Theoretical approach to the masses of the elementary fermions Nathalie Olivi-Tran To cite this version: Nathalie Olivi-Tran. Theoretical approach to the masses of the elementary fermions. Nuclear and Particle Physics Proceedings, Elsevier, 2020, 309-311C, pp.73-76. 10.1016/j.nuclphysbps.2019.11.013. hal-02322855 HAL Id: hal-02322855 https://hal.archives-ouvertes.fr/hal-02322855 Submitted on 21 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Nuclear and Particle Physics Proceedings Nuclear and Particle Physics Proceedings preprint (2019) 1–4 Theoretical approach to the masses of the elementary fermions N.Olivi-Trana aLaboratoire Charles Coulomb, C.N.R.S., Universite de Montpellier, place Eugene Bataillon, 34095 Montpellier cedex 5, France Abstract We made the hypothesis that, if spacetimep is composed of small hypercubes of one Planck length edge,p it exists elementary wavefunctions which are equal to 2 exp(ix j) if it corresponds to a space dimension or equal to 2 exp(it) if it corresponds to a time dimension. The masses of fermions belonging to the first family of fermions are equal to integer powers of 2 (in eV=c2) [1]. We make the hypothesis that the fermions of the 2nd and 3rd families are excited states of the fermions of the 1st family. Indeed, the fermions of the 2nd and 3rd families have masses equal to 2n:(p2)=2 where n is an integer [1] calculated for the first family of fermions and p is another integer. p is an integer which corresponds to the excited states of the elementary wavefunctions (the energy of the excited elementary wave functions are equal to p2=2; using normalized units). Keywords: 1. Introduction To obtain the remaining fermions (elementary parti- cles), one has to modify the quantum number p (similar I make the assumption that our threedimensional uni- to the quantum number of a particle in a box). Thus, the verse is embedded in a four dimensional euclidean remaining fermions of the Standard Model may be seen space. Time is a function of the fourth dimension of as excited states of the first fermion family. this space [2–4]. If we apply this hypothesis to particle Straightforwardly, we make the following hypothe- physics, we may say that elementary particles are four- ses: dimensional, threedimensional and twodimensional. The coordinates (x; y; z; t) are not orthonor- • Spacetime has an underlying hypersquare array of mal.Indeed, time t evolves as log(r) where r is the edge length ~ comoving distance in cosmology [2]. To obtain the first family of fermions from the Stan- • Elementary wave functions ( in (x; y; z; t) space)are dard Model ( i.e. quark up, electron, electron neutrino) eigenfunctions of a particlep in a square poten- one may say that [5]: ptial (reduced parameters) 2exp(−ix) for space 2exp(−it) for time • the electron is fourdimensional (t; x; y; z) • the quark is three dimensional (t; x; y). • The eigenvalues of the elementary wave functions are equal to p2=2 (with p an integer number) • finally, the electronic neutrino is twodimensional (t; x) and x; y and z are equivalent. 2. Masses of the electron, muon and tau ∗Talk given at 19th International Conference in Quantum Chromo- dynamics (QCD 19), 2 july - 5 july 2019, Montpellier - FR Email address: [email protected] For electrons, we have (N.Olivi-Tran) 1 µ Speaker, Corresponding author. iγ @µ = m (2) / Nuclear and Particle Physics Proceedings preprint (2019) 1–4 2 0 1 0 1 0 1 B γ1 0 0 0 0 0 0 0 0 0 C B @1Ψ C B Ψ C B C B C B C B 0 γ2 0 0 0 0 0 0 0 0 C B @2Ψ C B Ψ C B C B C B C B 0 0 γ3 0 0 0 0 0 0 0 C B @3Ψ C B Ψ C B C B C B C B 0 0 0 γ3 0 0 0 0 0 0 C B @1Ψ C B Ψ C B C B C B C B 0 0 0 0 γ1 0 0 0 0 0 C B @2Ψ C B Ψ C B C B C = m: B C (1) B 0 0 0 0 0 γ2 0 0 0 0 C B @3Ψ C B Ψ C B C B C B C B 0 0 0 0 0 0 γ2 0 0 0 C B @1Ψ C B Ψ C B C B C B C B 0 0 0 0 0 0 0 γ3 0 0 C B @2Ψ C B Ψ C B C B C B C B 0 0 0 0 0 0 0 0 γ1 0 C B @3Ψ C B Ψ C @B AC @B AC @B AC 0 0 0 0 0 0 0 0 0 σ0 @0Ψ Ψ particle Theoretical mass (eV=c2) Experimental mass (eV=c2) [6] electron 219 = 0:524MeV=c2 0:511MeV=c2 muon 219:202=2 = 104:8MeV=c2 105:6MeV=c2 tau 219:822=2 = 1:76GeV=c2 1:78GeV=c2 Table 1: Theoretical and experimental masses of the electron, muon and tau The Dirac matrices are representative of infinitesimal 0:524MeV=c2 ≈ 0.511 MeV=c2 rotations within the wavefunction of a given elementary particle. • The mass of the muon is equal to 19 2 19 22 22 22 22 52 1 2 16 22 16 Using combinatorial analyzis we obtain equation (1) 2 :20 =2=2 : 2 : 2 : 2 : 2 : 2 :( 2 ) :( 2 ) = 2 2 (using the fact that electrons are 4d [5] and that all space 104:8MeV=c ≈ 105.6 MeV=c dimensions are equivalent). • The mass of the tau is equal to There are 3 possibilities of arranging γ ; γ ; γ (the 2 2 2 2 2 1 2 3 219:822=2=219: 41 : 2 : 2 :( 1 )17:( 2 )17 = Dirac matrices) over x; y and z (all space dimensions are 2 2 2 2 2 1:76GeV=c2 ≈ 1.78 GeV=c2 equivalent) and one possibility to arrange σ0 (temporal Pauli matrix: half of γ ; because time does not go back- 0 The values in italic are the experimental masses [6]. ward). We see that for the tau particle, one of the eigenvalues The large matrix M (see equation (1) ) containing 2 ( 41 ) is much larger than the others. This may explain all combinations has a dimension 9X4 + 2 = 38. We 2 p the short lifetime of this particle. see that, with the coordinate vectors 2exp(−it) and p The masses (theoretical and experimental) of the 2exp(−ix) (eigenfunctions of a particle in a square po- electron, muon and tau are summarized in Table 1. tential), we have to multiply the modified Dirac equa- tion by the Jacobian correspondingp to these new coor- dinates. This Jacobian is equal to 238 where 38 is the 3. Masses of the quarks dimension of the large matrix [1]. We multiply the mass of the first particle of this family by the eigenvalues of For quarks, we have the eigenfunctions (of the particle). µ iγ @µ = m (4) We decompose the eigenvalues into prime numbers. The number of eigenvalues for the ground state (elec- The Dirac matrices are representative of infinitesimal tron) is 38 (the dimension of the large matrix M). For rotations within the wavefunction of a given elementary the other particles, we take into account the spinor particle. T (1; 0) corresponding to the σ0 Pauli matrix. So except Using combinatorial analyzis we obtain equation (3) for the electron, there are 37 eigenvalues for each parti- (using the fact that quarks are 3d [5] and that all space cle. dimensions are equivalent). There are 3 possibilities of arranging γ1; γ2; γ3 (the • pThe mass of the electron is equal to Dirac matrices) over x; y and z (all space dimensions 38 19 2 19 1 2 19 22 19 2 =2 eV=c =2 :( 2 ) :( 2 ) = are equivalent). There is one possibility to arrange σ0 / Nuclear and Particle Physics Proceedings preprint (2019) 1–4 3 0 γ 00000000000 1 0 @ Ψ 1 0 Ψ 1 B 1 C B 1 C B C B 0 γ 0 0 0 0 0 0 0 0 0 0 C B @ Ψ C B Ψ C B 2 C B 2 C B C B 0 0 γ 0 0 0 0 0 0 0 0 0 C B @ Ψ C B Ψ C B 3 C B 3 C B C B 0 0 0 σ 0 0 0 0 0 0 0 0 C B @ Ψ C B Ψ C B 0 C B 0 C B C B 0 0 0 0 γ 0 0 0 0 0 0 0 C B @ Ψ C B Ψ C B 2 C B 1 C B C B 0 0 0 0 0 γ 0 0 0 0 0 0 C B @ Ψ C B Ψ C B 3 C B 2 C = m B C (3) B 0 0 0 0 0 0 γ 0 0 0 0 0 C B @ Ψ C B Ψ C B 1 C B 3 C B C B 0 0 0 0 0 0 0 σ 0 0 0 0 C B @ Ψ C B Ψ C B 0 C B 0 C B C B 0 0 0 0 0 0 0 0 γ 0 0 0 C B @ Ψ C B Ψ C B 3 C B 1 C B C B 0 0 0 0 0 0 0 0 0 γ 0 0 C B @ Ψ C B Ψ C B 1 C B 2 C B C B 0 0 0 0 0 0 0 0 0 0 γ 0 C B @ Ψ C B Ψ C @B 2 AC @B 3 AC @B AC 00000000000 σ0 @0Ψ Ψ quark Theoretical mass (eV=c2) Experimental mass (eV=c2) [6] up 221 = 2:09Mev=c2 2:2MeV=c2 down 221:22=2 = 4:19MeV=c2 4:7MeV=c2 strange 221:92=2 = 84:9MeV=c2 96MeV=c2 charm 221:362=2 = 1:35GeV=c2 1:27GeV=c2 bottom 221:632=2 = 4:16GeV=c2 4:18GeV=c2 top 221:4052=2 = 171:9GeV=c2 173GeV=c2 Table 2: Theoretical and experimental masses of the quarks family.