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European Journal of Applied Sciences – Vol. 11, No. 1

Publication Date: February 25, 2023

DOI:10.14738/aivp.111.14114.

Fèvre, R. (2023). Elementary Particle Masses: Alternative Model to the BEH Mechanism. European Journal of Applied Sciences,

Vol - 11(1). 626-636.

Services for Science and Education – United Kingdom

Elementary Particle Masses: Alternative Model to the BEH

Mechanism

Raymond Fèvre

Abstract

In a first paper [1], we developed a model to calculate the masses of charged leptons

by quantifying the electrostatic field generated by these particles. In a second

article [2], we extended this model to weak and strong interactions in order to

calculate the masses of all elementary fermions. The present paper is a modified

version of the second article, adding a hypothesis on the mass of dark matter

particles. In this way, the model could be an alternative to the BEH mechanism of

the standard model.

INTRODUCTION

The BEH mechanism, called the "Higgs Field", was conceived in the 1960s to overcome

inconsistencies in the standard model. The standard model states that the masses of the

elementary particles are all null.

A quantum field was therefore introduced in order to impart a mass to the elementary particles,

first for the bosons W and Z, and then for the elementary fermions.

A particle must be associated with this quantum field. The particle theorists have thus assumed

the existence of a Higgs boson associated with the Higgs field. However, as the Higgs field is a

qualitative concept, it does not predict the mass of the boson and does not allow calculation of

the mass of elementary particles. Moreover, this theory postulates that the neutrino mass is

zero, which is false

The experimental discovery in 2012 at CERN of a boson with a mass close to 125 GeV was

presented almost unanimously as corresponding to the Higgs boson. But proof for this

identification was not provided. Now, the BEH model has been called into question by two

experimental results. The CMS and ATLAS detectors give two slightly different values (disjoint

error bars) for the H boson mass. The retroactive measurements of the W boson mass

(Fermilab) also give a different value from that measured so far.

The author of this article published a paper in JFAP on the subject in 2016, suggesting an

alternative for the Higgs boson: composite particles consisting of ultra-relativistic (UR)

interactions of tau leptons and /or bottom quarks [3]. A recent article [4] develops this subject

and shows that the W and Z bosons can also be the same type of composite particles

Furthermore, in 2014 we published an article in Physics Essays [1] suggesting an alternative to

the BEH mechanism in order to explain and calculate the mass of charged leptons. The model

infers the masses of these leptons from the quantification of the electrostatic field generated by

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Fèvre, R. (2023). Elementary Particle Masses: Alternative Model to the BEH Mechanism. European Journal of Applied Sciences, Vol - 11(1). 626-

636.

URL: http://dx.doi.org/10.14738/aivp.111.14114

the particles. Then, in another article [2], we extended the model to weak and strong

interactions in order to calculate the masses of all elementary fermions.

The present article resumes, in simpler form, the model used in the article cited above. It does

not repeat the calculations concerning cosmic neutrinos but introduces a hypothesis

concerning dark matter.

THE QUANTUM MODEL ON CHARGED LEPTONS

Each of the three charged leptons (electron, muon, tau) creates an electrostatic field. The

expression of the particle self-potential energy, if the electric charge is distributed over a

sphere of radius r is:

(r) =

αħc

(1a)

α is the fine structure constant:

α =

137.036

Assuming that the mass energy of the particle is of electrostatic origin, the principle of this

model is to define a quantization process of the difference between the electrostatic field energy

and the elementary particle mass energy.

(r) =

αħc

− mc

(1 b)

Then the model considers the quantum operator: Êd (r) associated with the energy defined

above and assumes that we can define a corresponding momentum operator by the following

relation:

Âc = ±4πθÊd

(r) (2)

This equation is the same type as that which connects the momentum vector and the energy of

the photons in the framework of the quantization of the free electromagnetic field (see below).

The parameter θ characterizes the particle and will be defined later for each particle.

We then pass to the equation of wave in spherical coordinates:

iħc

d(rψ)

rdr = ±4πθEd

(r)ψ (3)

Solutions to this differential equation appear immediately:

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rψ(r) ≺ exp (±

4πiθ

ħc

∫ Ed

(r)dr)

(4)

We assume that the lower integration range is very small, of the order of magnitude of a Planck

length.

Considering the real wave function, the sum of the function written above and its complex

conjugate, we obtain:

rψ(r) ≺ cos [

4πθ

ħc

∫ Ed

(r)dr]

(5)

The relation (3) shows that the derivative of the above function is zero when the total energy

defined in (1b) is zero, and therefore for the following value of the distance r to the particle:

rm =

αħ

2mc

(6)

For this value of r, called “classical radius of particle”, the electrostatic energy of particle

equals its mass energy, but it has not physical sense in classical physics.

The derivative of a function is null at its extrema, so the above value corresponds to an

extremum. Since the function defined in (eq.5) is a cosine function, this occurs when its

argument is a multiple of π and for the value of r given above, which defines a quantization

relation:

4πθ

ħc

∫ Ed

(r)dr = nπ

; n = integer (7)

Using (6) and (7), it becomes:

2θ

+ 1 with a =

= 137.036 (8)

It is interesting to transform (8) by introducing the particle mass according to relation (6), the

Planck mass and the Planck radius:

rp = √

Għ

3 mpcrp = ħ

We obtain thus:

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Fèvre, R. (2023). Elementary Particle Masses: Alternative Model to the BEH Mechanism. European Journal of Applied Sciences, Vol - 11(1). 626-

636.

URL: http://dx.doi.org/10.14738/aivp.111.14114

2θ + 1 + ln(2a) − ln

(9)

= exp (−

2θ + K) (10)

K = −1 − ln(2a) + ln

Therefore, the masses of charged leptons are functions of the elementary electrical charge.

Two parameters determine the mass of each of them: the quantum number n, deriving from the

characteristics of the wave function and the parameter θ which will be determined here in after.

It is known that the ratio between Planck mass and the electron (index 1) is such that:

= 51.53 (11)

This value is obtained with equation (9) by giving the parameters the following values:

n = 3; θ = 3√2 ; r0 =

35.5

(12)

The last datum means that the value of r corresponding to the lower integration range is less

than Planck's radius; it supposes, according to this model, that the core of the particle has a

dimension of the order of magnitude of the Planck radius. However, we show in two articles (cf.

section 3) that elementary particles can be considered as Schwarzschild photon spheres with a

radius r0 = rp/33.5 close to the value of eq.12. Thus, the lower integration range is not

arbitrary for this model.

The ratio between the mass of each of the two heavy leptons (of indices j = 2, 3) and that of the

electron is obtained from equation (9):

(

θ1

−

θj

) a = 137.036 (13)

This relationship will make it possible to calculate the mass of the muon and tau leptons, but it

will be necessary to take into account the radiative corrections (RC) to Coulomb's law as

described in the appendix.

ASSUMED STRUCTURE OF THE ELEMENTARY PARTICLES AND DETERMINATION OF THE

COEFFICIENTS θ

Fundamentals

We published an article in “Physics Essays” [5] showing that elementary particles can be

considered as self-gravitating photons on a Planck scale. In another article published in EJAS

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ref. [6], we develop this idea and show that each elementary lepton is composed of six

Schwarzschild photon spheres having the same center and radius. The model constructed from

this hypothesis makes it possible to calculate the value of the elementary electrical charge. Each

photon sphere carries one sixth of the elementary charge (plus or minus); one pair of photon

spheres carries one third (plus or minus) of the elementary charge or zero charge. Thus, we can

construct a representation of charged leptons and neutrinos.

Many authors have studied the spherical shells of self-gravitating photons within the

framework of general relativity, but without involving the quantum theory, which is necessary

to calculate the numerical value of the elementary electrical charge. The latest of the articles

concerned is cited in reference [7].

Representation of Leptons

We represent one S-photon sphere of charge –e/6 by the symbol Θ and that of charge +e/6 by

Charged leptons (electron, muon, tau): electric charge: -e

{ΘΘ| ΘΘ| ΘΘ}

Charged anti-leptons: electric charge: +e

{ΟΟ| ΟΟ| ΟΟ}

Neutrinos

Case 1: three zero pairs of S-photon spheres: {ΟΘ| ΟΘ| ΟΘ}

Case 2; one null pair and two pairs of opposite charges: {ΟΟ| ΟΘ| ΘΘ}

STRUCTURE AND MASSES OF THE THREE CHARGED LEPTONS

We have determined empirically that for charged leptons, the expression of the parameter θ in

eq.4 depends upon accounting for the three pairs of S-photon spheres, according to the

following coefficients: for one pair: √2; for a quadruplet of two pairs: 2 = √4; for the three

pairs together: √6

Electron

For the electron, the parameter θ is assumed to be the sum of the values corresponding to three

separate pairs:

θ1 = 3√2 (15)

The quantum number n = 3 gives the numerical result presented above.

Tau

For the tau, the parameter θ is assumed to be the sum of the value corresponding to a pair +

that corresponding to the quadruplet of the other two grouped pairs: θ3 = 2 + √2

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Fèvre, R. (2023). Elementary Particle Masses: Alternative Model to the BEH Mechanism. European Journal of Applied Sciences, Vol - 11(1). 626-

636.

URL: http://dx.doi.org/10.14738/aivp.111.14114

The value n = 2 of the quantum number makes it possible to obtain the mass of the tau as a

function of that of the electron with relation (13)

Muon

For the muon, the parameter θ is assumed to be the sum of the corresponding values to two

quadruplets of 2 pairs, subtracting the value corresponding to the common pair to the two

quadruplets, assumed to be equal to: √6/3. We obtain thus:

θ2 = √4 + √4 −

√6

= 4 −

√6

(14)

The value n = 2 of the quantum number makes it possible to obtain the mass of the muon with

relation (13).

The method is empirical but coherent because there are only three ways presented here to

combine the three pairs of photons, individually and / or by quadruplet of two pairs. This

justifies the existence of three, and only three, families of particles.

The calculation obtained from the relation (13) and taking into account the radiative

corrections give values very close to the experimental measurements:

Calculated mass of the muon: 105.87 MeV _ mass measured: 105.66 MeV

Calculated mass of the tau: 1781 MeV _ mass measured: 1777 MeV

STRUCTURE AND MASSES OF NEUTRINOS

Neutrinos are devoid of electrical charge. However, each fundamental element as defined

previously has an electric charge, positive or negative, equal in absolute value to |e|/6. Its

electrostatic energy is therefore equal, with respect to the potential energy defined in (1), to

Ep(r)/36. Reasoning thus, the potential energy of all these six elements would be: Ep(r)/6.

Moreover, if into defining the momentum operator all six permutations of the fundamental

elements are taken into account, we arrive at Ep(r).

We will study this hypothesis, which means using the model of charged leptons for neutrinos.

The calculation of the masses of neutrinos will therefore be done by means of a relation

identical to relation (13):

mνj

(

θe

−

nνj

θνj) j = 1,2,3 a = 137.036 (15)

Let us add that the radiative corrections will not intervene in this calculation because they

concern only the charged leptons whose mass is greater than that of the electron. Below we

give the value of θ and the mass of the three neutrinos.

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Electron Neutrino

It is assumed, as for the electron, that each pair is taken into account separately and that the

value of θ for a neutral pair is 1. For the three neutral pairs, θν1 = 3. Assuming the quantum

number 3, using eq.18 we obtain for the electron neutrino mass:

mν1 = 0. 983 meV (16)

Muon Neutrino

For the muon, only the quadruplets of 2 pairs are taken into account to calculate θ. The same

principle will be applied for its neutrino by considering two ways in the state 2 above to

consider the quadruplets (for one quadruplet θ = √3). We obtain after subtracting the value 1

of a neutral common pair:

θν2 = 3√3 + 1

With n=4 for the quantum number, we obtain for the muon neutrino mass:

mν2 = 8. 60 meV (17)

mν2

2 − mν1

2 = 7.3 × 10−5eV

The masses of neutrinos are unknown, only the differences between the squares of their masses

can be measured, or rather between the masses defined in the framework of the "neutrino

mixing paradigm" [6]

The experimental measurements give: Δm21

2 = 7.5 ± 0.3 x 10−5eV

Tau Neutrino

For the tau neutrino, we find empirically that the calculation of θ is based on the sum of 2 cases

of the state 1 above. Case1: similar to that of tau. Case2: analogous to that of the muon.

We consider the case 1 above; we have two possibilities to calculate θ. In the first one, the value

of the quadruplet of 2 pairs is assumed to be √2 and that of the pair 1; then θ1 = √2 + 1 . In the

second one, the value of a pair is subtracted from the sum of the 2 quadruplets θ2 = 2√2 − 1 .

The value of θ results from the sum of these two quantum states: θν3 = 3√2 ; n=4 is the value

retained for the quantum number. With eq.18 we obtain:

mν3 = 49. 5 meV (18)

mν3

2 − mν1

2 = 2.45 × 10−3eV

The experimental measurements give in the framework of the "neutrino mixing paradigm"

give:

Δ m31

2 = 2.5 ± 0.1 × 10−3 eV

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Fèvre, R. (2023). Elementary Particle Masses: Alternative Model to the BEH Mechanism. European Journal of Applied Sciences, Vol - 11(1). 626-

636.

URL: http://dx.doi.org/10.14738/aivp.111.14114

STRUCTURE AND MASSES OF QUARKS

The quarks carry an electrical charge: |e|/3 or 2|e|/3 and a color charge. In our model based on

self-gravitating photon pairs, we interpret this fact in the following way. It is known that in

quantum theory one cannot refer to the conventional notions of electrical vector and magnetic

vector for the photon. However, the notions of '’electric photon'’ and '’magnetic photon'’ are

distinguished, which suggests that the charge of color is related to the magnetic component of

the photons. Moreover, the experimental data indicate that the numerical value of the coupling

of the strong interaction tends to the constant α to the high energies (property of asymptotic

freedom). We will translate these remarks as follows for the quarks:

-- When the electric charge of the quark is |e|/3, a pair of photon spheres is electric, the

other 2 pairs are colored

-- When the electric charge of the quark is 2|e|/3, two pairs are electric, the third is

colored.

-- Thus, the model developed for leptons can also be applied to quarks, which means that

the calculation of the mass of a quark, function of the electron mass, will be done by applying a

relation of the same type as relation (13):

(

θe

−

θq

) (19)

The problem posed by the radiative corrections will be discussed below. The quarks can thus

be represented as follows; the color charge being represented by a triangle:

Quarks: up, charm, top: {ΟΟ| ΟΟ | ΔΔ}

Quarks: down, strange, bottom: {ΘΘ| ΔΔ| ΔΔ}

Masses of Quarks in the Electron Family

To obtain the parameter θ of the up and down quarks, we will proceed as for the electron, that

is, we will sum the coefficients of the three pairs, considered individually, assuming that the

coefficient of a colored pair is equal to 3. It is not possible for the quarks to take into account

the radiative corrections.

Down

--For the d quark: θd = 6 + √2; n = 5

The calculation with relation (24) gives: md = 4.81 MeV (20)

The QCD calculation carried out by Christine Davies [8] gives for its part: md = 4.79 ± 0.16 MeV

--For the u quark: θu = 3 + 2√2 n=4

The calculation with relation (24) gives: mu = 2. 13 MeV (21)

The calculation of QCD [8] gives for its part: mu = 2.01 ± O.14 MeV

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Masses of Quarks in the Muon and Tau Families:

In ref. [ ] we calculate the quarks masses of these two families, using a similar reasoning as that

for the muon and the tau. We give here only the results.

Strange: 115 MeV

Charm: 1200 MeV

Bottom: 4770 MeV

Top: 175 MeV

SUMMARY TABLE:

Electron 3√2 3 1.414 Reference 0.511 MeV

Muon 4 −

√6

2 1.592 105.87 MeV 105.66 MeV

Tau 2 + √2 2 1.707 1781 MeV 1777 MeV

Neutrino e 3 3 1 0.983 meV < 1 eV

Neutrino μ 3√3 + 1 6 1.0325 8.47 meV < 1 eV

Neutrino τ 3√2 4 1.0607 49.5 meV < 1 eV

Up 3 + 2√2 4 1.457 2.13 MeV 2.01 MeV (Davies)

Down 6 + √2 5 1.483 4.81 MeV 4.79 MeV (Davies)

DARK MATTER

In the article ref. [6] we present a hypothesis on dark matter considering the empirical

relationship below between the parameters θ and the quantum numbers n of the charged

particles in the first family:

θ(k, l) = √2k + 3l; n = k + 2l; k, l = [0,1,2,3]; k + l = 3;

This relationship allows us to suppose the existence of a dark quark (without electric charge)

with parameters which give the mass below:

k = 0; l = 3; => θ = 3 × 3; n = 6; => mdq = 8.163 MeV; (22)

We can assume now a neutral color dark baron composed of 3 dark quarks of 3 different colors.

As there are 3 pairs of dark quarks, then 3 possible movements, we can calculate the dark boson

mass as 3 times the mass of an elliptic UR dark quarkonium. Using the calculation of the mass

with the composite particle model used in [6]:

IC(dq, dq) =

6α

; mdb = 3 (

12α

+ 2) mdq = 2566 GeV; (23)

Let us calculate now the ratio:

2mdb

mp + me

= 5.467;

We can see that this ratio is very close to the ratio: dark matter/ordinary matter:

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Fèvre, R. (2023). Elementary Particle Masses: Alternative Model to the BEH Mechanism. European Journal of Applied Sciences, Vol - 11(1). 626-

636.

URL: http://dx.doi.org/10.14738/aivp.111.14114

% dark matter

% ordinary matter

26.8

4.9

= 5.469

Then the existence in the Universe of two dark baryons for one proton can explain the ratio

“dark matter/ ordinary matter”. We justify this point in [6].

CONCLUSION

The model presented is somewhat beyond the standard model; however, it cannot be termed

"new physics" since it appeals only to the simplest and best known concepts of quantum theory.

It only extends the notion of quantum momentum to a static field.

As we have pointed out, the determination of the parameter θ for each elementary particle,

which defines its mass, is the result of an empirical investigation. A more theoretical approach

will be necessary to tie the value of this parameter with particle properties. A beginning of such

an approach is the calculation of the masses of the three charged leptons and that of the three

neutrinos. Another one is the observed coherence between the values of θ in the first family of

elementary particles. This allows us to make a hypothesis on dark matter.

Appendix: Corrective radiation to Coulomb's law

Electrically charged virtual particles are the cause of a quantum vacuum polarization modifying

the effective value of an actual electric charge according to a function of the distance to this

charge. We have taken this effect into account, called "radiative corrections to the Coulomb law"

in our article cited in [2] for electron-positron virtual pairs.

The pairs of virtual particles e+ e- increase electric charges when r -> 0. Below the relation

defining the electrostatic potential of an electron at a distance r with respect to the charge

measured at infinity [9]:

φ(r) =

[1 +

2α

3π

(ln

mcr

− C −

)]; me

:electron mass

C=O.577... Euler constant

This effect is expressed in the model by the following expression of the energy (eq.1a) of the

electric field created by the particle and its mass energy.

(r) =

αħc

[1 − λln (

γmecr

)] − mc

λ =

2α

3π

mecr < ħ γ = exp (C +

)

C: Euler's constant = 0.577.

Continuing the calculation as before, we obtain for the mass of the electron and the muon:

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= K −

nea

2θe

−

ln2

;

mμ

(

θe

−

nμ

θμ

) −

2 (

2amμ

γme

)

To calculate the mass of the tau, it is necessary to take into account the radiative corrections

due to the virtual pairs of the muons-anti-muons, namely:

mτ

(

θe

−

nτ

θτ

) −

[ ln

2 (

2amτ

γme

) + ln

2 (

2amτ

γmμ

) ]

For the mass of the muon, we obtain: 105.87 MeV (for 105.66 MeV measured)

And for the mass of tau: 1781 MeV (for 1777 MeV measured)The value of radiative corrections

is important in the case of muon and tau. In their absence, we would have found a mass 7%

higher for the muon and 16% higher for the tau. On the other hand, there is no CR for neutrinos.

Concerning the quarks, the RC must be taken into account since these particles comprise a

fractional electrical charge. However, in this case the screen effects due to the strong interaction

also intervene, opposites to the RCs generated by the charged leptons. These effects are due to

complex QCD calculations that we have not carried out, limiting ourselves to gross estimates of

global RC.

References

[1] Raymond Fèvre; A Model of the Masses of Charged Leptons; Physics Essays; Vol 27 N ° 4 (December 2014) p

608_611

[2] Raymond Fèvre; Elementary Particle Masses: An Alternative to the Higgs Field, Chapter 1 in the book titled

“Leptons: Classes, Properties and Interactions”; Nova Science Publishers; Feb. 2019

[3] Raymond Fèvre; The Higgs Boson and the signal at 750 GeV, Composite Particles ? Journal of Fundamental AP

Vol3, N°1 (2016)

[4] Raymond Fèvre; The W Boson Mass: 80.433 GeV: A Result of a Composite H, W, Z, Bosons Model: Other

Results: H, Z, Proton, Neutron and Dark Matter Masses; European Journal of Applied Sciences-Vol.10, N°3;

June 25, 2022

[5] Raymond Fèvre; Photons Self-gravitating and Elementary Charge; Physics Essays; Vol 26 N ° 1; P 3_6 (March

2014)

[6] Raymond Fèvre; Cosmological Constant (Λ); Fine Structure Constant (α); Two Results of Friedmann-Planck- Schwarzschild Models on Vacuum and Elementary Particles; European Journal of Applied Sciences; Vol 10,

N°1 February 25, 2022

[7] Hakan Andrasson, David Fajam, Maximilian Thaller; Models for Self-gravitating Photon Shells and Geons;

Annales Henri Poincaré, Feb. 2017

[8] Christine Davies; Standard Model Heavy Flavor Physics on the lattice; ArXiv: 1203.3862v1 (17 March 2012)

[9] E.Lifchitz et L.Pitayevski; Théorie Quantique Relativiste ; Deuxième Partie ; p. 66