Phasic Structure of the Standard Model

Abstract

We present a physical relationship able to justify by itself the whole spectrum of fundamental particles of matter (fermions), that is, the hierarchical structure, the value of their masses, the basis of those values and the analytical and positional relation of the same. A hierarchy supported on an element of periodicity or sequence of differentiated states (phases) that makes it resemble a periodic table of elementary particles, which allows us to define material states not well defined until now and others unregistered, with which it is precisely evidenced that there has to exist a particle M=171.87 eV, or an equivalent energy entity, which represents a common zero generation to (anti)quarks and charged (anti)leptons, as well as a mediator particle in the connection of these with (anti)neutrinos, for which, by applying this methodology, we obtain a massive base ν1,ν2,ν3, consistent with the theoretical and experimental requirements for neutrino oscillation, which allows us to explain the CP violation. Previously, we obtained, by means of the quantum wave formulation, a symmetrized wave packet (SWP) and an associated energy transmutation equation, equivalent, although more general and precise, to the one obtained by means of a corpuscular treatment, this one that we will use and accounts for all the transitions between particles.

1. Introduction

The description of particles in the quantum mechanics environment is good for particles in a certain state of confinement, but not for free particles. In this case, by means of the wave functions, either of travelling waves or of waves groups, only a partial or not free correspondence has been achieved, and always by means of the use of wave quantum mechanics, since the matrix mechanics, despite the mathematical equivalence between one and the other, carried out by Von Neumann [1], is not applicable to these continuous states of energy or, said in another way, to the case of the free particle not subject to potentials, which does not lend itself to the application of the mathematical methods of Matrix Mechanics since its very essence is based on Fourier expansions valid for periodic systems that necessarily require the particle to be confined to a bound system [2]. It is precisely this non-applicability of the matrix quantum to the problem and its predominance, however, for many other issues as a consequence of its versatility, which has caused in a certain sense that research routes aimed at overcoming the problem are not sought, i.e. aimed at providing a true theoretical support to the wave-corpuscle duality, in accordance with the experimental evidence and the relevance of the question. To some extent, it has been decided to explore to satiety that which allows itself to be treated by matrix formalism, and what is not has been left aside, in spite of the fact that it is a particular area (unapproachable by other methods) capable of providing information analogous to reality, since the wave function is a physical entity and not a mathematical representation like that constructed with functions in matrix mechanics. Consequently, we know that particles are created by means of a creation operator, but we do not know what that creation act consists of because we know nothing about the created object.

It is for all these reasons that, going back a century, we are going to return to the path from the beginning, and give it another focus, another push, from the conviction that this reality, that of duality, is a superior (more primordial) reality by far to other realities or work schemes of physics that after many years of work know how to say many phenomenological things, but very few of the matter itself or its organization.

The theoretical pretensions of this exposition will be divided here into two objectives or developments: one in which by means of a wave treatment, we will find the same equations that govern corpuscular dynamics in all its energetic forms, but already related (as terms of a single equation) and with additional information of a wave character; and another development in which we will treat part of this additional information, referring mainly to the term of kinetic energy and the crucial role it plays in the energy conversion and in the creation-organization of the material particles of our universe or physical reality, while the study of mass (formed by means of wave constituents) and the rest of the energy forms associated with the other terms of the equation will be the object of another study. More specifically, we will demonstrate with this treatment and the aforementioned equation that all matter and all energy have a wave character, which is present both in their creation (which is precisely the process carried out from a wave packet in the first development) and in all processes or interactions, as a consequence of being precisely the resident wave component, which we call phase factor (the wave is not extinguished with the formation of matter), which allows them, or, to be more exact, because every energetic interaction and every possibility of constituting itself in particle is precisely an interaction carried out through that perpetual wave component.

In accordance with these purposes, we will see that in our itinerary we will arrive, indeed, as a first-order objective to the mentioned equation [Eq. (32)], but that it will lose protagonism (it will be developed term by term in another work) because the protagonism is taken by the phase factor of a phase $\varphi$ [defined from the first one through Eq. (37)], as the motor of all energetic transformation, which, however, also loses the protagonism (it will be developed for all the families of particles in detail in another work) because this importance is taken by the relation between phases $\chi$ defined in Eq. (39) from the previous one (being reduced to a formula without system constants), for being, according to the title, our main objective and of greater repercussion, or of a more contrastable impact due to the analytical character of what we want to postulate as true.

Indeed, the relationship between phases will allow us to establish by means of a simple formula between the masses of a triad (such as that of the three generations) a scheme or organization of all the material particles, that is, their masses, calculated analytically, and the distance between them in phase distances within that scheme, which will allow us to determine which points of it are likely to contain a mass (located or locatable).

In this context, besides verifying all the existing fermions, and bringing the adjustment of the masses of the massive neutrinos to experimental order with an unprecedented degree of theoretical precision, to the point of being able to determine their hierarchical ordering (which Katrin’s experimental precision has not resolved), we will find a particle or energy entity at a central or convergence point, necessary in this scheme, which curiously occupies the gap or energy jump of five orders of magnitude existing between the first two flavours of neutrinos. A central particle that will turn out to be the generating or intermediate element, in such a way that all the others are reached by it to a certain phase, as it happens with the set of three (locatable) particles or genuinely primordial masses that constitute a natural neutrino base with which we can compose the base of massive neutrinos already referred to (and a transformation matrix), in the same way that this is later transformed into the different flavours by another one.

A particle and a methodology provide answers to other diverse questions. Questions such as the seesaw mechanism or the CP violation, on which it is not intended to reach categorical conclusions within the scheme of work or to address it exhaustively, only to show its connection and potential, and to leave a note, advance or point of interest.

2. Symmetrized wave packet (SWP)

A free particle can be represented by a one-dimensional or elemental wave, which we call the plane wave, $\psi =A{\mathrm{e}}^{i\left(\mathit{kx}-\mathit{wt}\right)}$, which describes the possibility of the particle to be located in space in the interval $\left[-\infty ,+\infty \right]$, as occurs, for example, with its real part $\psi =\mathit{\text{ACos}}\left(\mathit{kx}-\mathit{wt}\right)$, called also travelling wave, which we can easily represent (see Figure 1), but which at the same time evidences the impossibility by means of it of determining in what position of space the particle is, as a consequence of the impossibility of normalizing the wave function in the mentioned $\left[-\infty ,+\infty \right]$ interval, that is, of integrating in the interval and finding a defined value for $A$, of which results an insufficient description. We could say, in accordance with the principle of indeterminacy, that if the quantity of motion ($p$) is fixed exactly, the position ($x$) of the particle is completely undefined. And we can say that it is this impossibility of being normalized that makes (beyond the Fourier reference) matrix mechanics or Neumann unification (isomorphism of mathematical spaces $L_2$ and ${\mathrm{\ell }}_2$) not applicable for plane waves associated with free particles, since it requires that the functions be square integrable, and the failed normalization process is nothing but the solution of an integral or sum of this type [3]. A question, by the way, that makes that, paradoxically, the vectors of that mathematical space (of quantum mechanics) do not belong to the space itself (see Section A-3 of Chapter 2 in [4]).

Figure 1.

Traveling wave.

This is somehow solved by the wave packets of different typology or distribution (Gaussians, Lorentzians) that can be obtained, since they are equivalent (see Chapter 20 of [5]), by constructing light pulses (wave train of finite length) or by superimposing many waves of approximate wave vector, that is, of different values $p$, and consequently of an undetermined value $p$, which restricts the indeterminacy of $x$ to an interval $\mathit{\Delta x}$, accounting for a defined location and a certain sense of occupation (the wave of matter of de Blogie).

These distributions are functional arrangements that make the original wave extinguish beyond a few boundaries on either side of a central point, so that it is a discrete spatial arrangement that can even evolve over time [6]. Explicitly, by applying the function $\theta =exp\left(-x^2/20\right)$ (see Figure 2), to the previous function $\psi$, we get (see Figure 3):

Figure 2.

Amplitude modulation.

Figure 3.

Wave packet.

This is what we also know by wave modulation or modulated wave used for the transmission of information associated precisely with a certain energy distribution. It is this way of distributing energy in the group that is intended to be used to identify the particle, as an evident form of energy information, in it.

The attempts, however, to associate this wave model with the real constitution of the elementary particle have failed, given that, contrary to what happened with the particles they try to represent, they are extinguished in time (see p. 64, complement G_I of [4]), which is unavoidable in the current paradigm.

In the first place, we are not going to try to overcome in our analysis the inescapable in the current paradigm, but to characterize corpuscular behaviour by using some kind of distribution. From there on, the results will be those that force us to interpret things in a certain way, among which there may be some paradigmatic. For this, and in particular way, we are going to use the superimposition of a set of plane waves of variable frequency and constant amplitude, such as:

$${\mathrm{\Psi }}_k\left(x,t\right)=\underset{-\infty }{\overset{+\infty }{\int }}{e}^{i\left(w_{k}t-\mathit{kx}\right)}\mathit{dk},$$E1

which for the cosine of the complex exponential, restricted to the interval $k=k_0\pm \mathit{\Delta k}/2$, gives rise to a modulated monochromatic wave of frequency $w_0$,

$${\mathrm{\Psi }}_k\left(x,t\right)=B\frac{sin\left[\mathrm{\Delta }k/2\left(\mathit{vt}-x\right)\right]}{\mathit{vt}-x}{e}^{i\left(w_{0}t-k_{0}x\right)},$$E2

that is, to a plane wave and a modulating or envelope factor accompanied by a normalization constant $B$, which includes the factor (×2) that should accompany the sine function. The function (1) has in reality the form (Taylor series):

$${\mathrm{\Psi }}_k\left(x,t\right)=e^{i\left(w_{0}t-k_{0}x\right)}\underset{k_0-\mathrm{\Delta }k/2}{\overset{k_0+\mathrm{\Delta }k/2}{\int }}{e}^{i\left[\left(k-k_0\right)/2\left(\mathit{vt}-x\right)\right]}\mathit{dk},$$E3

whose complete and mathematically most manageable solution is (see Chapter 12 of [5]):

$${\mathrm{\Psi }}_k\left(x,t\right)=B\left(i\right)\frac{e^{-i\left[\mathrm{\Delta }k/2\left(\mathit{vt}-x\right)\right]}}{\mathit{vt}-x}{e}^{i\left(w_{0}t-k_{0}x\right)},$$E4

to which we can later demand its real character, and we have added the factor $\left(i\right)$ precisely so that in this case, the real part is analogous to that presented in (2).

On this basis, we can build two functions that keep a certain symmetry. The first can be one similar to (4) but reversing the propagation of the travelling wave, that is:

$$\begin{array}{ll}{\mathrm{\Psi }}_1\left(x,t\right)& =B\left(i\right)\frac{e^{-i\left[\mathrm{\Delta }k/2\left(\mathit{vt}-x\right)\right]}}{\mathit{vt}-x}{e}^{-i\left(w_{0}t-k_{0}x\right)}\\ & =B\left(i\right)\frac{e^{-i\left(M/2\right)}}{\mathit{vt}-x}{e}^{-i\left(N\right)}=B\left(i\right)\frac{e^{-i\left(M/2\right)}}{\mathit{vt}-x}{\varphi }^{-1}\left(\mathrm{x},\mathrm{t}\right),\end{array}$$E5

a transformation that is analogous to having originally considered a function conjugated to the original (1) with the integration limits transposed:

$${\mathrm{\Psi }}_1\left(x,t\right)={\overleftrightarrow{\mathrm{\Psi }}}_k^{\ast }\left(x,t\right)=\underset{+\infty }{\overset{-\infty }{\int }}{e}^{-i\left(w_{k}t-\mathit{kx}\right)}\mathit{dk}.$$E6

The second, with respect to the original one (4), would be a wave function that preserves the propagation of the travelling wave but inverts the envelope:

$$\begin{array}{ll}{\mathrm{\Psi }}_2\left(x,t\right)& =B\left(-i\right)\frac{e^{i\left[\mathrm{\Delta }k/2\left(\mathit{vt}-x\right)\right]}}{\mathit{vt}-x}{e}^{i\left(w_{0}t-k_{0}x\right)}\\ & =B\left(-i\right)\frac{e^{i\left(M/2\right)}}{\mathit{vt}-x}{e}^{i\left(N\right)}=B\left(-i\right)\frac{e^{i\left(M/2\right)}}{\mathit{vt}-x}\varphi \left(\mathrm{x},\mathrm{t}\right),\end{array}$$E7

action that is equivalent (according to the corresponding Taylor series) to exchanging the limits of integration in the original function, that is:

$${\mathrm{\Psi }}_2\left(x,t\right)={\overleftrightarrow{\mathrm{\Psi }}}_k\left(x,t\right)=\underset{+\infty }{\overset{-\infty }{\int }}{e}^{i\left(w_{k}t-\mathit{kx}\right)}\mathit{dk},$$E8

The result will be two wave functions, (5) and (7), on which we will be able to carry out a combination, which will be nothing more than a process of symmetrization, that is to say, an adaptation similar to that carried out on certain quantum observables to ensure their classical validity. In this case, we will have two plane waves propagating in opposite directions that intersect and interfere into the mutual space of the envelopes, in a constructive way for the real component and destructive in what refers to the imaginary one, reaching its only real character naturally and not, as in (4), by means of physical criteria or impositions. A situation that we can associate to a real composition of the waves, that is, to the way of structuring themselves physically (that is, our thesis, in fact) and not only to a well-disposed operational model, which is also to participate in the operational advantages of the forming functions, already mentioned. Therefore,

$$\begin{array}{ll}\mathrm{\Psi }\left(x,t\right)& =\frac{\left({\mathrm{\Psi }}_1\left(x,t\right)+{\mathrm{\Psi }}_2\left(x,t\right)\right)}{\sqrt{2}}=B\frac{\left(i\right)e^{-i\left(M/2\right)}{e}^{-i\left(N\right)}+\left(-i\right)e^{i\left(M/2\right)}{e}^{i\left(N\right)}}{\sqrt{2}\left(\mathit{vt}-x\right)}\\ & =\left(i\right)B\frac{e^{-i\left(M/2\right)}{e}^{-i\left(N\right)}-e^{i\left(M/2\right)}{e}^{i\left(N\right)}}{\sqrt{2}\left(\mathit{vt}-x\right)}=\left(i\right)B\frac{e^{-i\left(M/2\right)}{\varphi }^{-1}\left(x,t\right)-e^{i\left(M/2\right)}\varphi \left(x,t\right)}{\sqrt{2}\left(\mathit{vt}-x\right)}\\ & =\left(i\right)B\frac{\left[{\vartheta }_1{\varphi }^{-1}\left(x,t\right)-{\vartheta }_2\varphi \left(x,t\right)\right]}{\sqrt{2}}=\left(i\right)B\frac{\left[{\vartheta }_1{\varphi }_1-{\vartheta }_2{\varphi }_2\right]}{\sqrt{2}},\end{array}$$E9

where ${\varphi }_2\equiv \varphi ,{\varphi }_1\equiv {\varphi }^{-1},$ the value $\sqrt{2}$ is a new normalization factor that leaves the initial $B$ factor intact, which we can calculate on the initial function (4), expressed in a more generic way (without the factor $i$, which was intentionally introduced), for $t=0$ and for $b\equiv 2/\mathrm{\Delta }k$, as:

$${\mathrm{\Psi }}_1\left(x\right)=B\frac{e^{i\left(x/b\right)}{e}^{i\left(k_{0}x\right)}}{x},$$E10

which we will be able to normalize,

$$\begin{array}{ll}\underset{-\infty }{\overset{+\infty }{\int }}\mathit{fdx}& =B^2\underset{-\infty }{\overset{+\infty }{\int }}{{\mathrm{\Psi }}_1}^{\ast }\left(x,0\right){\mathrm{\Psi }}_1\left(x,0\right)\mathit{dx}=B^2\underset{-\infty }{\overset{+\infty }{\int }}{\left|{\mathrm{\Psi }}_1\left(x\right)\right|}^2\mathit{dx}\\ & =B^2\underset{-\infty }{\overset{+\infty }{\int }}\frac{e^{i2\left(x/b\right)}}{x^2}{\varphi }^{\ast }\left(x,t\right)\varphi \left(x,t\right)\mathit{dx}=B^2\underset{-\infty }{\overset{+\infty }{\int }}\frac{e^{i2\left(x/b\right)}}{x^2}\mathit{dx}=1,\end{array}$$E11

and solve by means of the integral calculus¹:

$$B^2\oint \frac{e^{i2\left(\mathrm{z}/b\right)}}{z^2}\mathit{dz}=\underset{-\infty }{\overset{+\infty }{\int }}\mathit{fdx}+∲f=\underset{-\infty }{\overset{+\infty }{\int }}\mathit{fdx}=B^2\left(-\mathit{\pi i}\right)\frac{2i}{b}=B^2\frac{2\pi }{b}=1.$$E12

From which it results:

$$B={\left(\frac{b}{2\pi }\right)}^{1/2}={\left(\frac{1}{\mathit{\Delta k\pi }}\right)}^{1/2}$$E13

3. Energy form of the SWP

As the aim of our discussion is to obtain the corpuscular energy from the wave function, we can state that the target is to differentiate from the value of the total energy those values that are clearly undulatory, and incontrovertible, given the default undulatory character of our starting physical system, and to distinguish them from the remaining values, that is to enunciate, to separate the value of the energy in at least two energetic terms, the strictly undulatory and the one that is not. For this, let us again take the function of the symmetric wave packet (SWP) given in (9) and calculate with it the expectation value of the energy, which will be given by:

$$\overline{E}=\int \mathit{\text{ξPdx}}=\int {\mathrm{\Psi }}^{\ast }\left(x,t\right)i\mathrm{\hslash }{\partial }_t\mathrm{\Psi }\left(x,t\right)\mathit{dx},$$E14

and which must coincide with the energy of the system described by a free particle such as the electron, including that derived from its purely undulatory character.

The expected value is an application of the operator energy $\xi$ over a probability density $P$, which must be a positive real value. In fact, the normalization process (11) is intended by the conjugation of the function to achieve a positive real value coinciding with this density value. This is:

$$B^2\underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{\Psi }}^{\ast }\left(x,0\right)\mathrm{\Psi }\left(x,0\right)\mathit{dx}=B^2\underset{-\infty }{\overset{+\infty }{\int }}{\left|\mathrm{\Psi }\left(x\right)\right|}^2\mathit{dx}=\underset{-\infty }{\overset{+\infty }{\int }}\mathit{Pdx}=1$$E15

This directs us to reconsider the normalization process and observe that it is done through conjugation to achieve an integrable square function, which brings forth a positive real value. That square integrable function is, as seen in (11), the one that results from annulling the imaginary part of the original function through the relation ${\varphi }^{\ast }\left(x,t\right)\varphi \left(x,t\right)=1,$ being usually that resulting function is a real function since the imaginary factor has been weeded out by this method.

But neither does the resulting function necessarily have to be real, as in fact is the case in (11), nor does it necessarily have to be achieved by the conjugation of the complete original function as it also happens in our case. That is, the conjugation is a method (the most usual) but not an inescapable condition or an end in itself, the latter being to reach a positive real value for the probability density and the consequent normalization constant. For this very reason, and because of the possibility of solving the value of the complex integral as it is (because it has a solution), we have only needed to conjugate the plane wave and not the envelope. Such a resolution is a consequence of the extrapolation of the complex behaviour to the real one, or, in other words, of the possibility of carrying it out because the contribution of the imaginary part is null, as shown in $∲f$. That is, taking $\mathrm{\Psi }=\vartheta \times \varphi$, generically, we have not made ${\mathrm{\Psi }}^{\ast }={\vartheta }^{\ast }\times {\varphi }^{\ast }$ to obtain the real value, since this would have led us to ${\mathrm{\Psi }}^{\ast }\times \mathrm{\Psi }=1$, but ${\mathrm{\Psi }}^{\ast }=\vartheta \times {\varphi }^{\ast }$, being the idiosyncrasy of the integral itself that is responsible for accounting for the value (main Cauchy’s) where it has, that is, on the real line.

This real character that is derived from the properties of the integral in a natural way, and that we originally adopted in (5) and (7) by means of physical criteria, we will reach it in what follows revalidating ${\mathrm{\Psi }}^{\ast }=\vartheta \times {\varphi }^{\ast }$ and leaving (without imposing criteria) that the operative itself is the one in charge of rescuing the real value of the function where it corresponds without adulterating it, that is, without suppressing the imaginary part (forcing it to be real) nor conjugating it to obtain a real value in a forced way. All this can be summarized by saying that the probability density has the same form for any use of it.

With this in mind and taking into consideration what is already established for $M\equiv \left[\mathrm{\Delta }k\left(\mathit{vt}-x\right)\right]$, $N\equiv \left(w_{0}t-k_{0}x\right)$, $\varphi \left(x,t\right)\equiv e^{i\left(N\right)}={\varphi }_2={\varphi }_1^{-1}$ and ${\mathrm{\Psi }}_i={\vartheta }_i{\varphi }_i$, we can calculate Eq. (14). An equation what, according to the result obtained in Eq. (9), we can separate into its two components:

$$\begin{array}{ll}\overline{E}& =\int {\mathrm{\Psi }}^{\ast }\left(x,t\right)i\mathrm{\hslash }{\partial }_t\mathrm{\Psi }\left(x,t\right)\mathit{dx}=\left({\overline{E}}_1+{\overline{E}}_2\right)\\ & ={\left(i\right)}^2{B}^2\frac{\left[\int {\zeta }_1\left(x,t\right)\mathit{dx}-\int {\zeta }_2\left(x,t\right)\mathit{dx}\right]}{{\left(\sqrt{2}\right)}^2}\\ & =B^2\frac{\int \left({\zeta }_2\left(x,t\right)-{\zeta }_1\left(x,t\right)\right)\mathit{dx}}{2}=B^2\int \zeta \left(x,t\right)\mathit{dx},\end{array}$$E16

where it meets:

$$\begin{array}{l}\begin{array}{ll}{\zeta }_2\left(x,t\right)& ={{\mathrm{\Psi }}_2}^{\ast }\left(x,t\right)i\mathrm{\hslash }{\partial }_t{\mathrm{\Psi }}_2\left(x,t\right)={\vartheta }_2{{\varphi }_2}^{\ast }i\mathrm{\hslash }{\partial }_t\left({\vartheta }_2{\varphi }_2\right)\\ & ={\vartheta }_2{\varphi }^{-1}\left(x,t\right)i\mathrm{\hslash }\left[{\vartheta }_2{\partial }_t\varphi \left(x,t\right)+\varphi \left(x,t\right){\partial }_t{\vartheta }_2\right]\end{array}\\ =\frac{e^{i\left(M/2\right)}{e}^{-i\left(N\right)}}{\left(\mathit{vt}-x\right)}i\mathrm{\hslash }\left[\frac{e^{i\left(M/2\right)}{e}^{i\left(N\right)}}{\left(\mathit{vt}-x\right)}\right.\left({\mathit{iw}}_0+\frac{\mathit{iv}\mathrm{\Delta }k}{2}\right)-v\left.\frac{e^{i\left(M/2\right)}{e}^{i\left(N\right)}}{{\left(\mathit{vt}-x\right)}^2}\right]\\ =i\mathrm{\hslash }{e}^{i\left(M/2\right)}\left[\frac{e^{i\left(M/2\right)}}{{\left(\mathit{vt}-x\right)}^2}\right.\left({\mathit{iw}}_0+\frac{\mathit{iv}\mathrm{\Delta }k}{2}\right)+\left.\frac{e^{i\left(M/2\right)}}{{\left(\mathit{vt}-x\right)}^3}\left(-v\right)\right]\\ =\mathrm{\hslash }\left(\frac{e^{i\left(\mathit{vt}-x\right)\mathrm{\Delta }k}}{{\left(\mathit{vt}-x\right)}^2}\left(-w_0-\frac{\mathit{iv}\mathrm{\Delta }k}{2}\right)+\left.\left.\frac{e^{i\left(\mathit{vt}-x\right)\mathrm{\Delta }k}}{{\left(\mathit{vt}-x\right)}^3}\left(-\mathit{iv}\right)\right)\right]\right.,\end{array}$$E17

and

$$\begin{array}{l}\begin{array}{ll}-{\zeta }_1\left(x,\mathrm{t}\right)& =-{{\mathrm{\Psi }}_1}^{\ast }\left(x,t\right)i\mathrm{\hslash }{\partial }_t{\mathrm{\Psi }}_1\left(x,t\right)=-{\vartheta }_1{{\varphi }_1}^{\ast }i\mathrm{\hslash }{\partial }_t\left({\vartheta }_1{\varphi }_1\right)\\ & ={\vartheta }_1\varphi \left(x,t\right)i\mathrm{\hslash }\left[{\vartheta }_1{\partial }_t{\varphi }^{-1}\left(x,t\right)+{\varphi }^{-1}\left(x,t\right){\partial }_t{\vartheta }_1\right]\end{array}\\ =-\frac{e^{-i\left(M/2\right)}{e}^{i\left(N\right)}}{\left(\mathit{vt}-x\right)}i\mathrm{\hslash }\left[\frac{e^{-i\left(M/2\right)}{e}^{-i\left(N\right)}}{\left(\mathit{vt}-x\right)}\right.\left(-{\mathit{iw}}_0-\frac{\mathit{iv}\mathrm{\Delta }k}{2}\right)-\left.v\frac{e^{-i\left(M/2\right)}{e}^{-i\left(N\right)}}{{\left(\mathit{vt}-x\right)}^2}\right]\\ =i\mathrm{\hslash }{e}^{-i\left(M/2\right)}\left[\frac{e^{-i\left(M/2\right)}}{{\left(\mathit{vt}-x\right)}^2}\right.\left({\mathit{iw}}_0+\frac{\mathit{iv}\mathrm{\Delta }k}{2}\right)-\left.\frac{e^{-i\left(M/2\right)}}{{\left(\mathit{vt}-x\right)}^3}\left(-v\right)\right]\\ =\mathrm{\hslash }\left(\frac{e^{-i\left(\mathit{vt}-x\right)\mathrm{\Delta }k}}{{\left(\mathit{vt}-x\right)}^2}\left(-w_0-\frac{\mathit{iv}\mathrm{\Delta }k}{2}\right)-\left.\left.\frac{e^{-i\left(\mathit{vt}-x\right)\mathrm{\Delta }k}}{{\left(\mathit{vt}-x\right)}^3}\left(-\mathit{iv}\right)\right)\right]\right.\end{array}$$E18

From which results:

$$\begin{array}{ll}\zeta \left(x,t\right)& =\frac{{\zeta }_2\left(x,t\right)-{\zeta }_1\left(x,t\right)}{2}\\ & =\left[\frac{\mathrm{\hslash }vsin\left(\mathrm{\Delta }k\left(\mathit{vt}-x\right)\right)}{{\left(\mathit{vt}-x\right)}^3}-\frac{\mathrm{\hslash }v\mathrm{\Delta }kcos\left(\mathrm{\Delta }k\left(\mathit{vt}-x\right)\right)}{2{\left(\mathit{vt}-x\right)}^2}\right]\\ & +\left[-\frac{\mathrm{\hslash }{w}_{0}cos\left(\mathrm{\Delta }k\left(\mathit{vt}-x\right)\right)}{{\left(\mathit{vt}-x\right)}^2}\right]={\zeta }^m\left(x,t\right)+{\zeta }^{\varpi }\left(x,t\right),\end{array}$$E19

where we have taken into account the following relations:

$$2isinM=e^{\mathit{iM}}-e^{-\mathit{iM}}\mathrm{and}2cosM=e^{\mathit{iM}}+e^{-\mathit{iM}},$$E20

That is, it results an expression without imaginary components per se (which if we had not used SWP would have appeared and we would have had to discard them a posteriori as is usually done in physics), on which, by regrouping terms, the clearly undulatory part of the corpuscular one is clearly differentiated, as we have already mentioned, characterized by the parameters $w_0$ and $v$ respectively. Indeed, in the second term of the integrand, ${\zeta }^{\varpi }\left(x,t\right)$, the light characteristics of the integrand are revealed, and therefore, of the function as a whole, a matter that, given its nature, is normal. It will be, therefore, the term ${\zeta }^m\left(x,t\right)$ of the integrand that will be able to show, from its integral value, the double wave-corpuscle nature, despite having a light origin.

4. Energy treatment of the SWP

We have divided the proposed issue into several parts or actions. On the one hand, (which are actually two) to obtain a function with a particular distribution or configuration from which a plausible energetic form is derived, and on the other, which we are now addressing, to relate that energetic form to that of a particle. It is the moment, therefore, to face the task of obtaining a result, a corpuscular energetic expression from the previous one of undulatory origin. Put more precisely, it is time to obtain a recognizable corpuscular energy value, which is the only way to ensure that the wave function does indeed describe a particle.

The wave function is expressed on the variables $x$ and $t$, which is very good for the wave part in which we want to know its evolution in time and space on which its energy value depends. This does not happen for a free particle, which has an energy value, regardless of its position (unless there is a potential) or of time if it is at rest, and it is always at rest for its frame of reference. In addition to this, the calculation of energy as a function of the variable involves an error derived from the double consideration of the variable $x$, as a space for evolution and a space for occupation.

Although the question seems a priori unaddressable, we can nevertheless restrict to some extent this double consideration or ambiguity if we take into account that the wave function of the wave packet is practically cancelled out, that is, the probability that outside this interval a particle is represented (or, that the particle is outside this interval) and that outside this interval there are significant values of energy that contribute to the mean value of the particle is almost zero. Consequently, we can restrict this double consideration if we calculate the measured value to the interval of the group. This is:

$$\overline{E}=B^2\underset{-\infty }{\overset{+\infty }{\int }}\zeta \left(x,t\right)\mathit{dx}\cong B^2\underset{x_0-\mathrm{\Delta }x/2}{\overset{x_0+\mathrm{\Delta }x/2}{\int }}\left[{\zeta }^m\left(x,t\right)+{\zeta }^{\varpi }\left(x,t\right)\right]\mathit{dx},$$E21

However, we see that despite the approach we have not managed to overcome the duality of $x$ given that despite not existing outside the limits of integration (size) is still needed to assess the interior, also happening to the energy value differentiate positions internal that in no way can be differentiated into a particle, except for processes of formation of the same. In addition to this we see that, even assuming that in this way we were able to achieve a real energy value of the particle, at best it would only serve to evaluate the energy value in a state of rest (we have restricted the variable $x$ to size) but not for the dynamic states that we know as a function of the relative speed, which here is undefined, which forces us to go further in our way of reasoning the problem.

It is clear that in the case presented, the expected value is the energy value of the particle at rest ($t=0$), and that size $\mathrm{\Delta }x$ is coincident with the size of the particle in that situation. The question is whether, based on this, we might be able to find out the expected value in a general no-rest situation, that is, for another frame of reference. The answer is yes, and that it is so regardless of all the delimitations made above, if we consider the magnitude of that size $\mathrm{\Delta }x$ as the relativistic proper length of it, because, once the proper length $a=\mathrm{\Delta }x$ is defined, it is defined for all inertial systems by the Lorentz transform, in the consideration that any value $\mathit{vt}-x$ will be a fixed value for some speed $v$, and for any instant (being so, in particular, for $t=0$), that is, with

$$\mathit{vt}-x=a{\left(1-v^2/c^2\right)}^{1/2}=a{\gamma }^{-1},$$E22

we naturally move from a position space to a momentum space, transforming one nonspecific variable into another, $\upsilon$, of clear significance and better treatment.

Summarizing the approach made, it will only be possible to calculate the energy of a group of waves (that reside in their envelope) as a whole, if we consider it at a geometric level as a whole, that is, as an individual entity capable of representing or supporting the different states. In this case, and returning to the starting point of the original Eq. (14), known by integrating as a function of speed and the new limits of integration relative to the original $\pm \infty$, we only have to define the new integral for the expectation value of energy and calculate.

$$\begin{array}{ll}\overline{E}& =\underset{0}{\overset{c}{\int }}{\mathrm{\Psi }}^{\ast }\left(\upsilon \right)i\mathrm{\hslash }{\partial }_t\mathrm{\Psi }\left(\upsilon \right)\mathit{tdv}=\underset{0}{\overset{c}{\int }}{B}_c^2\zeta \left(\upsilon \right)\mathit{tdv}\\ & =B_c^2\left[T\right]\underset{0}{\overset{c}{\int }}\left[{\zeta }^m\left(\upsilon \right)+{\zeta }^{\varpi }\left(\upsilon \right)\right]\mathit{dv}=A^2\underset{0}{\overset{c}{\int }}\left[{\zeta }^m\left(\upsilon \right)+{\zeta }^{\varpi }\left(\upsilon \right)\right]\mathit{dv}\end{array}$$E23

where the normalization constant has been modified in two steps as a result of the change to the new variable and the integration interval assigned to it, comparable with the real half line, which, according to (13), meets:

$$B_c^2=2B^2=\frac{b}{\pi }=\frac{2}{\mathit{\Delta k\pi }}\text{and}{B}_c^2\left[T\right]=A^2$$E24

and where, considering (22), the integrands expressed in (19) take the form:

$${\zeta }^{\varpi }\left(\upsilon \right)=-\frac{\mathrm{\hslash }wcos\left(\mathrm{\Delta }k\left(\mathit{vt}-x\right)\right)}{{\left(\mathit{vt}-x\right)}^2}=-\frac{\mathrm{\hslash }wcos\left(\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right)}{a^2{\gamma }^{-2}}$$E25a

and

$$\begin{array}{ll}{\zeta }^m\left(\upsilon \right)& =\frac{\mathrm{\hslash }\upsilon sin\left(\mathrm{\Delta }k\left(\mathit{vt}-x\right)\right)}{{\left(\mathit{vt}-x\right)}^3}-\frac{\mathrm{\hslash }\upsilon \mathrm{\Delta }kcos\left(\mathrm{\Delta }k\left(\mathit{vt}-x\right)\right)}{2{\left(\mathit{vt}-x\right)}^2}\\ & =\frac{\mathrm{\hslash }\upsilon sin\left(\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right)}{a^3{\gamma }^{-3}}-\frac{\mathrm{\hslash }\upsilon \mathrm{\Delta }kcos\left(\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right)}{2a^2{\gamma }^{-2}},\end{array}$$E25b

that we can still put, according to (19), in a more compact way if we make: $\mathrm{\Phi }\equiv \left[\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right]$

$$\zeta \left(\upsilon \right)=\left[\frac{\mathrm{\hslash }\upsilon sin\left(\mathrm{\Phi }\right)}{a^3{\gamma }^{-3}}-\frac{\mathrm{\hslash }\upsilon \mathrm{\Delta }kcos\left(\mathrm{\Phi }\right)}{2a^2{\gamma }^{-2}}\right]+\left[-\frac{\mathrm{\hslash }wcos\left(\mathrm{\Phi }\right)}{a^2{\gamma }^{-2}}\right]$$E26

We can observe that after the change of variable, we have integrated time with the normalization constant ($A^2=B_c^2\left[T\right]$), which means that $t$ has no functionality as a variable either within or outside the integral, that is, that energy is not a function of time, and that the only functionality it has, which is assumed by the constant ($A^2$), is dimensional, that is, that of ensuring that energy remains dimensionally homogeneous. Time dimension $\left[T\right]$ which, according to (24), ultimately (for the moment) assumes $b$ through $A^2$. Resulting in this case:

$$A^2=\frac{b\left[T\right]}{\pi }=\frac{2\times \left[T\right]}{\mathit{\Delta k\pi }}\equiv \frac{\bcancel{b}}{\pi }$$E27

The reason is that although we are obliged to obtain the value of energy in a certain way, given that by definition $\xi \mathrm{\Psi }\left(x\right)=i\mathrm{\hslash }{\partial }_t\mathrm{\Psi }\left(x\right)$, any change of variable only needs to be dimensionally homogeneous, while the space in which it develops is arbitrary (not necessarily an extrapolation of the first), on the condition of having well-defined limits for normalization. In our case, the new space of study is not $x=x\left(\upsilon \right)$ but $\upsilon$ with its corresponding limits, which we could have set without the need to carry out all this transit. To realize this, we only have to remember that:

$$\begin{array}{ll}\overline{E}& =\underset{1}{\overset{2}{\int }}{\mathrm{\Psi }}^{\ast }\left(x\right)i\mathrm{\hslash }{\partial }_t\mathrm{\Psi }\left(x\right)\mathit{dx}=\underset{1}{\overset{2}{\int }}\frac{{\mathrm{\Psi }}^{\ast }\left(x\right)i\mathrm{\hslash }{\partial }_t\mathrm{\Psi }\left(x\right)\mathit{dx}}{{\mathrm{\Psi }}^{\ast }\left(x\right)\mathrm{\Psi }\left(x\right)\mathit{dx}}\\ & =\underset{1}{\overset{2}{\int }}\frac{{\mathrm{\Psi }}^{\ast }\left(x\right)i\mathrm{\hslash }{\partial }_t\mathrm{\Psi }\left(x\right)\mathit{tdv}}{{\mathrm{\Psi }}^{\ast }\left(x\right)\mathrm{\Psi }\left(x\right)\mathit{tdv}}=\underset{a}{\overset{b}{\int }}\frac{{\mathrm{\Psi }}^{\ast }\left(\upsilon \right)i\mathrm{\hslash }{\partial }_t\mathrm{\Psi }\left(\upsilon \right)\mathit{dv}}{{\mathrm{\Psi }}^{\ast }\left(\upsilon \right)\mathrm{\Psi }\left(\upsilon \right)\mathit{dv}},\end{array}$$E28

that is to say, that in the expectation or most probable value of a variable, elements that accompany it can be dispensed with if they are also eliminated in the denominator, that is, in the probability function (hidden by default). This can be done if in all cases it is normalized and complies with:

$$P=\underset{1}{\overset{2}{\int }}{\mathrm{\Psi }}^{\ast }\left(\upsilon \right)\mathrm{\Psi }\left(\upsilon \right)\mathit{tdv}=\underset{a}{\overset{b}{\int }}{\mathrm{\Psi }}^{\ast }\left(\upsilon \right)\mathrm{\Psi }\left(\upsilon \right)\mathit{dv}=1$$E29

In any case, and in a physical sense, we do not want to calculate $\overline{E}$ in the probabilistic space of the positions, but of the velocities, and this is not achieved by making a copy of the first, but by defining it correctly. Solving the terms of the integral in Eq. (23), expressed in Eqs. (25), as we do in Appendix A, we are left with a sum of integrals as follows:

$$\begin{array}{ll}\overline{E}& =A^2\int \zeta \left(\upsilon \right)\mathit{dv}=A^2\int {\zeta }^m\left(\upsilon \right)\mathit{dv}+{\zeta }^{\varpi }\left(\upsilon \right)\mathit{dv}\\ & =A^2\int \left({\zeta }_k\left(\upsilon \right)+{\zeta }_f\left(\upsilon \right)+{\zeta }_{\varpi }\left(\upsilon \right)\right)\mathit{dv}\\ & =A^2\left(\frac{\mathrm{\hslash }}{a^3}\right)sin{\left[\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right]}_{\upsilon }\underset{{\upsilon }_o}{\overset{{\upsilon }_f}{\int }}\frac{\upsilon }{{\gamma }^{-3}}\mathit{dv}\\ & -A^2\left(\frac{\mathrm{\hslash \Delta }k}{2a^2}\right)c^2\int \frac{cos\left[\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right]}{{\gamma }^{-1}}d\left({\gamma }^{-1}\right)\\ & +A^2\left(\frac{\mathrm{\hslash }w}{a^2}\right)c\int \frac{cos\left[\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right]}{{\left(1-{\gamma }^{-2}\right)}^{1/2}{\gamma }^{-1}}d\left({\gamma }^{-1}\right),\end{array}$$E30

in which the mass term of the integrand has been separated into two from another species, one kinetic and the other corpuscular or formation.

Expression that, taking into account the normalization factor (24), clarifying the notation of the variable by using $\nu \equiv {\gamma }^{-1}$ in the last two terms and distinguishing the arguments of the trigonometric functions in them through variable, with respect to that defined for (26), despite being mathematically equal in Eq. (30),

$$\mathrm{\Phi }\equiv \left[\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right]=\left[\mathrm{\Delta }k\left(\mathit{a\nu }\right)\right]\equiv {\mathrm{\Phi }}_{\nu },$$E31

to note that such arguments are affected by the integral through that variable, we can write as:

$$\begin{array}{ll}\overline{E}& =A^2\int {\zeta }^m\left(\upsilon \right)\mathit{dv}+{\zeta }^{\varpi }\left(\upsilon \right)\mathit{dv}\\ & =A^2\int \left({\zeta }_k\left(\upsilon \right)+{\zeta }_f\left(\upsilon \right)+{\zeta }_{\varpi }\left(\upsilon \right)\right)\mathit{dv}={\overline{E}}_k+{\overline{E}}_f+{\overline{E}}_{\varpi }\\ & =\left(\frac{\mathrm{\hslash }\bcancel{b}}{\pi a^3}\right)sin{\left[\mathrm{\Phi }\right]}_{\upsilon }\underset{{\upsilon }_o}{\overset{{\upsilon }_f}{\int }}\frac{\upsilon }{{\gamma }^{-3}}\mathit{dv}\left(a\right)\\ & -\left(\frac{\mathrm{\hslash \Delta }k\bcancel{b}}{2\pi a^2}\right)c^2\int \frac{cos\left[{\mathrm{\Phi }}_{\nu }\right]}{\nu }\mathit{d\nu }\left(b\right)\\ & +\left(\frac{\mathrm{\hslash }w\bcancel{b}}{\pi a^2}\right)c\int \frac{cos\left[{\mathrm{\Phi }}_{\nu }\right]}{{\left(1-{\nu }^2\right)}^{1/2}\nu }\mathit{d\nu },\left(c\right)\end{array}$$E32

that we can assimilate to the pro-corpuscular energy in its three forms of expression, kinetic, material and electromagnetic and to the (more general and precise) energy transmutation equation (ETE).

We will not go into the analysis of all the terms here, which will be the subject of another work. But we are going to put in evidence for what matters to us in this one that they are already beginning to take on, in effect, a corpuscular appearance, a question that we will appreciate more notably in the first of them if, dispensing for now with the factor $\mathrm{sen}{\left[\mathrm{\Phi }\right]}_{\upsilon }$, we develop it between the limit ${\upsilon }_o=0$ and any final $\upsilon$:

$$\begin{array}{ll}{\overline{E}}_k& =\left(\frac{\mathrm{\hslash }\bcancel{b}}{\pi a^3}\right)\underset{0}{\overset{\upsilon }{\int }}\frac{\upsilon }{{\gamma }^{-3}}\mathit{dv}=\left(\frac{\mathrm{\hslash }\bcancel{b}}{\pi a^3}\right)\underset{0}{\overset{\upsilon }{\int }}\frac{\upsilon }{{\left(1-{\upsilon }^2/c^2\right)}^{3/2}}\mathit{dv}\\ & =\left(\frac{\mathrm{\hslash }\bcancel{b}}{\pi a^3}\right)c^2\left(\frac{1}{{\left(1-{\upsilon }^2/c^2\right)}^{1/2}}-1\right)=m_r{c}^2\left(\frac{1}{{\left(1-{\upsilon }^2/c^2\right)}^{1/2}}-1\right)\\ & =m_r{c}^2\left(\gamma -1\right)=m_r\gamma c^2-m_r{c}^2=E_t-E_r=E_k,\end{array}$$E33

since it corresponds to the expression describing the relativistic kinetic energy of a particle of mass $m_r$, which in this case, taking into consideration (27), is:

$$m_r=A^2\frac{\mathrm{\hslash }}{a^3}=\frac{\mathrm{\hslash }\bcancel{b}}{\pi a^3}=\frac{\mathrm{\hslash }b\left[T\right]}{\pi a^3}=\frac{2\mathrm{\hslash }\times \left[T\right]}{\left(\mathrm{\Delta }k\right)\pi a^3}=\frac{2\bcancel{\mathrm{\hslash }}}{\left(\mathrm{\Delta }k\right)\pi a^3},$$E34

that is, a mass expressed in wave terms or, more accurately, expressed by its wave constituents, which as a whole validates the treatment carried out for $\left[T\right]$. Result (33) that implicitly establishes that the other two terms of Eq. (32) correspond to the energetic part of the process that is not kinetic, and that is necessarily related to the corpuscular formation or energetic transit between the initial and final objects of the (dis)encapsulated material, which will require an appropriate and differentiated definition of the limits of integration, but what for the moment we can associate with $E_r$ in Eq. (33) itself and to a pure electromagnetic form or pulse (32b) and (32c) respectively.

In this context, the term (32a) represents an extension or generalization of the relativistic dynamic expression (33), through the presence of an undulatory term $\mathrm{sen}{\left[\mathrm{\Phi }\right]}_{\upsilon }$ that is there, but is not perceived because its participation in phenomenology is null except for those training processes that culminate precisely, as we will see, with (because of) its annulment or cancellation.

Having achieved with this, at the expense of being properly characterized, what we initially intended, that is, a recognizable corpuscular energy expression from the wave expression, and even more so if we realize that (32) is of the form:

$$-E_{\mathit{em}}=-\mathit{hv}={\mathit{mc}}^2=m_0{c}^2+E_k.$$E35

which brings us face to face with the mechanism of energy transmutation and, more precisely, with the need for this additional element to achieve it, as the only one capable of making it possible for kinetic energy to dissipate suddenly from a certain value or, in other words, to lose the possibility of expressing itself in this way and be forced to take (reconfiguring itself in the other terms) another more essential (wave) or more optimized (mass) energy form, which is what gives rise, as we will study, to the whole spectrum of elementary particles.

An additional element by which it is evidenced that matter is not inert, since the process of formation of the corpuscle or material encapsulation represented in (32) does not annihilate the wave outside the limits of the particle or does not assimilate all of it in the process of integration (mathematically carried out in the last two terms) but leaves the “suplus” oscillating part as a vestige, and recovery mechanism, of its wave nature in the kinetic term, or in the mass itself that serves as support.

An additional element or factor that will allow us to reach, consequently, many of the fundamentals of Eq. (32), without the need to treat it, through the knowledge or the correct interpretation of the transition processes for each one of the arguments $\mathrm{\Phi }$, that takes us to the concept of phase, which we can put for $m_r$ by means of (34) in (31).

$$\begin{array}{ll}\left[\mathrm{\Phi }\right]& =\left[\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right]=\left[2b^{-1}\left(a{\gamma }^{-1}\right)\right]=\left[2b^{-1}\left(b^{1/3}{\left(\frac{\mathrm{\hslash }}{\mathit{\pi m}}\right)}^{1/3}{\gamma }^{-1}\right)\right]\\ & =\left[2b^{-2/3}\left({\left(\frac{\mathrm{\hslash }}{\mathit{\pi m}}\right)}^{1/3}{\gamma }^{-1}\right)\right]=\left[{\beta }_i\left({\left(\frac{1}{m}\right)}^{1/3}{\gamma }^{-1}\right)\right]\end{array}$$E36

from which will be derived, in agreement with certain conditions all phenomenology of the matter, this is, its creation and transformation, as we will show, in correspondence with the phase changes associated with said processes of transition.

To be more precise, it is the more general Eq. (37) that will allow us to account for all these processes,

$$\varphi ={\left[\mathrm{\Phi }\right]}^{-q}={\left[{\beta }_i\left({\left(\frac{1}{m}\right)}^{1/3}{\gamma }^{-1}\right)\right]}^{-q},$$E37

from which we conclude that all our previous development, with which we arrive at Eq. (36), is valid for the particles of charge $q=-1$, being through a generalization of that development (which here we obviate and we place to another work in order not to deviate from the pretensions of this one) how we would reach the last expression.

In a particular way we can find a value ${\mathrm{\Phi }}_1$ for one transition (${\gamma }_1$) and a value ${\mathrm{\Phi }}_2$ for another transition (${\gamma }_2$), being ${\gamma }_i$ the same as that which would circumstantially appear in the root Eq. (32a), and the relationship (ratio) between both transitions the relationship that really interests us for this purpose, by allowing us to achieve, through the simplification of common parameters, a scheme of all possible transitions, i.e. phases and particles.

It will be as a consequence of reaching this recognizable phase structure through a single condition

$$\mathrm{sen}{\left[\mathrm{\Phi }\right]}_{\upsilon }=0,$$E38

in the phase factor, which entails various conditions or states in the phase $\mathrm{\Phi }$, that we can certify what role this factor plays (already theoretically argued) in all the transitions and, properly speaking, its existence, that is to say, to assure the validity of the term Eq. (32a), as promoter, and that of the Eq. (32) of starting point (as well as the pertinence of the estimations carried out to obtain it), insofar as the equation certifies the non-casual character of said phase structure or the physical nature (reality) of the same, sustained in the energetic transit, and in the capacity to reach discrete material states by means of it.

In other words, the equation establishes simple phase relations that are only satisfied by the masses of the elementary particles that exist in our universe, which thus serve as an experimental verification.

5. Sole and simple relation for all particles in the standard model

Consequently, and based on the above expression, we will indeed look at a simple relationship among all particles of the Standard Model that has to do with an even simpler relation in a certain phase or state diagram, whereby the different generations of particles are in different phases from $\varphi \equiv \left({\varphi }_0,{\varphi }_1,{\varphi }_2,\dots \right)$ each other (which gives an idea of periodicity), and consequently phasic distances among them, which are determined by the limit value of the speed they can develop in the same, which in turn is determined by the inertial mass of the particle in question by means of the Lorentz factor $\gamma$. The above represents the outline of a theoretical framework derived from the one we have just seen and from the potential of the relationship itself. Here we will start from the relationship, taken as a formula, i.e. treated as if it were empirical, without going into theoretical questions beyond what is strictly necessary to the understanding of the factors that make it up, because by means of this relationship we are establishing a structured and hierarchical relation of the particles or organization of all matter with an argument as free of analysis or manipulation, simple and unequivocal, as that provided by the equation of a straight line. This approach, on the other hand, will allow us to carry out a summary treatment and achieve an overall view of the entire particle system.

Consequently, and now explicitly, we will develop what has already been expressed according to the following statement:

All fundamental particles of matter (fermions) in the SM satisfy the relationship of phases expressed as follows:

$$\begin{array}{l}\chi =\frac{{\varphi }_2}{{\varphi }_1}=\frac{{\left[{\beta }_2{\left(\frac{1}{m_2}\right)}^{1/3}{\gamma }_2^{-1}\right]}^{-q}}{{\left[{\beta }_1{\left(\frac{1}{m_1}\right)}^{1/3}{\gamma }_1^{-1}\right]}^{-q}}={\left[\frac{{\beta }_2}{{\beta }_1}{\left(\frac{\left(m_2^2/m_3^3\right)}{\left(m_1^2/m_2^3\right)}\right)}^{1/3}\right]}^{-q}\\ ={\left[{\theta }_0^2{\left(\frac{m_2^5}{m_1^2{m}_3^3}\right)}^{1/3}\right]}^{-q}={\left[\alpha {\left(\frac{m_2^5}{m_1^2{m}_3^3}\right)}^{1/3}\right]}^{-q}={\alpha }^{-q}\times {{\Re }_m}^{-q},\\ \text{for phase}{\varphi }_2=\mathit{p\pi }\text{and}{\varphi }_1=\mathit{q\pi }:p,q\in \mathbb{N},\\ \text{with}{\gamma }_1=m_2/m_1\text{and}{\gamma }_2=m_3/m_2,\end{array}$$E39

with m₁, m₂ and m₃ being the triad or natural sequence of particles of a given class, i.e. the generations of that class, of electric charge q, susceptible of being related by means of Lorentz factors γ₁ and γ₂ according to Eq. (37), for the numerator and denominator.

That is, Eq. (39) establishes that, just as there is a phase through (37) for the first two particles of a triad, and there is a phase through (37) for the second two (the central one participates in both), there is a simple ratio $\chi$ between both relationships (37). A simple ratio that will not only allow us to verify the relationship between the known particles but also to propose a third one (as we will see later), either in ascending or descending sequence, knowing two others.

The relationship is simpler, if possible, with respect to the one presented if we take into account that ${\beta }_2\approx {\beta }_1$, and that, consequently, $\alpha \cong 1$. That is, according to (36)${\beta }_i$ depends on $b$, associated with the normalization constant $B$ in (13), which from (24) evolved as the constant $A$, so that it has a single value for all particles of the same value $q$, except for the fact that the integral (23) from which the constant $A$ derives is not established for each particle in the interval $\left[0,c\right]$ defined there but for the one defined by the final $\upsilon$, according to ${\gamma }_1$ and ${\gamma }_2$, which satisfies (38), which may introduce a small deviation. It can be verified with the cases solved in the following sections that this deviation is only significant in one of them (that of the charged leptons, with $1/{\alpha }^{-q}\cong 1.04$), being defined, in any case, through $\alpha ={\theta }_0^2$ as the established ratio between the angle subtended by the speed v₂, associated with γ₂, and the one subtended by $c$, that is, ${\theta }_0^2\equiv {\theta }_2/{\theta }_0\equiv \theta \left({\upsilon }_2\right)/\theta \left(c\right)$. The latter will be well understood if we pay attention to the fact that β₁ is characterized by the transit defined by ${\gamma }_1$, and β₂ by the one defined by γ₂ on the first, so that $\beta {\theta }_0^1={\beta }_1$ and $\beta {\theta }_0^1{\theta }_0^2={\beta }_1{\theta }_0^2={\beta }_2$, and consequently $\alpha ={\beta }_2/{\beta }_1={\theta }_0^2.$

On the other hand, and as a consequence of the previous simplification, we will be able to apply Eq. (39) without needing to know the phases. That is, we can apply it and find the value $\chi$ (valid for different numerator and denominator phase values) that relates to three given particles of a triad. In fact, the reason and difference of introducing and working with Eq. (39) are the suppression of the internal constants or variables (undulatory) and the possibility of making an external treatment to the particles, unlike the internal one carried out for each one of them through (37) or, being more exact, of (36) for being the one that supports all those variables.

6. The charged particles

The requirement presented can be made explicitly for each of the types of charged particles and assigned to each of them according to Eq. (39) a simple theoretical $\chi$ value compatible with the value of its masses:

6.1 Leptons

$$\chi =\frac{{\varphi }_2}{{\varphi }_1}=\left[\alpha {\left(\frac{m_2^5}{m_1^2{m}_3^3}\right)}^{1/3}\right]=\left[\alpha {\left(\frac{m_{\mu }^5}{m_e^2{m}_{\tau }^3}\right)}^{1/3}\right]=2\left(\propto \frac{2\pi }{1\pi }\right),$$E40a

Value, in this case, is achieved by ${\alpha }^{-q}=\alpha =0.962$ and ${\Re }_m^{-q}={\Re }_m$ for the values $m_1,m_2,m_3$ known. In effect, with:

$$\begin{array}{l}{\gamma }_2=m_3/m_2=16.82\left(\Rightarrow {\upsilon }_2/c=0.99823\equiv sin{\theta }_2\right),\\ \text{and}\alpha ={\theta }_0^2=\frac{arcsin\left(sin{\theta }_2\right)}{{\theta }_0}=\frac{86,{59}^o}{{90}^o}=0.962,\\ \text{we have}\chi =\alpha \times {\Re }_m=0.962\times 2.079=2.\end{array}$$

Treatment with which, as we have already mentioned, we do not obtain the phases but a simplified ratio $\chi$ between them, on which we can establish different equivalent proportionality relationships, which we can only solve or make unique with the knowledge of ${\beta }_i$ or assuming that the simplest relationship is the correct one, which will allow us to calculate ${\beta }_i$ for each of the two phases applied to Eq. (37).

6.2 Quarks (2/3 q)

$$\begin{array}{l}\chi =\frac{{\varphi }_2}{{\varphi }_1}={\left[\alpha {\left(\frac{m_2^5}{m_1^2{m}_3^3}\right)}^{1/3}\right]}^{-2/3}={\left[\alpha {\left(\frac{m_c^5}{m_u^2{m}_t^3}\right)}^{1/3}\right]}^{-2/3}=1.5\left(\propto \frac{3\pi }{2\pi }\right),\\ \left(\text{with}{\alpha }^{-2/3}=1.005\cong 1\right)\end{array}$$E40b

Analogous to the comment made above, for ${\beta }_2={\beta }_1=\beta$ in this case, with which it could be shown, inversely and known $\beta$, that by means of Eq. (37) we can reach phase ${\varphi }_1=2\pi$ when the rest mass $m_1=m_u$ reaches the value $m_2=m_c$ as inertial mass, and phase ${\varphi }_2=3\pi$ when the rest mass $m_2=m_c$ reaches the value $m_3=m_t$ as inertial mass.

6.3 Quarks (−1/3 q)

$$\begin{array}{l}\chi =\frac{{\varphi }_2}{{\varphi }_1}={\left[\alpha {\left(\frac{m_2^5}{m_1^2{m}_3^3}\right)}^{1/3}\right]}^{1/3}={\left[\alpha {\left(\frac{m_s^5}{m_d^2{m}_b^3}\right)}^{1/3}\right]}^{1/3}=0.5\left(\propto \frac{1\pi }{2\pi }=\frac{2\pi }{4\pi }\right),\\ \left(\text{with}{\alpha }^{1/3}=1.004\cong 1\right)\end{array}$$E40c

Analogous comment to the previous case, for their respective phases and masses, since the treatment (which we will exemplify in the following section) is universal and valid for all particle triads.

6.4 Summary

The above results are shown in Table 1, where it is more clearly evidenced that we have been able to adjust the $\chi$ values with certain values, chosen (except for the leptons, which are known and well defined) from among the possible experimental ones, for the masses of the different particles (see Appendix B) for reasons that we will show in the following section, derived from the ability to propose the masses of the particles already mentioned.

Triad	Particle	Mass (MeV)	$\chi \left({\varphi }_2/{\varphi }_1\right)$
m0	e	0.511	2
m1	μ	105.658
m2	τ	1776.840
m0	u	2.368	1.5
m1	c	1361.698
m2	t	172,950
m0	d	5.395	0.5
m1	s	83.883
m2	b	4180.425

Table 1.

Value $\chi$, according to Eqs. (40), for each triad of values m₁, m₂, m₃, corresponding to the three generations of particles, for each of the three classes of charged particles.

Results that are equally applicable and valid for the antiparticles, which, with the same values for the masses and charge of the opposite sign, will give as result in Eq. (39) a phase inversion with respect to the particles, fulfilling that:

$${\chi }_i{\overline{\chi }}_i=1.$$E41

7. The $\mathfrak{M}$ particle

Although in a first evaluation, we did not say that expression (39) determined the value of the masses, but that the value of the masses is regulated by (meets) the Eq. (39), in a second evaluation, we will be able to discriminate the values by means of it, that is, to overcome the indetermination of these experimental data or to correct them. And the latter as a consequence of the fact that the relationship (39) participates in another relational element, by which it can be stated that for all types of charged particles we can find a particle of previous hierarchy $m_0$ which equally fulfils the expression (39),

$${\chi }_0^L=\frac{{\varphi }_1}{{\varphi }_0}={\left[\alpha {\left(\frac{m_1^5}{m_0^2{m}_2^3}\right)}^{1/3}\right]}^{-q}=\left[{\left(\frac{m_e^5}{m_0^2{m}_{\mu }^3}\right)}^{1/3}\right]=1,$$E42a

$${\chi }_0^{q+}=\frac{{\varphi }_1}{{\varphi }_0}={\left[\alpha {\left(\frac{m_1^5}{m_0^2{m}_2^3}\right)}^{1/3}\right]}^{-q}={\left[{\left(\frac{m_u^5}{m_0^2{m}_c^3}\right)}^{1/3}\right]}^{-2/3}=1,$$E42b

$${\chi }_0^{q-}=\frac{{\varphi }_1}{{\varphi }_0}={\left[\alpha {\left(\frac{m_1^5}{m_0^2{m}_2^3}\right)}^{1/3}\right]}^{-q}={\left[{\left(\frac{m_d^5}{m_0^2{m}_s^3}\right)}^{1/3}\right]}^{1/3}=4.$$E42c

with the particularity that such a particle is only one for all of them, or of equal mass, which we will now identify as $\mathfrak{M}=171.87$eV, for the different masses $m_1,m_2$ previously defined, with which it forms a triplet of new triads, being by means of the first, Eq. (42a), as we have calculated the exact value of $\mathfrak{M}$ through the exact value of $e$ and $\mu$, which gives rise to well-defined ${\chi }_0^{q\pm }$ values for the other expressions (42).

Triad, in which, according to the position of the particles and the equations, the ${\varphi }_1$ value for $\chi$ of Table 1 takes the position of the numerator, while the new denominator is ${\varphi }_0$, as we present in Table 2.

Triad	Particle	Mass (MeV)	$\chi \left({\varphi }_1/{\varphi }_0\right)$
m0	$\mathfrak{M}$	0.00017187	1/1
m1	e	0.511
m2	μ	105.658
m0	$\mathfrak{M}$	0.00017187	2/2
m1	u	2.368
m2	c	1361.698
m0	$\mathfrak{M}$	0.00017187	4/1
m1	d	5.395
m2	s	83.883

Table 2.

Phase, according to Eq. (42), established between two first generations of particles $m_1$, $m_2$ for each of the three families with charged, and a particle of a zero generation shared, $m_0$, named as $\mathfrak{M}$.

In conclusion, taking the ϕ phases as distances, what we are saying is not only that the different charged fermions are found one from the other at defined distances, quantized or marked by the natural numbers, but that following the quantization process we find the same particle defined in the vicinity of each of the three differentiated electric charges, even equidistant (${\varphi }_0={\varphi }_1$) for two of the three classes of particles. From this point on, being particles of the same value, we can think, as we have done initially, that they are the same and common particle $\mathfrak{M}$, or an equivalent energetic entity, which is at a level prior to the first generation for the three classes of elementary charged particles, which, consequently, can be taken as a zero generation or precursor particle and as evidence of a shared origin.

A common particle $m_0$ of lower hierarchical order for the establishment of a new triad, in accordance with Eq. (39) and for $\alpha \cong 1$, that, on the other hand and by virtue of the exact definition of its value, conditions the values of the particles $m_1,m_2$, i.e. their choice from the whole experimental spectrum, as we have already mentioned (Section 6) in connection with the choice of these values in the previous triad.

Expressed in a more exhaustive form, we see that $m_0$ imposes an additional condition on (39), so that it sets $m_1^5/m_2^3$ by $\chi$ at Eqs. (42), according to the experimental values ($m_1,m_2$), which in turn determines $m_2^5/m_1^2$, which in turn determines exactly $m_3$ at Eqs. (40), and vice versa. In this case, if we alter the $m_1,m_2$ values while keeping the $m_1^5/m_2^3$ ratio at Eqs. (42), the $m_2^5/m_1^2$ ratio is changed to Eqs. (40) and consequently $m_3$. It follows that only one pair of values fulfils the double condition, and that this pair cannot fluctuate in their values more than the fluctuation in the value of $m_3$, that is, its experimental error.

We can see this in another way, if we notice that associated to the expression (39) are the separate conditions about $\varphi$, that is, ${\varphi }_1$ and ${\varphi }_2$, conditions that, particularized to expressions (40) and (42) take for each type of charge the following forms, which are simply the corresponding Eq. (37) with the ${\gamma }_i$ developed:

$$\begin{array}{l}{\varphi }_2={\left[\beta {\left(m_2^2/m_3^3\right)}^{1/3}\right]}^{-q},{\varphi }_1={\left[\beta {\left(m_1^2/m_2^3\right)}^{1/3}\right]}^{-q},\left(\mathrm{a}\right)\\ {\varphi }_1={\left[\beta {\left(m_1^2/m_2^3\right)}^{1/3}\right]}^{-q},{\varphi }_0={\left[\beta {\left(m_0^2/m_1^3\right)}^{1/3}\right]}^{-q},\left(\mathrm{b}\right)\end{array}$$E43

Consequently, we have a system of three equations (${\varphi }_0,{\varphi }_1,{\varphi }_2$) with three unknown values ($m_1,m_2,m_3$), which has a single solution (a triad of values), and a free parameter $\beta$ (for the more general case $\alpha =1\Rightarrow \beta ={\beta }_1={\beta }_2$), which adjusts this solution within the experimental values or its neighbourhood, in a solidary way (by the dependence reached through the equations) which in fact supposes the restriction of the experimental spectrum or exact definition of its values. Specifically, from $m_0$ and fixed a value $\beta$ we have the whole itinerary until $m_3$, we only have, therefore, to choose the value $\beta$ that reaches our value $m_3\pm {\epsilon }_3$, and we will have $m_1\pm {\epsilon }_1$ and $m_2\pm {\epsilon }_2$, which we can particularize for a negatively charged quark which satisfies its transitions for a fixed value $\beta =5411406.96$, characteristic of the family of particles $\left(-1/3\right)$-quarks, in this case, for which it is not difficult to calculate the speeds for which such transitions occur (see Figure 4), according with ${\gamma }_i=m_{i+1}/m_i$, just like we did for the leptons in Eq. (40a). We will be able to see the Itinerary in detail in Appendix C.

Figure 4.

Phase transitions for negatively charged quarks $\left(-1/3\right)$. Evolution of phase as a function of inertial mass according to Eqs. (5) and concatenation of these equations by converting the inertial mass (kinetic energy) into matter (formation energy) of a higher generational order particle at the point of equivalence. Where we have taken $m_1$ as the reference and, consequently, ${\varphi }_0$, negative.

We can understand all the transitions described by Eqs. (43), and the original Eq. (37), like the equation of a curved line of curvature $\beta$ which passes over a point B, and with origin in A. In this case, the origin A is $m_0$, and the point B on which it passes is $m_1$, which is the origin of the following equation, as we have represented in Figure 4, where it is shown that the phase is, in fact, the evolution of a particle as a function of speed up to the limit, for a determined speed, in which it stops being that particle and is another one (phase change), what, taken to an extreme, could be understood as a process of energetic recombination, that is, of real conversion of kinetic energy into formation energy, demonstrating that both energies are the same thing, nothing extraordinary if we take into account that these phases $\varphi \equiv \left({\varphi }_0,{\varphi }_1,{\varphi }_2,\dots \right)=\mathit{n\pi }$, or maximum value of $\upsilon$, correspond (see Eq. (38)) with the zeros ($\mathrm{sen}{\left[\varphi \right]}_{\upsilon }=0$) of the kinetic energy for the same values ${\gamma }_i$ developed in Eq. (43) for ${\varphi }_i$, while the expression (39) itself already determines that the size of the phases are defined through the velocities at which the respective Lorentz factors make the inertial mass of a particle coincide with the real mass of the hierarchically subsequent one, evidencing that this inertial mass can only reach this maximum value, and that another value higher than this belongs to another phase, whether the interface is presented as a place of transit or as a differentiation and separation of states (Appendix D).

In this Appendix D where the physical reality underlying the introduction of the particle $\mathfrak{M}$ is evaluated, that is to say, on the one hand, it is emphasized that the phase structure is governed by ${\gamma }_i$, in such a way that whenever we talk about a particle we talk about the one reached, for a phase, as inertial mass of another, fulfilling (38), and with this that ${\gamma }_i$ is the only way to relate the candidates to particles, and the phases ${\varphi }_i$ the way to measure their energy distance between them, which introduces a new limitation (discretization) of the values. This allows to establish or filter the possible values (a discrete set) from a whole universe of values. On the other hand, we study the degree of convergence and, consequently, the impossibility that, according to the phasic structure, a supposed and unknown particle $\mathfrak{M}$ (which underlies the others) is not there: it must be that one, in case there is one, and at those phasic distances.

We say in case there is one because it can be a particle that has fulfilled a function, and no longer exists, or an energetic entity that conforms, given its characteristics, the dark matter and all the genesis of the other (we start from the SWP process). From there, we cannot assure its existence beyond the fact that it is through it, through the phase reached, and repeating the process, that we also reach particles of which we do have knowledge, as we will study in the next section.

8. Neutrinos

We see that all first-generation particles converge on the particle $m_0$ i.e. the values increasingly diverge from it, differentiated by charge ($e$, u, d), for different phase values. Reciprocally, it is possible to consider the existence of a divergent and decreasing previous generation from $m_0$ that posed in the line of progress of charged leptons and/or those not charged involve, for different phases, values lower-order particles that we could identify with neutrinos.

In that consideration, we find in the progress line of charged leptons, also used in Eq. (42a), a number of candidates $m_{\nu }$ in the role of $m_{-1}$, that is, those that would be reached by varying the values ${\varphi }_{\nu }=\left(1,2,3,4,5\dots \right)\times \pi$ with respect to ϕ₀ in relationship (39) for $m_0=\mathfrak{M}$ and from which we will have to choose those that could truly represent our neutrinos. That is, in this case it is not relevant to find the points (possible neutrino masses) by means of Eq. (39), but to verify that some of them, quantized or marked by means of the natural numbers by ${\varphi }_{\nu }$ in (39) in the neighbourhood of $m_0$, truly correspond to those neutrino masses, a verification that can only be done by means of physical criteria.

This verification is achieved in a first order as a consequence of the fact that the number of points reached by the line of progress of charged leptons is restricted (as we will see) by the line of progress of uncharged leptons, from which it results that ${\nu }_a,{\nu }_b,$$\text{and}{\nu }_c$ (by ${\varphi }_{\nu }=3\pi ,6\pi ,9\pi$), represent a set of previous (and necessary) generations for charged leptons, that is (with $\alpha \cong 1$ and ${\nu }_i\equiv \text{mass of}{\nu }_i$):

$${\chi }_a^e=\left[\alpha {\left(\frac{m_0^5}{{\nu }_a^2{m}_1^3}\right)}^{1/3}\right]=\left[{\left(\frac{{\mathfrak{M}}^5}{{\nu }_a^2{m}_e^3}\right)}^{1/3}\right]=1/3,$$E44a

$${\chi }_b^e=\left[\alpha {\left(\frac{m_0^5}{{\nu }_b^2{m}_1^3}\right)}^{1/3}\right]=\left[{\left(\frac{{\mathfrak{M}}^5}{{\nu }_b^2{m}_e^3}\right)}^{1/3}\right]=1/6,$$E44b

$${\chi }_c^e=\left[\alpha {\left(\frac{m_0^5}{{\nu }_c^2{m}_1^3}\right)}^{1/3}\right]=\left[{\left(\frac{{\mathfrak{M}}^5}{{\nu }_c^2{m}_e^3}\right)}^{1/3}\right]=1/9.$$E44c

for $m_a=0.00551\mathrm{eV},m_b=0.01559\mathrm{eV},m_c=0.02863\mathrm{eV}.$

While ${\nu }_d={\nu }_{12}=0.0448\mathrm{eV}$ for ${\varphi }_{\nu }=12\pi$ would be plausible but not functionally necessary (it is not, as we shall see in next section, to construct a basis), and the rest of the subscripts, not plausible or invalidated by the experimental reality derived from neutrino transitions in decays or, as we said, of the progress line of uncharged leptons. Assertions that require to demonstrate that the same treatment can be carried out with neutrinos as that already carried out with charged particles, and that, taking as a reference the values supplied by the disintegration processes, which we will assume to be reliable, we will demonstrate, even though they are not accompanied by a lower limit:

$${\nu }_e\cong 0\mathrm{eV}{\nu }_{\mu }\le 0.17\mathrm{MeV}\text{and}{\nu }_{\tau }\le 17\mathrm{MeV},$$E45

and although it is evident that our main working relationship is not fulfilled for these values, given the energetic distance between them, and mainly the five orders of magnitude between the first two.

It is precisely because of this disproportionate distance that we contemplate the possibility that there is an intermediate particle intercalated between ${\nu }_e$ and ${\nu }_{\mu }$ that allows the transit and verification of Eq. (39). It does indeed exist, and not only that, but it is also, surprisingly, the particle $m_0$, by means of which not only one solution for (39) is reached but, as we have already anticipated, three (${\varphi }_{\nu }=1,2,3$), satisfying some equations equivalent to Eq. (44) for the same (${\mathrm{\nu }}_a,{\mathrm{\nu }}_b\mathrm{y}{\mathrm{\nu }}_c$), as candidates for ${\nu }_e$, and the particle ${\nu }_{\mu }$ (with $\alpha \cong 1$ and ${\nu }_{\mu }\equiv \text{mass of}{\nu }_{\mu }$):

$${\chi }_a^{\nu }=\left[\alpha {\left(\frac{m_0^5}{{\nu }_a^2{m}_1^3}\right)}^{1/3}\right]=\left[{\left(\frac{{\mathfrak{M}}^5}{{\nu }_a^2{\nu }_{\mu }^3}\right)}^{1/3}\right]=\frac{{\varphi }_{\mu }}{{\varphi }_{\nu }}=3/3\left(\propto 1/1\right),$$E46a

$${\chi }_b^{\nu }=\left[\alpha {\left(\frac{m_0^5}{{\nu }_a^2{m}_1^3}\right)}^{1/3}\right]=\left[{\left(\frac{{\mathfrak{M}}^5}{{\nu }_b^2{\nu }_{\mu }^3}\right)}^{1/3}\right]=\frac{{\varphi }_{\mu }}{{\varphi }_{\nu }}=3/6\left(\propto 1/2\right),$$E46b

$${\chi }_c^{\nu }=\left[\alpha {\left(\frac{m_0^5}{{\nu }_a^2{m}_1^3}\right)}^{1/3}\right]=\left[{\left(\frac{{\mathfrak{M}}^5}{{\nu }_c^2{\nu }_{\mu }^3}\right)}^{1/3}\right]=\frac{{\varphi }_{\mu }}{{\varphi }_{\nu }}=3/9\left(\propto 1/3\right).$$E46c

which correlate with a subsequent triad, for which $m_0$ is also a solution, i.e. which also satisfies Eq. (39) for the same in another hierarchical situation, thus demonstrating that it is truly the concatenation element of the sequence:

$${\chi }_0^{\nu }=\left[\alpha {\left(\frac{m_1^5}{m_0^2{m}_2^3}\right)}^{1/3}\right]=\left[{\left(\frac{{\nu }_{\mu }^5}{{\mathfrak{M}}^2{\nu }_{\tau }^3}\right)}^{1/3}\right]=\frac{{\varphi }_{\tau }}{{\varphi }_{\mu }}=3/3\left(\propto 1/1\right),$$E47

and in which the last equivalence (member) guarantees in Eq. (46) that the values ${\varphi }_{\nu }$ reached (which we know what they are because they are associated with the masses ${\nu }_a,{\nu }_b\mathrm{y}{\nu }_c$) correspond to those obtained in the Eqs. (44), what, on the other hand, determines the real value of phase ${\varphi }_{\mu }$ of connection with $m_0$ for ${\nu }_{\mu }$ in (46), with respect to phase ${\varphi }_0$ for $e$ in Eqs. (44) for these same values ${\varphi }_{\nu }=3,6,9$. Results that we can put in respective Tables 3 and 4:

Triad	Particle	Mass (eV)	$\chi \left({\varphi }_0/{\varphi }_{\nu }\right)$
mν1	νa	0.00551	1/3
m0	$\mathfrak{M}$	171.87
m1	e	511,000
mν1	νb	0.01558	1/6
m0	$\mathfrak{M}$	171.87
m1	e	511,000
mν1	νc	0.02863	1/9
m0	$\mathfrak{M}$	171.87
m1	e	511,000

Table 3.

Phases and values of corresponding to those phases, according to Eq. (44), for the sequence ${\nu }_1$, $m_i$_,$e^{-}$, corresponding to the first generation of each type of lepton and an intermediate particle $m_i$ represented by $\mathfrak{M}$.

Triad	Particle	Mass (eV)	$\chi \left({\varphi }_{\mu }/{\varphi }_{\nu }\right)$
mν1	νa	0.0056	3/3 = 1/1
m0	$\mathfrak{M}$	171.87
mν2	νμ	168,564
m ν1	νb	0.0158	3/6 = 1/2
m0	$\mathfrak{M}$	171.87
mν2	νμ	168,564
mν1	νc	0.0291	3/9 = 1/3
m0	$\mathfrak{M}$	171.87
mν2	νμ	168,564

Table 4.

Phases and values of corresponding to those phases, according to (46), for the sequence ${\nu }_1$, $m_i$_,${\nu }_2$, corresponding to the first two generations of neutrinos and to an intermediate particle represented by $\mathfrak{M}$.

where the values reached for ${\nu }_a,{\nu }_b\mathrm{y}{\nu }_c$ can be considered as theoretical and exact in Table 3, since the other two factors of the triad are exact, while those obtained in Table 4 (and the triad (47) for ${\nu }_{\tau }=16636417\mathrm{eV}$) can be considered as experimental or subject to the imprecise experimental limits reflected in Eq. (45). A difference of values that we will later contextualize and revalidate in another way (Section 9).

We see, in conclusion, that the infinite generations or possibilities of neutrinos are restricted to three in the line or chain of charged leptons in correspondence with the first three generations (${\varphi }_{\nu }=1,2,3$), for ${\varphi }_{\mu }=1$, in the chain of uncharged leptons. Likewise, we see that, just as there is the sequence $m_0,m_1,m_2,m_3$, for charged particles, there is for uncharged particles in the form ${\nu }_e$, $m_0$, ${\nu }_{\mu }$, ${\nu }_{\tau }$ with its two corresponding triads. One of the triads is Eq. (47) and the other, the resultant of the Eqs. (46), in which ${\nu }_e$ does not appear, but ${\nu }_a$, ${\nu }_b$, ${\nu }_c$, which we can understand as a natural base of neutrinos (supplied by the chain of charged leptons) with which we can build, as we will see later, a massive base and, where appropriate, a flavour (${\nu }_e$). That is, the natural neutrino base is created, associated to $e$, through $m_0$ by the charged lepton chain ($m_{\nu },m_0,m_e$), and then progresses (occasionally with the generational change of e itself) with certain configurations (flavours) through $m_0$ by its chain of distribution (${\nu }_{\mu }$, ${\nu }_{\tau }$), which involves a modification with respect to the massive base (oscillations).

We also see that the above creation corresponds to a seesaw mechanism [7] that does not require the mass of a charged fermion (electron) under a large mass (GeV-TeV) $M_R$ ($m_e^2={\nu }_i{M}_R$), but an intermediate mass $m_0$ ($\mathfrak{M}$), which had a reserved energy space (between $\nu$ and $e$), under the medium value of the electron mass, for the potential function developed in the phase relation (39), which we can write as some arms of the seesaw:

$$m_0^5={\nu }_i^2{\left(m_e{\chi }_i\right)}^3\to m_0^2={\nu }_i^{4/5}{\left(m_e{\chi }_i\right)}^{6/5}$$E48

which is not due to the exchange of bosons or fermions referred to in this paradigm [8], but to the energy conversion process itself (fermion exchange, in this case) represented by Eq. (37), together with the conceptual and practical differentiation, which we will not discuss here, between electron and free electron.

On another level, Eqs. (44) and (46) clear any doubt about some kind of coincidence or chance in (42) and put us in the idea that truly such a particle $\mathfrak{M}$ exists, either as a particle or as a circumstantial element of energy distribution, since it is valid for all types of particles and antiparticles, charged and uncharged, or even as the foundation of all present and non-present matter (dark matter), through which the entire spectrum of particles is systematically distributed in phase units, and every particle in this spectrum defined by the phase reached, and by its mass as an evolutionary parameter. A situation that we can represent (Figure 5):

Figure 5.

Phases diagram for 2/3 charge quarks, −1/3 charge quarks and charged leptons. Each generation of a class of particles is nothing but a phase or differentiated state of the same particle in its evolution or progress (of phase) from the fundamental state $\mathfrak{M}$ (as already shown in Figure 4). Likewise, charged leptons progress through $\mathfrak{M}$ by means of Eq. (39) towards the natural base of neutrinos ${\nu }_a$, ${\nu }_b$, ${\nu }_c$. The same ones can then form the massive base of neutrinos and progress in this way through $\mathfrak{M}$ with any state of oscillation, and do so, in particular, in one of the eigenstates that we call flavours, associated to each of the charged leptons, from the initial ${\nu }_e$.

9. The massive base

We cannot address the issue of the neutrinos base without presenting the corresponding data to antineutrinos:

$${\overline{\chi }}_a^{\nu }={\left[{\left(\frac{m_0^5}{{\overline{\nu }}_a^2{\overline{\nu }}_{\mu }^3}\right)}^{1/3}\right]}^{-1}=\frac{{\overline{\varphi }}_{\mu }}{{\overline{\varphi }}_{\nu }}=3/3={\left({\chi }_a^{\nu }\right)}^{-1},$$E49a

$${\overline{\chi }}_b^{\nu }={\left[{\left(\frac{m_0^5}{{\overline{\nu }}_b^2{\overline{\nu }}_{\mu }^3}\right)}^{1/3}\right]}^{-1}=\frac{{\overline{\varphi }}_{\mu }}{{\overline{\varphi }}_{\nu }}=3/2\ne {\left({\chi }_b^{\nu }\right)}^{-1},$$E49b

$${\overline{\chi }}_c^{\nu }={\left[{\left(\frac{m_0^5}{{\overline{\nu }}_c^2{\overline{\nu }}_{\mu }^3}\right)}^{1/3}\right]}^{-1}=\frac{{\overline{\varphi }}_{\mu }}{{\overline{\varphi }}_{\nu }}=3/1={\left({\chi }_c^{\nu }\right)}^{-1}.$$E49c

We can realize that as a consequence of the change of sign of the exponent, we switch from a multiplier phase relation to a divider phase relation, and so while for neutrinos the phase space ${\varphi }_{\mu }=\pi$ implies ${\varphi }_{\nu }=\pi ,2\pi ,3\pi$, for antineutrinos the sequence is inverse over the largest ${\varphi }_{\nu }$ taken as ${\overline{\varphi }}_{\mu }$ (${\overline{\varphi }}_{\mu }=3\pi$ implies ${\overline{\varphi }}_{\nu }=3\pi ,2\pi ,\pi$), and that, in consequence, the phase ${\overline{\chi }}_b^{\nu }$ corresponding to the second antineutrino of the natural base, and consequently the mass, differs from that of the neutrino. Mass that can be calculated by Eq. (49b) or more precisely and generally, as we did with the neutrinos, from ${\overline{\chi }}_{\nu }^e$ corresponding to Eqs. (44). This is (${\overline{\varphi }}_0=9\pi$ implies ${\overline{\varphi }}_{\nu }=3\pi ,2\pi ,\pi$):

$${\overline{\chi }}_a^e={\left[{\left(\frac{m_0^5}{{\overline{\nu }}_a^2{m}_{\overline{e}}^3}\right)}^{1/3}\right]}^{-1}=\frac{{\overline{\varphi }}_0}{{\overline{\varphi }}_{\nu }}=9/3={\left({\chi }_a^e\right)}^{-1},$$E50a

$${\overline{\chi }}_b^e={\left[{\left(\frac{m_0^5}{{\overline{\nu }}_b^2{m}_{\overline{e}}^3}\right)}^{1/3}\right]}^{-1}=\frac{{\overline{\varphi }}_0}{{\overline{\varphi }}_{\nu }}=9/2\ne {\left({\chi }_b^e\right)}^{-1},$$E50b

$${\overline{\chi }}_c^e={\left[{\left(\frac{m_0^5}{{\overline{\nu }}_c^2{m}_{\overline{e}}^3}\right)}^{1/3}\right]}^{-1}=\frac{{\overline{\varphi }}_0}{{\overline{\varphi }}_{\nu }}=9/1={\left({\chi }_c^e\right)}^{-1},$$E50c

from what it turns out:

$${\overline{\nu }}_a=0.00551\mathrm{eV},{\overline{\nu }}_b=0.01008\mathrm{eV},\text{and}{\overline{\nu }}_c=0.02863\mathrm{eV},$$

which represents a natural base of antineutrinos different from that of neutrinos, which, however, allows the creation of identical massive bases:

$$\begin{array}{ll}{\nu }_1& ={\nu }_a=0.00551\mathrm{eV}={\overline{\nu }}_a,\\ {\nu }_2& ={\nu }_b-{\nu }_a=\left(0.01559-0.00551\right)\mathrm{eV}=0.01008\mathrm{eV}={\overline{\nu }}_b,\\ {\nu }_3& ={\nu }_a+{\nu }_b+{\nu }_c=\left(0.00551+0.01559+0.02863\right)\mathrm{eV}\\ & =0.04973\mathrm{eV}=2{\overline{\nu }}_a+{\overline{\nu }}_b+{\overline{\nu }}_c,\end{array}$$E51

With (as required experimentally):

$${\sum }_{\nu }{m}_{\nu }={\nu }_1+{\nu }_2+{\nu }_3=0.06532<0.1\mathrm{eV}.$$E52

Values that clear any uncertainty and ignorance about them, the hierarchy of particles and related issues (see Ref. [9], p. 301), being fulfilled for the oscillations of the solar and atmospheric neutrinos, respectively:

$$\begin{array}{l}\mathrm{\Delta }{m}_{21}^2=7.113\times {10}^{-5}{\mathrm{eV}}^2\text{and}\mathrm{\Delta }{m}_{31}^2=2.442\times {10}^{-3}{\mathrm{eV}}^2,\\ \text{with}\mathrm{\Delta }{m}_{21}^2=m_2^2-m_1^2\text{and}\mathrm{\Delta }{m}_{31}^2=m_3^2-m_1^2\end{array}$$E53

Results congruent with the experimental data [10], which oblige, on the other hand, a range of values for the masses, congruent with those reached in (51). Results that, nevertheless, can be placed in Table 5 together with other close ones (including the experimental ones) in order to appreciate the real gap and evaluate our calculations.

R	Mass νa (eV)	$\mathrm{\Delta }{m}_{21}^2/{10}^{-5}$	$\mathrm{\Delta }{m}_{31}^2/{10}^{-3}$
1	0.00551	7.113	2.442
2	0.00554	7.189	2.468
3	0.00558	7.296	2.505
4	0.00559	7.323	2.514
5	0.00560	7.347	2.522
6	0.00561	7.375	2.532
7	0.00562	7.402	2.541
8	0.00563	7.427	2.549
9	0.00564	7.455	2.559

Table 5.

Difference of squares for solar and atmospheric mass depending on the massive value of ν₁, where we have highlighted the (approx.) best experimental fit, written in red the theoretical values, and marked (blue) the most plausible row.

First, we have obtained the table by infinitesimal variations of in (44) and adjusted as in (51) again. That said, we see that the experimental data (highlighted) are reached for masses different from the initial theoretical masses, which are additionally different for solar (R-8) and atmospheric (R-5) neutrinos, with the particularity that the latter remain with this mismatch (row) even considering their experimental error, while the former can vary $\pm {5.10}^{-5}$ eV considering theirs, and may be placed with the value 7.347 in the same row R-5 (0.00560 eV) as the latter. Giving the latter as less uncertain, the former could not only occupy the same row R-5 but would have to occupy it since, from our approach, $∆m_{21}^2$ and $∆m_{31}^2$ have linked evolution through ${\nu }_1$ (each row is a state defined: there is no mass that gives rise to the two experimental values or row) by ${\chi }_i^e$$\left[{\nu }_{b,c}=\nu \left({\nu }_a={\nu }_1\right)\right]$ in (44).

Consequently, the plausible ${\nu }_1$ value for the experimental values, $∆m_{21}^2$ and $∆m_{31}^2$ is 0.00560 eV which we have likewise considered as experimental (theoretical and not exact of Table 4) with respect to the exact, strictly theoretical and exact value 0.00551 eV of Table 3. Being in any case the possible error $<{10}^{-4}$eV.

From there, and taking into consideration that our theoretical calculation is practically arithmetical in Eq. (42a), it would remain to determine whether this experimental deviation is from the ideal situation (that of the model), or on the contrary is attributable to the methodology used (which contains theoretical and indirect elements) to calculate the $∆m_{21}^2$ and $∆m_{31}^2$ values

10. CP violation

A physical system with this configuration presents an asymmetry that would justify the structural nonexistence of antimatter in the known universe, or supremacy of matter over antimatter as a consequence of the different number of construction elements in both species, and the concurrence of fewer of them (the particles $\mathfrak{M}$ from which the natural base starts) in a process of mutual annihilation:

$$\begin{array}{l}{\sum }_{\nu }{m}_{\nu }={\nu }_1+{\nu }_2+{\nu }_3={\nu }_a+2{\nu }_b+{\nu }_c,\\ {\sum }_{\overline{\nu }}{m}_{\overline{\nu }}={\overline{\nu }}_1+{\overline{\nu }}_2+{\overline{\nu }}_3=3{\overline{\nu }}_a+2{\overline{\nu }}_b+{\overline{\nu }}_c.\end{array}$$E54

In reality, this differential of elements is the extrapolation of the differential that we could already see between the element ${\nu }_b$ and the element ${\overline{\nu }}_b$, which showed that under equal conditions there is more matter than antimatter, and all of this, ultimately, due to the relation between phases resulting from the charge of the electron in Eq. (39) as opposed to that of the positron.

We could elude this asymmetry or consider an alternative configuration assuming a four-element base, with which we would obtain an asymmetric element (sterile neutrino ${\nu }_c$ for ${\varphi }_1=3\pi$), and three symmetric ones, sufficient to create a symmetric base of matter and antimatter, that if they coexist would be annihilated and if they do not coexist would not require any type of transformation to constitute a mass base:

$$\begin{array}{l}{\nu }_1={\nu }_a={\overline{\nu }}_a\simeq 0.0056\mathrm{eV},\\ {\nu }_2={\nu }_b={\overline{\nu }}_b\simeq 0.0158\mathrm{eV},\\ {\nu }_3={\nu }_d={\overline{\nu }}_d\simeq 0.0448\mathrm{eV},\end{array}$$E55

but which would nevertheless report for differences in mass squares, results that are notably distant from the experimental data:

$$\begin{array}{l}\mathrm{\Delta }{m}_{21}^2=\mathrm{\Delta }{m}_{\mathit{ba}}^2=21.951\times {10}^{-5}{\mathrm{eV}}^2,\\ \mathrm{\Delta }{m}_{31}^2=\mathrm{\Delta }{m}_{\mathit{da}}^2=1.903\times {10}^{-3}{\mathrm{eV}}^2.\end{array}$$E56

11. Summary and discussion

In this article, we have studied the relationship (39) defined for all the material particles of the Standard Model, by which it is evident that the different generations of particles are in different phases from each other, which have their own size associated to the limit value of the velocity, that is, to the value at which the particle reaches an inertial mass equivalent to the real mass of the particle of the next generation and at which, consequently, it would reach the phase of that particle. This not only regulates the distance between one particle and another in a simple measurement pattern such as the phase, expressed in units $\pi$, but also grounds or gives meaning to the dispersed and apparently chaotic values of the masses of the whole spectrum and defines other hitherto undefined ones (neutrinos), since it is these values that culminate or resolve satisfactorily (39).

This means that even if we ignore the original physical reasons for Eq. (39), they arise spontaneously since we are applying an energetic equivalence as an ordering principle which, given the results, could be showing that there is indeed such an equivalence, that is, a higher-order identity between one and another form of energy, an idea that would be reinforced by the impossibility of energetically exceeding this limit, that is, that the same energy value can exist associated with two generations.

The idea of progression or advance through the phase to the limit leads us to another one of transmutation of some particles into others or to the identification, at least practically, of both, and the latter to that of generation, that is, to that of real and effective generation of the different generations of particles from the preceding ones, until reaching generation zero as precursor and, at the same time, candidate for dark matter, as necessary responsible for the whole spectrum and variety of particles (matter and antimatter, with charge and without it), their common nexus. That common nexus or point of convergence is what is reached here with $\mathfrak{M}$, that is, that by means of it we not only reach an organization of the whole spectrum but it completes or resolves it, revealing the unification or logical organization described through the structural, that is, of the structured, quantized and hierarchical relation of the particles from the first one through the development of a single rhythmic or jump pattern defined by Eq. (39), by which, above any other consideration, such as the generational energetic transmutation, a phasic progression of the particle $\mathfrak{M}$ is evidenced, for which the other particles would be but the expression, through the different phases, of each of its states or forms in which $\mathfrak{M}$ itself can express itself.

A. Appendix

We are going to calculate the integral over $\upsilon$ of the integrand²:

$$\begin{array}{ll}\zeta \left(\upsilon \right)& =\frac{\mathrm{\hslash }\upsilon sin\left(\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right)}{a^3{\gamma }^{-3}}-\frac{\mathrm{\hslash }\upsilon \mathrm{\Delta }kcos\left(\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right)}{2a^2{\gamma }^{-2}}-\frac{\mathrm{\hslash }wcos\left(\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right)}{a^2{\gamma }^{-2}}\\ & =\frac{\mathrm{\hslash }\upsilon sin\left[\mathrm{\Delta }k\left(a{\left(1-{\upsilon }^2/c^2\right)}^{1/2}\right)\right]}{a^3{\left(1-{\upsilon }^2/c^2\right)}^{3/2}}-\frac{\mathrm{\hslash }\upsilon \mathrm{\Delta }kcos\left[\mathrm{\Delta }k\left(a{\left(1-{\upsilon }^2/c^2\right)}^{1/2}\right)\right]}{2a^2\left(1-{\upsilon }^2/c^2\right)}\\ & -\frac{\mathrm{\hslash }wcos\left[\mathrm{\Delta }k\left(a{\left(1-{\upsilon }^2/c^2\right)}^{1/2}\right)\right]}{a^2\left(1-{\upsilon }^2/c^2\right)}\end{array}$$EA.1

FIRST TERM- The term (A.1a) is left as:

$$\begin{array}{l}\int \frac{\mathrm{\hslash }\upsilon sin\left[\mathrm{\Delta }k\left(a{\left(1-{\upsilon }^2/c^2\right)}^{1/2}\right)\right]}{a^3{\left(1-{\upsilon }^2/c^2\right)}^{3/2}}\mathit{dv}\\ =\frac{\mathrm{\hslash }}{a^3}\int \frac{\upsilon sin\left[\mathrm{\Delta }k\left(a{\left(1-{\upsilon }^2/c^2\right)}^{1/2}\right)\right]}{{\left(1-{\upsilon }^2/c^2\right)}^{3/2}}\mathit{dv}\\ =\frac{\mathrm{\hslash }}{a^3}{\left[sin\left[\mathrm{\Delta }k\left(a{\left(1-{\upsilon }^2/c^2\right)}^{1/2}\right)\right]\int \frac{\upsilon }{{\left(1-{\upsilon }^2/c^2\right)}^{3/2}}\mathit{dv}\right[}_{{\upsilon }_o}^{{\upsilon }_f}\\ +\frac{\mathrm{\hslash \Delta }k}{a^2}\int \frac{\upsilon cos\left[\mathrm{\Delta }k\left(a{\left(1-{\upsilon }^2/c^2\right)}^{1/2}\right)\right]}{\left(1-{\upsilon }^2/c^2\right)}\mathit{dv}\end{array}$$EA.2

The term (A.2a) is will be the TERM-1 of the general solution, which we can put in a more elegant way as:

$$\frac{\mathrm{\hslash }}{a^3}sin{\left[\mathrm{\Delta }k\left(a{\left(1-{\upsilon }^2/c^2\right)}^{1/2}\right)\right]}_{{\upsilon }_o}^{{\upsilon }_f}\underset{{\upsilon }_o}{\overset{{\upsilon }_f}{\int }}\frac{\upsilon }{{\left(1-{\upsilon }^2/c^2\right)}^{3/2}}\mathit{dv}=\frac{\mathrm{\hslash }}{a^3}sin{\left[\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right]}_{\upsilon }\underset{{\upsilon }_o}{\overset{{\upsilon }_f}{\int }}\frac{\upsilon }{{\gamma }^{-3}}\mathit{dv}$$EA.3

SECOND TERM- The term (A.2b) has the same form as the term (A.1b) so we add it for analysis.

$$\begin{array}{l}\frac{\mathrm{\hslash \Delta }k}{a^2}\int \frac{\nu cos\left[\mathrm{\Delta }k\left(a{\left(1-{\upsilon }^2/c^2\right)}^{1/2}\right)\right]}{\left(1-{\upsilon }^2/c^2\right)}\mathit{dv}-\frac{\mathrm{\hslash \Delta }k}{2a^2}\int \frac{\upsilon cos\left[\mathrm{\Delta }k\left(a{\left(1-{\upsilon }^2/c^2\right)}^{1/2}\right)\right]}{\left(1-{\upsilon }^2/c^2\right)}\mathit{dv}\\ =\frac{\mathrm{\hslash \Delta }k}{2a^2}\int \frac{\upsilon cos\left[\mathrm{\Delta }k\left(a{\left(1-{\upsilon }^2/c^2\right)}^{1/2}\right)\right]}{\left(1-{\upsilon }^2/c^2\right)}\mathit{dv}\end{array}$$EA.4

(A.4) we can put it as:

$$\begin{array}{l}\frac{\mathrm{\hslash \Delta }k}{2a^2}\int \frac{\upsilon cos\left[\mathrm{\Delta }k\left(a{\left(1-{\upsilon }^2/c^2\right)}^{1/2}\right)\right]}{\left(1-{\upsilon }^2/c^2\right)}\mathit{dv}\\ =\frac{\mathrm{\hslash \Delta }k}{2a^2}\int \frac{\upsilon cos\left[\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right]}{{\gamma }^{-2}}\mathit{dv}\\ =-\frac{\mathrm{\hslash \Delta }k}{2a^2}{c}^2\int \frac{cos\left[\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right]}{{\gamma }^{-1}}d\left({\gamma }^{-1}\right)\end{array}$$EA.5

which will be the TERM-2 of the general solution.

THIRD TERM- The term (A.1c), is left as:

$$\begin{array}{l}-\frac{\mathrm{\hslash }w}{a^2}\int \frac{cos\left[\mathrm{\Delta }k\left(a{\left(1-{\upsilon }^2/c^2\right)}^{1/2}\right)\right]}{\left(1-{\upsilon }^2/c^2\right)}\mathit{dv}\\ =-\frac{\mathrm{\hslash }w}{a^2}\int \frac{cos\left[\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right]}{{\gamma }^{-2}}\mathit{dv}\\ =\frac{\mathrm{\hslash }w}{a^2}\int \frac{ccos\left[\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right]}{{\left(1-{\gamma }^{-2}\right)}^{1/2}\left({\gamma }^{-1}\right)}d\left({\gamma }^{-1}\right)\end{array}$$EA.6

which will be the TERM-3 of the general solution. That, taking into account the three results, is:

$$\begin{array}{ll}I& =\int \zeta \left(\upsilon \right)\mathit{dv}\\ & =\left(\frac{\mathrm{\hslash }}{a^3}\right)sin{\left[\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right]}_{\upsilon }\underset{{\upsilon }_o}{\overset{{\upsilon }_f}{\int }}\frac{\upsilon }{{\gamma }^{-3}}\mathit{dv}\\ & -\left(\frac{\mathrm{\hslash \Delta }k}{2a^2}\right)c^2\int \frac{cos\left[\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right]}{{\gamma }^{-1}}d\left({\gamma }^{-1}\right)\\ & +\left(\frac{\mathrm{\hslash }w}{a^2}\right)c\int \frac{cos\left[\mathrm{\Delta }k\left(a{\gamma }^{-1}\right)\right]}{{\left(1-{\gamma }^{-2}\right)}^{1/2}{\gamma }^{-1}}d\left({\gamma }^{-1}\right),\end{array}$$EA.7

B. Appendix

The adjustment to Eqs. (40), presented in Table 1, with the tabulated experimental data [9] is unique and incontrovertible for the case of leptons, but it is not, however, for the other charged particles. Thus, we have that the best evaluation for negative quarks is $\chi =0.517$ with these data, i.e. with the data intervals provided by the best experimental fit of the data:

$$m_d={4.67}_{-0.17}^{+0.48}\mathrm{MeV},m_s={93}_{-5}^{+11}\mathrm{MeV},m_b={4.18}_{-0.02}^{+0.03}\mathrm{GeV},$$EB.1

while for the intervals of the positives:

$$m_u={2.16}_{-0.26}^{+0.49}\mathrm{MeV},m_c=1.27\pm 0.02\mathrm{GeV},m_t=172.90\pm 0.04\mathrm{GeV},$$EB.2

the intended value $\chi =1.5$ is satisfied for some combination of values $\left(\text{with}{m}_u=2.00\mathrm{MeV}\right)$, which however we have discarded.

Inexact or indefinite adjustment as the case may be, but nevertheless sufficient to ensure that in case Eq. (39) is met, i.e. satisfied by a relation of natural numbers, it must be (plausibility criterion) with the values for $\chi$ defined in Eqs. (40).

In neither of the two cases, however, are we guaranteed certainty (neither to endorse the fit nor to discard the maladjustment) because we do not have that certainty about the experimental data which, coming from indirect measurements, are not totally reliable, and which are susceptible, therefore, to be corrected to other experimental values, less recognized, by means of some criterion methodology or tool (such as the one resulting from our proposal).

Consequently, we are not proposing, and presenting in Table 1, a simple and timely correction of data but an exact theoretical solution starting from (39), from which all certainty regarding the starting data will be derived.

Data that we can represent in a more visual way (Figures 6–8) on those already existing and presented in the Data listings (Ref. [9]), where we can verify a certain disparity with respect to some of the best fitted values for A which, however, as we have already mentioned, are in accordance, respectively, with the experimental results [11, 12], [13, 14] and [15, 16], among others, which we can also find in summary form [9].

Figure 6.

u-Quark and d-Quark. Status of our data (Table 1) with respect to the accredited data.

Figure 7.

s-Quark and c-Quark. Status of our data (Table 1) with respect to the accredited data.

Figure 8.

b-Quark and t-Quark. Status of our data (Table 1) with respect to the accredited data.

C. Appendix

Starting from (39), we are going to express a generic Eq. (43) in an alternative form with more manageable and differentiated parameters, that is

$$\begin{array}{ll}{\varphi }_i& ={\left[\beta {\left(\frac{1}{m_i}\right)}^{1/3}{\gamma }_i^{-1}\right]}^{-q}={\left[Z{\left(\mathrm{\hslash }/\pi \right)}^{1/3}{\left(\frac{1}{m_i}\right)}^{1/3}{\gamma }_i^{-1}\right]}^{-q}\\ & ={\left[Z{\left(\frac{\mathrm{\hslash }}{\pi m_i}\right)}^{1/3}{\gamma }_i^{-1}\right]}^{-q}={\left[Z{\delta }_i{\gamma }_i^{-1}\right]}^{-q}={\varphi }_0,{\varphi }_1,{\varphi }_2\dots \end{array}$$EC.1

Parameters that remain fixed in each phase of the different transitions and that, according to (C.1), fulfil:

$${\gamma }_i=\frac{m_{i+1}}{m_i}=\left(Z{\delta }_i\right)/{\varphi }_i\text{for}{\delta }_i={\left(\frac{\mathrm{\hslash }}{\pi m_i}\right)}^{1/3},$$EC.2

which allows to determine ${\gamma }_i$ for a phase, from the value of $Z$ and the initial mass, and then, as we said in Section 7, the mass or starting point of the next transit (and vice versa with two masses), and allows us to progress through these transitions in a simple and reproducible way, establishing a progressive sequence of well-defined ternaries, $m_0,{\delta }_0,{\gamma }_0-m_1,{\delta }_1,{\gamma }_1-m_2,{\delta }_2,{\gamma }_2$, from the input values.

Input values that, consequently, serve us to modulate or see the behaviour depending on them, that is, varying $Z$ or $m_0$, and check to what point they are what they should be unequivocally, as well as the initial expression that motivates all these relations and values. Verification that is done quickly and easily by means of an Excel table, such as the one represented in Figure 9.

Figure 9.

Phases transitions for negatively charged quarks $\left(-1/3\right)$.

On the represented table we see that, in fact, starting from the input values we reach in A9 the true initial value of the sequence, $m_0$, which, consequently, could be entered directly as a numerical input value (171.87). By doing it this way, we have managed to start the sequence from the value of the first-generation particle (known) and obtain ${\gamma }_0$ (E7) from it, that is, in reverse, although it is then recalculated, according to the equations and the progression of the transitions, in C11.

For this particular case, we have used the sequence $4\pi ,2\pi ,\pi \left(?\right)$ in A15 and D15, and could have used, for another proportional value of $Z$, the sequence $2\pi ,\pi ,$ which would not allow a transition to a fourth generation. It can be seen that for the antiparticle an identical progression is reached, in which the phases change the order of the sequence $\pi ,2\pi ,4\pi \left(?\right)$ and become divisor (by the sign of the charge in the exponent) for a value $\overline{Z}=33.0431$, which can be calculated directly by Eq. (C.2) or through its relation with $Z$.

$$\begin{array}{ll}\frac{\mathit{Z\delta }}{{\left(4\pi \right)}^3}& =\gamma ={\left(\pi \right)}^3\overline{Z}\delta \Rightarrow \frac{Z}{\overline{Z}}=\left[{\left(4\pi \right)}^3\right]{\left(\pi \right)}^3\\ & ={\left(4{\pi }^2\right)}^3\Rightarrow \overline{Z}={\left(61528.9\right)}^{-1}Z\end{array}$$EC.3

Analogously, we can establish the expression (39) in its final form $\chi ={\Re }^{-q}$ and establish its values for the two triads. Following the same case we would have an exponent (1/3) in $\Re$ and another (1/3) associated to $-q$, as detailed in the Excel table represented in Figure 10, with which we can vary any of the inputs and observe how it affects the default values $\chi$ exactly. On which, finally, we can replace the numerical input values C3-F3 by the values A11-D11-B15-E15 of the preceding table in order to control all the masses and other variables with the input of this one.

Figure 10.

Adjustment of Eq. (39) for negatively charged quarks (−1/3).

D. The physical issue

We can think, in spite of all the above, that the expression (39) lacks validity because it is not supported by physical presuppositions or even downplay the importance of the correspondence presented and consider it simply cunning. The truth is that although these presuppositions have not been presented (for the reasons given above), they appear or ooze from the expression itself, while it could be said, as we shall see, that the correspondence is impossible without this underlying physical requirement or a mathematical one of equal importance.

Regarding the latter, we have already said, Section 5, that our analysis was as simple and unequivocal as that derived from the equation of a straight line. That is its importance.

In effect, if we think of any three straight lines defined by their respective pair of points, for example B and C, and we ask ourselves about the probability that the three converge at a single point A, or about the requirement for them to do so, we will conclude that the lines will pass through point A if the definition of the lines includes passing through that point, that is, if the lines are defined by point A and the respective point B, so that point C is as unnecessary as any other point D. Something similar happens with the point X, with respect to the different points P, although in appearance (even by the way of introducing it) it is presented as fortuitous and unexpected. That is to say, although in our case they are not straight lines but curves, and the equation therefore is not that of a straight line but the equation of the relativistic mass, defined on the Lorentz factor that defines the trajectory between B-C and C-D, if we ask ourselves the same question, the answer is identical: the points B-C-D that following the trajectory reach the points A are exclusively those that, starting from A, reach by means of the trajectory the points B-C-D for the different equations. A statement that in the present case, unlike that of the straight lines, needs to be developed, A statement that in this case, unlike that of the straight lines, needs to be developed, for which we will make use of Figure 11, in which we have also replaced the curves by straight lines using the logarithmic scale for better visualization, equivalence and understanding, where it is highlighted not only the confluence, or common and logically unique point of intersection of the three straight lines, but also that it occurs despite the restriction of the domain of application to those points (${\varphi }_0=\pi ,2\pi ,3\pi \dots$), among all the real ones, in which it is possible, as we shall see, to find a particle, for being those that conform a phase or represent a node according to the conditions given in Eq. (39).

Figure 11.

Development or prolongation of the phases (${\varphi }_0^{\mathit{xi}}$) prior to $\mathfrak{M}$ for each of the families (L, +Q, −Q).

In a first approximation, although we have not followed this itinerary for the presentation of results, the phases are defined (the most simplified ratio of natural numbers) because $\chi$ is defined from an approximate version of $\chi$ itself, reached by means of the plausibility criterion applied to the masses of the particles (see Appendix C), which in turn defines Z for those values; or if desired by means of Eq. (C.2) and the factors ${\gamma }_1$ and ${\gamma }_2$ obtained from $m_1,m_2,m_3$. Having defined Z we have found ${\varphi }_0$ for each particle chain or electric charge (L, +Q, −Q) by the corresponding Eq. (C.1) or the analogous (43) as a function of $\beta$,

$${\varphi }_0={\left[\beta {\left(m_0^2/m_1^3\right)}^{1/3}\right]}^{-q}={\left[Z{\delta }_0{\gamma }_0^{-1}\right]}^{-q}$$ED.1

which gives rise to a common particle $m_0=171.87\mathrm{eV}$ (that we found in (42), for ${\alpha }^{-q}\cong 1$, without needing to know the Z value) that holds a factor ${\gamma }_0$ with the first generation particle $m_1$ in each family.

Thus, we have expressed it, when the reality is that once defined Z, we find ${\varphi }_0^1,{\varphi }_0^2,{\varphi }_0^3\dots$ by means of (D.1) that would give rise for ${\gamma }_0^1,{\gamma }_0^2,{\gamma }_0^3\dots$ to some masses $m_0^1,m_0^2,m_0^3\dots$ in each one of those chains or electric charges, that is to say, we would reach the corresponding sequences $m_0^{L1},m_0^{L2},m_0^{L3}\dots$, $m_0^{+Q1},m_0^{+Q2},m_0^{+Q3}\dots$ and $m_0^{-Q1},m_0^{-Q2},m_0^{-Q3}\dots$ dispersed and discordant, that only coincide in the singular value $m_0$ for the phases ${\varphi }_0^{L1},{\varphi }_0^{+Q2},{\varphi }_0^{-Q1}$ (to which we have generically called ${\varphi }_0$ obviating the sequences), from which the values follow different courses.

We can concretize this on Figure 9 for -Q, where it is seen (Box E5-6-7) that ${\gamma }_0={\gamma }_0^1$ decreases in the ratio $\left[{\left(1/n^3\right)}^{3/2}\right]=n^{-9/2}$ as we vary $n$ (initially to 1) in agreement with ${{\varphi }_0}_{-\mathit{Qi}}^{}=\left(1,2,3\dots \right)\times \pi$, consequently increasing the mass (Box A9-10-11) in the form $m_0^{-\mathit{Qi}}=\left(171.87,3888.94,24112.48\dots \right)\mathrm{eV}$, the same occurring, as we will detail later, for $n^{-3/2}$ in the chain L with $m_0^{\mathit{Li}}=\left(171.87,343.85,515.78\dots \right)\mathrm{eV}$, while +Q responds to $n^{9/4}$ and would lead us $m_0^{+\mathit{Qi}}=\left(857.54,171.87,69.02\dots \right)\mathrm{eV}$ for ${{\varphi }_0}_{+\mathit{Qi}}^{}=\pi ,2\pi ,3\pi \dots$, where we can apply an initial phase difference $\pi$, by redefining the starting “straight line”, so that they all find the common value at ${\varphi }_0=\pi$, as a didactic adjustment (since it is not a requirement).

That is we see that there is not only one pro-precursor particle but many as a consequence of the fact that there is not only one ${\gamma }_0$ but ${\gamma }_0^i$ applicable to each chain, seeing also that neither the different masses $m_0^1,m_0^2,m_0^3\dots$ nor the generic mass $m_0$ chosen among them, are defined by a sort of adjustment or approximation but on the contrary they are subject to the sequence of values of a series, which we can even define in a specific way as a consequence of the fact that (and here is the key of the matter) responds to an analytical relationship between ${\gamma }_1$ and each ${\gamma }_0^i$. A relationship that we will establish initially for -Q and then generalize. From Eq. (C.1) for ${\varphi }_1$ and the d-quark we have:

$${\left[Z_d{\delta }_d{{\gamma }_d}^{-1}\right]}^{1/3}={\varphi }_1=4\pi \Rightarrow {\gamma }_d=\frac{Z_d{\delta }_d}{{\left(4\pi \right)}^3}$$ED.2

Now we are looking for:

$${{\left[Z_0{\delta }_0{{\gamma }_0}^{-1}\right]}_d}^{1/3}={\left[Z_d{\delta }_0{{\gamma }_0}^{-1}\right]}^{1/3}={\varphi }_0=\pi$$ED.3

Given that:

$${\delta }_0={\left(\frac{\mathrm{\hslash }}{\pi m_0}\right)}^{1/3}={\left(\frac{\mathrm{\hslash }}{\pi m_d{{\gamma }_0}^{-1}}\right)}^{1/3}={\left(\frac{\mathrm{\hslash }}{\pi m_d}\right)}^{1/3}{{\gamma }_0}^{1/3}={\delta }_d{{\gamma }_0}^{1/3},$$ED.4

we have:

$${\left[Z_d{\delta }_0{{\gamma }_0}^{-1}\right]}^{1/3}={\left[Z_d\left({\delta }_d{{\gamma }_0}^{1/3}\right){{\gamma }_0}^{-1}\right]}^{1/3}={\left[Z_d{\delta }_d{{\gamma }_0}^{-2/3}\right]}^{1/3}={\varphi }_0$$ED.5

From which we obtain, for $Z_d=Z_s=Z_b=Z^{-Q}$, that:

$${{\gamma }_0}^{2/3}=\frac{Z^{-Q}{\delta }_d}{{\left({\varphi }_0\right)}^3}\Rightarrow {\gamma }_0={\left(\frac{Z^{-Q}{\delta }_d}{{\pi }^3}\right)}^{3/2}=31390.28.$$ED.6

which is the result presented in Figure 9, with which, from $m_d$, we can obtain $m_0=171.87\mathrm{eV}$. From here we only have to generalize it for the three classes of charged particles (Z), for any first generation (${\delta }_1$) of reference (although it is true that it is applicable to any two contiguous generations), and for ${\varphi }_0^1={\varphi }_0,{\varphi }_0^2,{\varphi }_0^3\dots$, to finally, that was our objective, put it as a function of the Lorentz factor of that reference (${\gamma }_1$), that is:

$${\gamma }_0^i={\left(\frac{Z{\delta }_1}{{\left({\varphi }_0^i\right)}^{-1/q}}\right)}^{3/2}={\left(\frac{{{\varphi }_1}^{-1/q}}{{\left({\varphi }_0^i\right)}^{-1/q}}\right)}^{3/2}{\left(\frac{Z{\delta }_1}{{{\varphi }_1}^{-1/q}}\right)}^{3/2}=\mathrm{\Theta }\times {{\gamma }_1}^{3/2}$$ED.7

Being in our starting case, for ${\gamma }_d=15.548$:

$$\begin{array}{ll}{\gamma }_0^1& =\mathrm{\Theta }\times {{\gamma }_1}^{3/2}={\left[{\left(\frac{4\pi }{\pi }\right)}^3\right]}^{3/2}{\left(\frac{Z^{-Q}{\delta }_d}{{{\varphi }_1}^{-1/q}}\right)}^{3/2}\\ & ={\left(64\right)}^{3/2}\times {{\gamma }_d}^{3/2}=512\times 61.309=31390.28\end{array}$$ED.8

While in the other classes for $m_0=171.87\mathrm{eV}$, with ${\varphi }_0\equiv {\varphi }_0^{L1}={\varphi }_1$ and ${\varphi }_0\equiv {\varphi }_0^{+Q1}={\varphi }_1$, the result is $\mathrm{\Theta }=1$, which in all cases will be affected by taking ${\varphi }_0^2,{\varphi }_0^3$, affecting ${\gamma }_0^2,{\gamma }_0^3$ and consequently affecting $m_0^2,m_0^3$.

Beyond the evidence of the behaviour derived from the data, and saving the mentioned didactic adjustment, on the figure it is clearly seen that the lines -Q and L intersect at a point (as any pair of non-parallel lines) to then progress in a divergent, essentially discrete and exponentially dispersed way, in such a way that it is highly improbable that any other value $m_0^{-\mathit{Qi}}$ reached by a node coincides with one $m_0^{\mathit{Li}}$ reached by another node, highly improbable that it has coincided, as it happens, with a value $m_0^{+\mathit{Qi}}$ (for a node or point of the application domain), and already impossible that it does so (given its decreasing sequence) with another $m_0^{+\mathit{Qi}}$, from which we conclude that the only and exceptionally possible common value of mass is $m_0=171.87\mathrm{eV}$, which is the truly relevant fact, and the one that has physical significance, being that significance the basis for such an improbable fact occurs: as we said (and now we have substantiated) when a series of straight lines intersect at a point $A$ (and we have converted our curves into straight lines) it is because they are defined by $A$ and their respective and differentiated slopes (governed in this case by physical criteria), everything else being a consequence: it is not possible to explain a convergence of this degree, if not as a specular process of a divergence.

In our real physical case, our straight lines are curves, and in them, in effect, the condition is only fulfilled for the curves that fulfil the above, that is to say, that in origin fulfil Eq. (C.1) for $m_i=m_0$, because these are the only ones that can count with $m_0$ among their points. We speak of curves and more specifically of the curves described by the Lorentz factor, that is to say, by their development. The factor $\gamma$ describes the physical relation that exists between the mass of a particle and its inertial or relativistic mass, i.e., if we wanted to find another function that unites these two masses taken as points for any value of the latter (any speed) we would not find it because it is the one that describes the physical relation that is occurring. This is important because it supposes a first limitation with respect to the family of possible curves. From there, the Z value is defining a family of curves among them: the one that starting from $m_1$ reaches $m_2$ and $m_3$ by means of some concrete phases, according to (43), then shown by the ratio of those masses. That is the true physical condition, the one expressed in (C.1), which makes of those masses not any masses but some concrete masses, that is to say, that they are these and not others.

These are masses that have proven their real existence (although only $m_1$ in a natural and generalized way) while others such as $m_0^1,m_0^2,m_0^3\dots$ only have it guaranteed within the family of curves, just as a hypothetical fourth generation $m_4$ would have it through Z for the corresponding ${\varphi }_3$. Apart from this, Z is defining them equally, that is, it is saying what characteristics would have some preceding and/or precursor particles subjected to the same pattern, governed by the same Z, regardless of what Z is. This said for each type of charge implies that sequence of infinite possibilities, restricting ourselves instead to the candidates that fulfil a valid pattern for the three classes of particles, it turns out that only one particle exists, that is, only one particle intersects in the projection of the three chains, only one is viable as a precursor, only one has the capacity to evolve (Figure 4), according to three different Z-values, in these three different ways and reach the whole material spectrum, and justify even other material forms.. That is the immediate physics of (39), which does not need to presupposes, but rather creates presuppositions or new physics environments.