 7 ( r ) ] ) ⋅ λ 3 / 2 ⋅ m h 1 + ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ r ⋅ [ A 5 ⋅ h 8 ( r ) ] ) ⋅ λ 2 ] (6)

We will fill out the A i terms in the next block of equations as follows:

A 1 ( r ) = 5 α ˜ 2 r − ( 3 α ˜ 3 + 2 α ˜ r 2 ) A 2 ( r ) = ( ( 15 / 2 ) + 15 2 ) α ˜ 2 r − ( ( 12 / 2 ) α ˜ 3 + ( ( 1 / 2 ) + 6 2 ) α ˜ r 2 ) A 3 ( r ) = ( 81 / 2 ) α ˜ 2 r − ( 33 α ˜ 3 + ( 47 / 2 ) α ˜ r 2 ) A 4 ( r ) = ( ( 24 / 2 ) + 20 2 ) α ˜ 2 r − ( ( 21 / 2 ) α ˜ 3 + ( 20 / 2 ) α ˜ r 2 ) A 5 ( r ) = 6 α ˜ 2 r − ( 4 α ˜ 3 + 2 α ˜ r 2 ) . (7)

This leads to a decomposition of the mass, as looking like what is given in Equation (1) with the following coefficients filled in for Equation (1)

A ˜ 1 = ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ r ⋅ [ A 1 ( r ) ⋅ e − 4 α ˜ r ] ) ⋅ λ 0 A ˜ 2 = ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ r ⋅ [ A 2 ( r ) ⋅ e − 5 α ˜ r ] ) ⋅ λ 1 / 2 A ˜ 3 = ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ r ⋅ [ A 3 ( r ) ⋅ e − 6 α ˜ r ] ) ⋅ λ 1 A ˜ 4 = ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ r ⋅ [ A 4 ( r ) ⋅ e − 7 α ˜ r ] ) ⋅ λ 3 / 2 − ( d 2 z d t 2 ) acceleration / 4 π A ˜ 5 = ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ r ⋅ [ A 5 ( r ) ⋅ e − 8 α ˜ r ] ) ⋅ λ 2 (8)

The sheer amount of algebra is, in this situation, unbelievable, and this will be sourced to the following three integrals in from [

The three integrals, from [

∫ d r ⋅ r ⋅ exp ( − B ˜ 1 ⋅ r ) = exp ( − B ˜ 1 ⋅ r ) B ˜ 1 2 ⋅ ( − B ˜ 1 ⋅ r − 1 ) ∫ d r ⋅ exp ( − B ˜ 1 ⋅ r ) = exp ( − B ˜ 1 ⋅ r ) − B ˜ 1 ∫ d r ⋅ r − 1 ⋅ exp ( − B ˜ 1 ⋅ r ) = [ log ( r ) ] − B ˜ 1 ⋅ r 1 ! + ( B ˜ 1 ⋅ r ) 2 2 ⋅ 2 ! − ⋯ . (9)

Having said, this we should in the next section say a bit as to the physics behind the formula

m h ( d 2 z d t 2 ) acceleration = − 4 π ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ r 2 ⋅ { } { } = ( 1 r 2 ⋅ ∂ ∂ r ⋅ r 2 ⋅ ∂ ∂ r ⋅ ℑ V ( r ) ) ⋅ ( 1 r 2 ⋅ ∂ ∂ r ⋅ r 2 ⋅ ℑ V ( r ) ) . (10)

We are, here, implying use of symmetry, in our ideas, and making the creation of the mass of the Higgs, as seen above, dependent upon a regime of space-time roughly contingent upon the regime of the electro weak regime. In doing so we will be making the following assumptions [

Quote from [

In physical cosmology, the electroweak epoch was the period in the evolution of the early universe when the temperature of the universe was high enough to merge electromagnetism and the weak interaction into a single electroweak interaction (>100 GeV). The electroweak epoch began when the strong force separated from the electroweak interaction. Some cosmologists place this event at the start of the inflationary epoch, approximately 10^{−}^{36} seconds after the Big Bang. Others place it at approximately 10^{−32} seconds after the Big Bang.

End of quote.

We will, as a start, begin with the assumption that r ( electroweak ) ~ planck length l P and that the expression Δ r is connected to the Heisenberg uncertainty principle, i.e. we will be approximately using the following for making sense of the HUP with the following arguments, Afterward, we will then begin to ascertain what is to be done with the integrals, as for the input for Equation (8) and Equation (9). But before that we will say something of the HUP to be used, in this input into Equation (9), so to begin our process, we will give a review of HUP Basics. The HUP basics are extremely important in terms of giving an upper bound to the variance of Δ r which is of extreme importance, especially in our modeling of if we have Tachyon’s in the Pre Planckian Space time era, and then regular Higgs physics from after the Electroweak era to today. So a lot of basic physics is covered in this supposition.

Starting with background given in [

( Δ l ) i j = δ g i j g i j ⋅ l 2 ( Δ p ) i j = Δ T i j ⋅ δ t ⋅ Δ A ϕ ~ ξ + ≪ M Planck . (11)

Then

If we use the following, from the Roberson-Walker metric [

g t t = 1 g r r = − a 2 ( t ) 1 − k ⋅ r 2 g θ θ = − a 2 ( t ) ⋅ r 2 g ϕ ϕ = − a 2 ( t ) ⋅ sin 2 θ ⋅ d ϕ 2 . (12)

Following Unruh [

a 2 ( t ) min ~ 10 − 110 , r ∼ l P ∼ 10 − 35 meters . (13)

Then, if Δ T t t ~ Δ ρ [

V ( 4 ) = δ t ⋅ Δ A ⋅ r δ g t t ⋅ Δ T t t ⋅ δ t ⋅ Δ A ⋅ r 2 ≥ ℏ 2 ⇔ δ g t t ⋅ Δ T t t ≥ ℏ V ( 4 ) . (14)

This Equation (14) is such that we can extract, up to a point the HUP principle for uncertainty in time and energy, with one very large caveat added, namely if we use the fluid approximation of space-time [

T i i = d i a g ( ρ , − p , − p , − p ) . (15)

Then by [

Δ T t t ~ Δ ρ ~ Δ E V ( 3 ) . (16)

Then, by [

δ t Δ E ≥ ℏ δ g t t ≠ ℏ 2 Unless δ g t t ~ O ( 1 ) . (17)

We will then if we make the following approximations have a connection to the Δ r value used in Equation (8)

Δ r ≈ ( Δ l ) i j δ i j = δ g i i g i i ⋅ l planck 2 → i → t = timeinPlanckregime δ g t t g t t ⋅ l planck 2 . (18)

Use, then

δ t Δ E ≈ ℏ δ g t t ⇒ Δ r ≈ | δ g t t g t t | ⋅ l planck 2 ≈ | ℏ δ t Δ E | ⋅ l planck 2 . (19)

It means that the spatial fluctuation, if δ t is of the order of Planck time, and Δ E an emergent energy value, is then crucially dependent upon what we chose for Δ E , then we will have a way to input values into Equation (8).

Due to the smallness of the value Δ r ≈ | ℏ δ t Δ E | ⋅ l planck 2 which will be evaluated,

the following is a default choice, i.e. Gauss quadrature [

∫ r 0 r 0 + Δ r f ( r ) d r ≈ Δ r 2 ⋅ [ f ( r 0 + ( 1 2 − 3 6 ) Δ r ) + f ( r 0 + ( 1 2 + 3 6 ) Δ r ) ] + Ο ( ( Δ r ) 4 ) (20)

In this case, the values of function f will be dependent upon the details of what is chosen for the different integrands in the five integrals given in Equation (8),

but due to the smallness of Δ r ≈ | ℏ δ t Δ E | ⋅ l planck 2 , we claim that this will put a

premium upon the evaluation of a suitable Δ E value, and we will next, in order to address this, detail procedures as to isolating optimal Δ E values in order to ascertain, optimal strategies for implementing Equation (20) strategies on the integrals given in Equation (8) above.

We will look at what is given in [

E mean = ℏ ω 0 e ℏ ω 0 / k B T ( temperature ) − 1 + ℏ ω 0 2 . (21)

Assume that T, temperature is of the order of over 100 GeV, and that we will also have frequency ω 0 , for the background as to the creation of particles in the electroweak era, i.e. our estimate is, then that we will be able to assert, then that with temperature of a very large magnitude, and with frequency ~1/(time for initiation of electro weak regime) that if we have say

E mean = ℏ ω 0 e ℏ ω 0 / k B T ( temperature ) − 1 + ℏ ω 0 2 ℏ ω 0 ≪ k B T ( temperatureElectroweak ) Δ E ≈ E mean − ℏ ω 0 2 ≐ ℏ ω 0 e ℏ ω 0 / k B T ( temperature ) − 1 ~ very big . (22)

This would push down the magnitude of Δ r

Δ r ≈ | ℏ δ t Δ E | ⋅ l planck 2 ≈ | e ℏ ω 0 / k B T ( temperature ) − 1 δ t ⋅ ω 0 | ⋅ l planck 2 ≈ | e ℏ / ( timeforinitiationofelectroweakregime ) ⋅ k B T ( temperature ) − 1 δ t / ( timeforinitiationofelectroweakregime ) | ⋅ l planck 2 . (23)

What we are doing is to group the terms, in terms of powers of m h , and to do this we have a block of five inputs into Equation (1) above which we write up as follows in the following equation.

A ˜ 1 = ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ [ [ ( 5 ) ⋅ α ˜ 2 ] − ( 2 ) ⋅ α ˜ 3 r − [ ( 3 ) ⋅ α ˜ / r ] ] e − 4 α ˜ r ) A ˜ 2 ∝ ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ [ [ ( [ 15 / 2 ] + 15 2 ) ⋅ α ˜ 2 ] − ( 12 / 2 ) ⋅ α ˜ 3 r − [ ( [ 1 / 2 ] + 6 2 ) ⋅ α ˜ / r ] ] e − 5 α ˜ r ) ⋅ λ 1 / 2 A ˜ 3 = ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ [ [ ( 81 / 2 ) ⋅ α ˜ 2 ] − ( 47 / 2 ) ⋅ α ˜ 3 r − [ ( 33 ) ⋅ α ˜ / r ] ] e − 6 α ˜ r ) ⋅ λ A ˜ 5 ∝ ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ [ ( 6 α ˜ 2 − 4 α ˜ 3 r − [ 2 α ˜ / r ] ) ⋅ e − 8 α ˜ r ] ) ⋅ λ 2 (24)

Integrals of the form due to the smallness of Δ r would likely be deal with by Gaussian Quadrature, as for using Equation (20) on Equation (25)

∫ d r ⋅ r − 1 ⋅ exp ( − B ˜ 1 ⋅ r ) = [ log ( r ) ] − B ˜ 1 ⋅ r 1 ! + ( B ˜ 1 ⋅ r ) 2 2 ⋅ 2 ! − ⋯ . (25)

The other integrals, in Equation (9) would most likely be dealt with via their integral table values, but if we wish to group the integrals from Equation (24) via their integrands, according to r, 1 and 1/r we observe the following pattern, i.e. it is in essence a recreation of the driven harmonic oscillator model, which in terms of physics, especially after Gauge invariance is invoked one of the work horses of elementary particle physics, i.e. observe the following Equation (26). And in spirit it is not that dissimilar to [

In doing so, we are also aware that this is so far akin to much of the phenomenology of the standard model, but with the outstanding difference which we have put in the prior paragraph.

C 0 ∝ ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ α ˜ 2 ⋅ [ ( 6 ) ⋅ e − 8 α ˜ r λ 2 + ( [ 24 2 ] + [ 20 / 2 ] ) ⋅ e − 7 α ˜ r λ 3 / 2 m h + ( 81 / 2 ) ⋅ e − 6 α ˜ r λ 1 m h 2 + ( [ 15 / 2 ] + [ 15 2 ] ) ⋅ e − 5 α ˜ r λ 1 / 2 m h 3 + ( 5 ) ⋅ e − 4 α ˜ r λ 0 m h 4 ] )

C 1 ∝ − ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ r ⋅ α ˜ 3 ⋅ [ ( 4 ) ⋅ e − 8 α ˜ r λ 2 + ( 21 / 2 ) ⋅ e − 7 α ˜ r λ 3 / 2 m h + ( 81 / 2 ) ⋅ e − 6 α ˜ r λ 1 m h 2 + ( 47 / 2 ) ⋅ e − 5 α ˜ r λ 1 / 2 m h 3 + ( 2 ) ⋅ e − 4 α ˜ r λ 0 m h 4 ] )

C − 1 ∝ − ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ r − 1 ⋅ α ˜ 1 ⋅ [ ( 2 ) ⋅ e − 8 α ˜ r λ 2 + ( 20 / 2 ) ⋅ e − 7 α ˜ r λ 3 / 2 m h + ( 33 ) ⋅ e − 6 α ˜ r λ 1 m h 3 + ( [ 1 / 2 ] + 6 2 ) ⋅ e − 5 α ˜ r λ 1 / 2 m h 3 + ( 3 ) ⋅ e − 4 α ˜ r λ 0 m h 4 ] ) (26)

This is further grouped as

m h ⋅ ( d 2 z / d t 2 ) / 4 π = C 1 + C 0 + C − 1 . (27)

In order to solve this in terms of powers of m h , Equation (1) and Equation (24) are preferred. This grouping in Equation (26) and Equation (27) is done for ease in terms of the rules of integration, alone.

Doing this means we will be considering an optimal set of procedures so as to find a numerical protocol as to solve for a fourth order polynomial for iteration of mass of m h .

Simple on line root finders abound. Here is one of them [

If we wish to solve for a fourth order equation and we are examining to obtain real values for m h one of the very few papers giving the Cardano solution is sourced here, i.e. [

In other words, if Equation (1) and its inputs are defined properly and sourced mathematically, one has the probability one is looking at a real and complex part of a solution to m h .

Fortunately, we are not completely without guidance, i.e. in Wikipedia, there is a solution [

Quote [

In complex analysis, a branch of mathematics, the Gauss-Lucas theorem gives a geometrical relation between the roots of a polynomial P and the roots of its derivative P'. The set of roots of a real or complex polynomial is a set of points in the complex plane. The theorem states that the roots of P' all lie within the convex hull of the roots of P that is the smallest convex polygon containing the roots of P. When P has a single root then this convex hull is a single point and when the roots lie on a line then the convex hull is a segment of this line. The Gauss-Lucas theorem, named after Carl Friedrich Gauss and Félix Lucas, is similar in spirit to Rolle’s theorem.

I.e. we can look at the derivative, [

A ˜ 1 m h 4 + A ˜ 2 m h 3 + A ˜ 3 m h 2 + A ˜ 4 m h 1 + A ˜ 5 = 0 ⇒ 4 A ˜ 1 m h 3 + 3 A ˜ 2 m h 2 + 2 A ˜ 3 m h 1 + A ˜ 4 = 0 ⇒ A ˜ 1 m h 3 + 3 4 A ˜ 2 m h 2 + 1 2 A ˜ 3 m h 1 + 1 4 A ˜ 4 = 0. (28)

Beyer, [

y 3 − b y 2 + ( a c − b d ) y − a 2 d + 4 b d − c 2 = 0 (29)

where

a = A ˜ 2 A ˜ 1 , b = A ˜ 3 A ˜ 1 , c = A ˜ 4 A ˜ 1 , d = A ˜ 5 A ˜ 1 . (30)

The terms in Equation (24) are huge numerical monsters in their own right. And of course we will be requiring that in Equation (31), below, is that to solve for Equation (1) in full generality, we will almost certainly be observing complex valued solutions to m h . If we solve for m h and obtain complex valued solutions to the Higgs mass, as obtained this way, we then have some fundamental questions to answer.

Before we do that, let us review below, in Equation (31) the full generality of our answer. It is to put it very specifically, a huge mess. Why would we do this or want to do it?

Note that Equation (31) is, superficially like a driven harmonic oscillator. The program for identification of the Higgs mass, has been to identify terms in the potential which proportionality are linkable to the mass of a Harmonic oscillator, i.e. go to [

We have been ascertaining possible solutions to the Higgs, and seeking solutions to beyond the standard model of the Higgs, for decades, i.e. see [

The solutions as to what was brought up in [

Due to how nonstandard all this is, we will commence in our section past Equation (31) below to comment upon the implications of real and complex parts to mass terms, especially in something as foundational as the Higgs mass.

We wish to also, afterwards, go to the 2^{nd} part of this paper, which will be to make linkage to the dark matter model, which may involve the Higgs, in modified form, as given in [

Part of the problem especially even with extensions as to the Higgs models, is still the fix upon derivations of matter and reality as based squarely on modifications of the Harmonic oscillator. While it has served us very well in the last 100 years, extensions beyond it may require a re think of basic assumptions. We will be revisiting some of these assumptions in the second half of our paper.

To make the case for possible complex parts to the Higgs mass, we reference Equation (31) below and also then will use it as to going to the foundational question in the next part which is what happens if we have not solely real valued mass terms. What are we in for?

A ˜ 1 m h 4 + A ˜ 2 m h 3 + A ˜ 3 m h 2 + A ˜ 4 m h 1 + A ˜ 5 = 0 ⇒ m h 4 + A ˜ 2 A ˜ 1 m h 3 + A ˜ 3 A ˜ 1 m h 2 + A ˜ 4 A ˜ 1 m h 1 + A ˜ 5 A ˜ 1 = 0 islinkableto x 4 + a x 3 + b x 2 + c x + d = 0 x = m h 1 , a = A ˜ 2 A ˜ 1 , b = A ˜ 3 A ˜ 1 , c = A ˜ 4 A ˜ 1 , d = A ˜ 5 A ˜ 1

Then , to y 3 − b y 2 + ( a c − b d ) y − a 2 d + 4 b d − c 2 = 0 x = − ( a / 4 ) + R / 2 ± ( D / 2 ) or x = − ( a / 4 ) − R / 2 ± ( E / 2 ) where R = a 2 4 − b + y

If R ≠ 0 D = 3 a 2 4 − R 2 − 2 b + ( 4 a b − 8 c − a 3 4 R ) E = 3 a 2 4 − R 2 − 2 b − ( 4 a b − 8 c − a 3 4 R )

If R = 0 D = 3 a 2 4 − 2 b + 2 y 2 − 4 d 2 E = 3 a 2 4 − 2 b − 2 y 2 − 4 d 2 (31)

What we can expect if the Higgs, say has

m h = Re ( m h ) + i ⋅ Im ( m h ) . (32)

Baez says that if the mass of a particle is purely imaginary that, in [

According to Baez, [

What we are supposing, taking a line from [

Note that in [

Our supposition is as follows, i.e. to a large degree what we are predicting is that solving Equation (31) would lead to [

m h = Re ( m h ) + i ⋅ Im ( m h ) → PriorToElectroweakera i ⋅ Im ( m h ) & 'Quantum-entanglement ( information-transfer ) m h = Re ( m h ) + i ⋅ Im ( m h ) → Electroweakeratotoday Re ( m h ) & StandardModelHiggsmassformationproperties (33)

Getting a description of this modeling would require extensive numerical and analytical treatment of Equation (1), Equation (31), and of also reviewing what we think we know of how information could be transferred through the ideas of finding EPR style entangled wave function states from a prior to the present universe. The readers are encouraged to review [

Having made our predictions, as far as a goal to be investigated, we remind the readers to see what entanglement may be doing in the Pre Planckian to Planckian era, as in [

We will make brief reference, to what Equation (1) has as a solution to We will before moving on, make brief reference as to what the value of Equation (1) is if we take the derivative of it, with respect to We will before moving on, make brief reference as to what the value of Equation (1) is if we take the derivative of it, with respect to m h is, and then state that this is a brief foray into what would be needed even if we merely had the DERIVATIVE of Equation (1) with respect to m h , set equal to zero. What would we have to have so that the mass, m h would have a definite imaginary component? i.e. we will be looking at in the case of a DERVIATIVE of Equation (1) with respect to the presumed Higgs mass the following procedure which may happen in the Pre Planckian to Planckian regime of space-time about the Electroweak spatial regime the following procedure as seen in the next page.

Keep in mind the following that this is to illustrate the complexities involved, as far as how to isolate in the spatial regime, to the beginning of the electro weak era how complex values of the mass could arrive.

In doing so, it is important to note that this is for the DERIVATIVE of Equation (1) with respect to Higgs mass, and that this is in order to appeal to this well-known example, i.e.

This is from the Wiki discussion given in [

Quote: from [

Special Cases:

It is easy to see that if P(x) = ax^{2} + bx + c is a second degree polynomial, the zero of P'(x) = 2ax + b is the average of the roots of P. In that case, the convex hull is the line segment with the two roots as endpoints and it is clear that the average of the roots is the middle point of the segment.

End of quote.

What we did below is to appeal to a very approximate method in order to isolate conditions in which we could have an imaginary component to the Higgs mass. In doing so, we do this with the expectation that in the regime of space-time so written out, that we have the dreaded by String theorists Tachyon solution, which is going faster than light.

In the case of the solution given in Equation (34) below, we have an imaginary component to the Higgs mass as specified in the spatial regime for which the Electro weak processes occur in cosmology, i.e. sometime after a distance of at least Planck length from the boundary of creation of a new universe. The results of this are given below. We will then, say something as to the restriction as to having what we call C2 as nearly zero, in order to draw out this approximate imaginary behavior, for Higgs mass, which is what we will be appealing to

m h 4 + A ˜ 2 A ˜ 1 m h 3 + A ˜ 3 A ˜ 1 m h 2 + A ˜ 4 A ˜ 1 m h 1 + A ˜ 5 A ˜ 1 = 0 Take − derivative − w . r . t . − m h 1

⇒ m h 3 + 3 A ˜ 2 4 A ˜ 1 m h 2 + A ˜ 3 2 A ˜ 1 m h 1 + A ˜ 4 4 A ˜ 1 = 0 a 1 = 3 A ˜ 2 4 A ˜ 1 , a 2 = A ˜ 3 2 A ˜ 1 , a 3 = A ˜ 4 4 A ˜ 1 ⇒ m h 3 + a 1 m h 2 + a 2 m h 1 + a 3 = 0

Then rewrite y 3 + c 1 ⋅ y + c 2 = 0 m h 1 = y − ( a 1 / 4 ) c 1 = 1 3 ⋅ ( 3 2 a 2 − [ 3 4 a 1 ] 2 ) c 2 = 1 27 ⋅ ( 2 [ 3 4 a 1 ] 2 − 27 8 ⋅ a 1 ⋅ a 2 + 27 4 ⋅ a 3 )

A = ( − ( c 2 ) 2 + ( c 2 ) 2 4 + ( c 1 ) 27 ) 1 / 3 , B = − ( − ( c 2 ) 2 + ( c 2 ) 2 4 + ( c 1 ) 27 ) 1 / 3 y = A + B , − ( A + B 2 ) + ( A − B 2 ) ⋅ − 3 , − ( A + B 2 ) − ( A − B 2 ) ⋅ − 3 & m h 1 = A + B − ( a 1 / 4 ) , − ( A + B 2 ) + ( A − B 2 ) ⋅ − 3 − ( a 1 / 4 ) , and − ( A + B 2 ) − ( A − B 2 ) ⋅ − 3 − ( a 1 / 4 )

then c 2 ≈ ε + ⇒ m h 1 ≈ ( A − B 2 ) ⋅ − 3 − ( a 1 / 4 ) ≈ 2 ( + ( c 1 ) 27 ) 1 / 3 ⋅ − 3 − ( a 1 / 4 ) ; m h 1 ≈ 2 ( + 1 3 ⋅ ( 3 2 a 2 − [ 3 4 a 1 ] 2 ) 27 ) 1 / 3 ⋅ − 3 − ( a 1 / 4 ) (34)

Here, C2 nearly zero is if we can make the following identification, i.e. this is what would be necessary, i.e.

If A ˜ 1 = ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ [ [ ( 5 ) ⋅ α ˜ 2 ] − ( 2 ) ⋅ α ˜ 3 r − [ ( 3 ) ⋅ α ˜ / r ] ] e − 4 α ˜ r ) A ˜ 2 ∝ ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ [ [ ( [ 15 / 2 ] + 15 2 ) ⋅ α ˜ 2 ] − ( 12 / 2 ) ⋅ α ˜ 3 r − [ ( [ 1 / 2 ] + 6 2 ) ⋅ α ˜ / r ] ] e − 5 α ˜ r ) ⋅ λ 1 / 2 A ˜ 3 = ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ [ [ ( 81 / 2 ) ⋅ α ˜ 2 ] − ( 47 / 2 ) ⋅ α ˜ 3 r − [ ( 33 ) ⋅ α ˜ / r ] ] e − 6 α ˜ r ) ⋅ λ

A ˜ 4 ∝ ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ [ [ ( 24 2 + 20 / 2 ) α ˜ 2 ] − ( 21 / 2 ) α ˜ 3 r − [ ( 20 / 2 ) α ˜ / r ] ] e − 7 α ˜ r ) ⋅ λ 3 / 2 − ( d 2 z / d t 2 ) / 4π A ˜ 5 ∝ ( ∫ r ( electroweak ) r ( electroweak ) + Δ r d r ⋅ [ ( 6 α ˜ 2 − 4 α ˜ 3 r − [ 2 α ˜ / r ] ) ⋅ e − 8 α ˜ r ] ) ⋅ λ 2

Then c 2 = 1 27 ⋅ ( 2 [ 3 4 a 1 ] 2 − 27 8 ⋅ a 1 ⋅ a 2 + 27 4 ⋅ a 3 ) ≈ ε + ⇔ 27 8 ⋅ a 1 ⋅ a 2 ≈ 2 ⋅ [ 3 4 a 1 ] 2 + 27 4 ⋅ a 3 & a 1 = 3 A ˜ 2 4 A ˜ 1 , a 2 = A ˜ 3 2 A ˜ 1 , a 3 = A ˜ 4 4 A ˜ 1 (35)

What we could do would be to try to ascertain if this is possible in the regime of space-time going up to the electroweak regime in cosmology. Our supposition as to the transfer between the regime of imaginary mass, which may involve Faster than light transference of “information” to the Higgs we believe may be similar to the physics, given in page 70 of Ohanian, and Ruffini [

∂ v T u v = f u (36)

i.e. the relativistic correction to Newton’s second law. The force, f, in Newtonian physics becomes force density, or four force per unit volume arising from forces external to the system.

This is the trick we are using, i.e. assuming that there is an external to the initial universe recycling of matter-energy, with the idea of mass, initially re injected into the system. To get an idea of what the author is suggesting, we recommend that readers peruse [

Having said this, and specifically enjoining readers that this discussion has to be confirmed and vetted with numerical studies, we then will go to how we think this set of ideas may impact a model of Dark matter, called “inert doublet Dark matter” which is the next section of our manuscript. We will first, in [

In [

V ( H , S ) = m 2 2 ⋅ H † H + λ 4 ⋅ ( H † H ) 2 + δ 1 2 ⋅ H † H S + δ 2 2 ⋅ H † H S 2 + δ 1 2 ⋅ m 2 λ S + κ 2 2 S 2 + κ 3 3 S 3 + κ 4 4 S 4 . (37)

This leads to a total Lagrangian for the Higgs singlet model

ℑ = ℑ S M + 1 2 ∂ μ S ∂ μ S − ( m 2 2 ⋅ H † H + λ 4 ⋅ ( H † H ) 2 + δ 1 2 ⋅ H † H S + δ 2 2 ⋅ H † H S 2 + δ 1 2 ⋅ m 2 λ S + κ 2 2 S 2 + κ 3 3 S 3 + κ 4 4 S 4 ) . (38)

Here, we have that h is the physical Higgs field, v = 246 GeV , and the vacuum expectation value (vev) of scalar H is defined by m and λ as − 2 m 2 / λ , with H defined as given below

H = 1 2 ( 0 v + h ( x ) ) . (39)

Furthermore, [

δ 2 2 ⋅ H † H S 2 = δ 2 2 ⋅ ( v 2 S 2 2 + v h S 2 + h 2 S 2 2 ) (40)

where we also have a scalar mass term defined by, (if λ = δ 2 v 2 2 is a coupling between two scalars and the Higgs field, and δ 2 2 ⋅ H † H S 2 the interaction term

between the two scalars and the Higgs)

V mass = 1 2 ⋅ [ ( m h 2 = − m 2 = λ v 2 2 ) + ( m s 2 = κ 2 + λ v 2 2 ) ] . (41)

Our supposition, is that if we have an initially imaginary mass, that this will lead to

m 2 | initial = λ v 2 2 ⇒ λ = 2 m 2 | initial v 2 = δ 2 v 2 2 . (42)

The consequence would be in having

δ 2 = 4 m 2 | initial v 4 . (43)

Leading to a recasting of the coupling of the two scalars and the Higgs field, to read as follows

λ = 2 m 2 | initial v 2 . (44)

What we are saying, is that when we have this, that we will be referring to m 2 | initial , which is due to the imaginary mass contribution being dominant, as a direct result of information transfer, a.k.a. the same way we have quantum entanglement. So, to highlight the importance of this idea, we will next revisit what we can say from [

From [

Quote, pages 38-39 of [

We begin with a non-standard representation of mass from Plebasnki and Krasiniski [

If one assumes a metric given by [

d S 2 = [ exp ( C [ t , r ] ) ] ⋅ d t 2 − [ exp ( A [ t , r ] ) ] ⋅ d r 2 − R 2 ( t , r ) ⋅ [ d ϑ 2 + ( sin 2 ϑ ) ⋅ d ϕ 2 ]

(45)

Pick, in this case, R is equal to r, usual spatial distance, due to the following argument given below

S ( surface ) = 4 π ⋅ R 2 ( x , t ) ⇔ R 2 ( x , t ) = square , of areal radius ⇔ R 2 ( x , t ) = r 2 ( x , t ) . (46)

Leading to an effective mass which we can define via page 296 of [

m ( effective ) = C 2 ( r , t ) 2 G ⋅ ( R + ( exp ( − C ( r , t ) ) ⋅ R ⋅ ( ∂ R ∂ t ) 2 − ( exp ( − A ( r , t ) ) ) ⋅ R ⋅ ( ∂ R ∂ r ) 2 + Λ ⋅ R 3 3 ) . (47)

This expression for an effective mass would be zero if the R goes to zero, but we are presuming due to Beckwith, [

Furthermore, to get this in terms of R = r, that the above is, then at (Friedman equation Hubble parameter H, with H = 0 re written as, if R = r = Planck length),

m ( effective ) | H ( Hubble ) = 0 = C 2 ( r , t ) 2 G ⋅ ( r 2 − ( exp ( − A ( r , t ) ) ) ⋅ r + Λ ⋅ r 3 3 ) | r ∝ l ( Planck ) ≈ C 2 ( r = l Planck , t Planck ) 2 G ⋅ ( l Planck 2 − ( exp ( − A ( l Planck , t Planck ) ) ⋅ l Planck + Λ ⋅ l Planck 3 3 ) | r ∝ l ( Planck ) (48)

End of quote.

We say that in the very early universe, perhaps before the electroweak era, that the above expression for mass, would have to be equivalent in “information” with respect to Equation (42), i.e. note that in [

Quote, [

We claim that this effective mass should be put into the following wave function at the boundaries of the H = 0 causal boundary, as outlined by Beckwith, in [

ψ Entangled ∝ 1 8π ⋅ { 1 − 1 3 ! n A 2 ⋅ π 2 ⋅ r Prior 2 ⋅ ℏ 2 2 m E 0 } ⊗ { 1 − 1 3 ! n B 2 ⋅ π 2 ⋅ r Present 2 ⋅ ℏ 2 2 m E 0 } + 1 8π ⋅ { 1 − 1 3 ! n A 2 ⋅ π 2 ⋅ r Present 2 ⋅ ℏ 2 2 m E 0 } ⊗ { 1 − 1 3 ! n B 2 ⋅ π 2 ⋅ r Prior 2 ⋅ ℏ 2 2 m E 0 } . (49)

End of quote.

What we are suggesting is that we may have something very similar as to the formation of mass problem, for a Higgs, with the positions r(prior) referring to prior universe positions, and r(present) referring to positions as of about the formation(start) of the electro weak era. The indices (A) and (B) refer to information states in the prior to present universe which would be relatively instantaneously transferred from the prior to the present universe, i.e. the position r(present) would be about at the start of the electroweak era in the present universe, i.e. an “imaginary” mass corresponding to “information” packet which is almost instantaneously transferred.

The entanglement protocol is also described in [

Our supposition is that imaginary mass, not being measurable in our space-time, would be in its transfer from a present to a prior universe, similar in the entanglement information exchange as referenced in [

This is a detail which has to be worked out. The numbers n(A) and n(B) in this case would refer to specifically numbered states of matter-energy packets whose information would be exchanged almost instantaneously from a prior to a present universe. In this case, our supposition is that any would be imaginary mass, going in between the prior to present universe, would probably be unmeasurable, until they went to being a real valued traditional mass value.

This is our supposition. And what we would be working with. Our next goal will be to see if there is any linkage of this topic with that as to gravity waves and gravitation.

So far, we have discussed the formation of a Higgs mass, with, prior to and then up to the start of the Electro weak era having at least some imaginary mass component.

To rephrase our question, what would happen if the mass went from an unmeasurable by physics instruments imaginary value to real value, say as in the middle of the electroweak transformation? We assert that such an emergence, from imaginary to real values, as we assert could occur through evaluation of Equation (1) would in itself create turbulence in space-time which may lead to dark energy, i.e. to see what we are referring to, we go to [

Our supposition is that the emergence of Higgs mass due to the following transformation created local space-time turbulence, hence thereby generation of GW, and what we suppose are gravitons.

So we will make the following linkage between a change from imaginary (Higgs boson mass), to real (Higgs boson mass), and then the generation of GW, from what would be a phase transition induced in the Electro-weak era, as we go directly to real valued Higgs boson mass, from an earlier imaginary valued Higgs boson mass. This would induce turbulence, in local space time which would also then induce GW, and by default, massive gravitons.

Chiara Caprinia, Ruth Durrer and G’eraldine Servant [

Quote, page 2 of [

Although the first GW detections will come from astrophysical processes, such as merging of black holes, another mission of GW astronomy will be to search for a stochastic background of GWs of primordial origin. An important mechanism for generating such a stochastic GW background is a relativistic first-order phase transition [

End of quote.

Note that in [

β ˜ ˜ ⋅ h ˜ ˜ ~ 8 π G T ˜ ˜ (50)

h ˜ ˜ = AmplitudeofGWproducingPerturbation 1 / β ˜ ˜ = CharacteristictimewhichGWproducingPerturbationoccurs ∝ 1 / ( ω GW = Gravitationalwavefrequency ) T ˜ ˜ = EnergymomentumtensorofsourceforcreationofGW ρ GW = ( GWenergydensity ) = ( 1 / β ˜ ˜ ) ⋅ 8 π ⋅ G ⋅ T ˜ ˜

In [

What we will be doing as a future project will be to make a convincing argument as to what T ˜ ˜ should be following an emergence of imaginary (initially) Higgs bosons, transferred to real Higgs bosons, in the middle of the Electroweak era.

We submit that the emergence from imaginary to real space time of such Higgs bosons, would precipitate the creation of the bubbles in space time which would subsequently collide, thereby generating gravitational waves and turbulence, with the energy of such collisions, and other factors of Gravitational generation leading to massive graviton production, i.e. this is our prediction,

Prediction. Should the author in careful analysis, create due to the emergence of space-time bubbles, co currently with Higgs bosons emerging from imaginary to real space time, in the electroweak era be able to write out T ˜ ˜ as a suitable mass-energy tensor, then this analysis will be co currently solving not only where the information (similar to the [

First of all, this paper supposes that if there was implementation of Equation (1) of this document, there would be largely, due to the smallness of the spatial dimensions, a regime in space-time where solving the Higgs boson mass could lead to imaginary Higgs masses, i.e. an argument sourced from Baez [

1) 1^{st} question

Can rigorous numerical simulations verify in fact, what we are supposing as to how mass could become at least in part imaginary, as solutions to the Equation (1) nonlinear Higgs mass solver? Qualitative solutions to this, in terms of what is known about the numerical procedures for solution to cubic and quartic equations were referenced. We also state that many of the integrals in the coefficients in Equation (1) will be needed to be solved via Gaussian quadrature, as given in Equation (20) which puts a premium upon Δ r .

2) 2^{nd} question

To vet and to analyze is how far we can go to solve Δ r . This means confirmation

of Δ r ≈ | ℏ δ t Δ E | ⋅ l planck 2 . Note that this is akin to, with caveats to picking, in the

case of Equation (50), i.e. we think that the energy momentum tensor, as talked about in Equation (50).

3) 3^{rd} question

If we can solve for the shift in energy, and we discuss that about Equation (21) and Equation (22) we will be taking steps to obtain the energy momentum tensor of Equation (50). Can this be done?

I.e. in addition, what we did, is to invert the procedure as to solve the Higgs mass problem and to argue that its mass emergence from imaginary, to real values would be the template as to create in local space time the bubbles, which can collide in the electroweak era to create gravitational physics, which could create relic GW and hopefully massive gravitons.

4) 4^{th} question

Can we state that emergence of imaginary Higgs mass to real Higgs mass create bubbles of space-time in the electro weak era?

5) 5^{th} question

We have argued in terms of entanglement, in terms of transferal of information from a past to a present universe. Is entanglement linkable to the following supposition?

Our supposition is that entanglement instantaneous transfer of information, would be akin to having Faster than Light Tachyons go from the prior to the present universe, with a dump of that information into real mass valued Higgs bosons, in the electro weak era.

Can a rigorous mathematical model of an answer to this question above be ascertained?

Being able to confirm this would be largely updating Equation (49), which is from [

6) 6^{th} question

Is of DM, i.e. the singlet DM hypothesis, i.e. the key question centers about what we call Equation (44), i.e. is Equation (44) actually allowable?

We claim that if Equation (44) is true, that we will have CMBR signatures to pick up and that this will have a lot to do with.

7) 7^{th} question

How reliable is [

In addition, how could we link our inquiry into issues brought up in [

Corda’s document, [

Finally is the question of extra dimensions, i.e. [

This work is supported in part by National Nature Science Foundation of China grant No. 11375279.

Beckwith, A.W. (2018) New Procedure for Delineating the Mass of a Higgs Boson, While Interpolating Properties of the Scalar Singlet Dark Matter Model. Journal of High Energy Physics, Gravitation and Cosmology, 4, 96-122. https://doi.org/10.4236/jhepgc.2018.41010