The Masses of the First Family of Fermions and of the Higgs Boson Are Equal to Integer Powers of 2 Nathalie Olivi-Tran

The masses of the first family of fermions and of the Higgs boson are equal to integer powers of 2 Nathalie Olivi-Tran

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Nathalie Olivi-Tran. The masses of the first family of fermions and of the Higgs boson are equal to integer powers of 2. QCD14, S.Narison, Jun 2014, MONTPELLIER, France. pp.272-275, 10.1016/j.nuclphysbps.2015.01.057. hal-01186623

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The masses of the first family of fermions and of the Higgs boson are equal to integer powers of 2

ARTICLE · JANUARY 2015 DOI: 10.1016/j.nuclphysbps.2015.01.057

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Nathalie Olivi-Tran Université de Montpellier

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Available from: Nathalie Olivi-Tran Retrieved on: 27 August 2015 The masses of the ﬁrst family of fermions and of the Higgs boson are equal to integer powers of 2

a, Nathalie Olivi-Tran ∗

aLaboratoire Charles Coulomb, UMR CNRS 5221, cc. 074, Place Eugène Bataillon, 34095 - Montpellier Cedex 05, France. Laboratoire Charles Coulomb, UniversitéMontpellier 2, cc. 074, Place Eugène Bataillon, 34095 - Montpellier Cedex 05, France.

Abstract We noticed that the ﬁrst family of fermions and the Higgs boson have masses which are equal to integer powers of 2 in eV/c2 units (i.e. in the Planck length units). We made the hypothesis that, if spacetime is composed of small hypercubes of one Planck length edge, it exists elementary wavefunctions which are equal to √2 exp(ikxi) if it corresponds to a space dimension or equal to √2 exp(iωt) if it corresponds to a time dimension. By using the Dirac propagation equation and combinatorics we showed that the electron has a mass of 219eV/c2, the quark has a mass of 221eV/c2 and the electron neutrino has a mass of 2eV/c2. Finally, the Higgs boson is showed to have a mass of 237eV/c2 Keywords: theoretical masses of elementary particles, fourdimensional space, real space theory

1. Introduction the electron is fourdimensional (t, x, y, z) • I make the assumption that our threedimensional uni- the quark is three dimensional (t, x, y). verse is embedded in a four dimensional euclidean • space. Time is a function of the fourth dimension of this space [1–3]. If we apply this hypothesis to particle finally, the electronic neutrino is twodimensional • physics, we may say that elementary particles are four- (t, x) and x, y and z are equivalent. It is only dimensional, threedimensional and twodimensional. due to an equivalent propagation equation to the The coordinates (x, y, z, t) are not orthonormal. In- Fermi Dirac equation that when this neutrino prop- deed, time t evolves as log(r) where r is the comoving agates, there are infinitesimal rotations between the distance in cosmology [1]. characteristic coordinates (leading to flavor oscilla- Let us make the additional assumption that for each of tions). these four dimensions there are functions like exp(ikxi) with (xi = x, y, z, t) which vibrate (like in string theory). To obtain the remaining fermions (elementary parti- So, our reasoning is simply the description of how to cles), one has to modify the quantum number n (similar distribute these functions in the fourdimensional space. to the quantum number during oscillation in a hypercu- In the following, the reasoning applies in real space. bic box). Thus, the remaining fermions of the Standard A previous paper of mine (see [5]) predicts that the Model may be seen as excited states of the first fermion Higgs potential in real space is a hypercubic box in our family. fourdimensional space. So, we made a classification of elementary particles To obtain the first family of fermions from the Stan- with respect to their dimensions [4]. To verify these dard Model ( i.e. quark up, electron, electron neutrino) conclusions, we analyze here the masses of the first fam- one may say that: ily of fermions. One may remark that the masses of the first family of fermions (elementary particles) and of the Higgs boson ∗Speaker 2 Email address: [email protected] are integer powers of 2 if expressed in eV/c . For details (Nathalie Olivi-Tran) on these masses, see table I.

Preprint submitted to Nuc. Phys. (Proc. Suppl.) September 3, 2014 Name measured mass 2n A hint is to write equation (6) with the use of one unique in eV/c2 matrix M containing all Dirac matrices. Moreover, one Electron 511.103 219 524.103 has to take into account the fact that this large matrix has ≈ ≈ Quark up 2.3.106 221 2.1.106 to contain all possible combinations of Dirac matrices . ≈ ≈ Electron neutrino 2 21 Hence, we modify the Dirac equation by putting more ≈ Higgs boson 126.109 237 137.109 constraints: all space dimensions have to be equivalent ≈ ≈ and time cannot go backwards. Table 1: Masses of the ﬁrst family of fermions and of the Higgs boson

iM∂µψ = mψ (7) In order to explain this property, let us analyze the Dirac equation which is the propagation equation of part Thus there are 3 possibilities of arranging γ1, γ2, γ3; all of the first family of fermions (except the electron neu- space dimensions are thus equivalent. Because time trino). cannot go backwards, we take into account half of the The Dirac equation may be written: matrix γ0 which is equal to the Pauli matrix σ0. The large matrix M containing all combinations of γµ matri- iγµ∂ ψ mψ = 0 (1) µ − ces over x, y and z is then: with ψ the wavefunction, m the mass of the fermion and γ 000000000 with the Dirac matrices: 1  0 γ2 00000000    0 = I2 0  0 0 γ3 0000000  γ , (2)   0 I2 !  0 0 0 γ3 000000  −    0 0 0 0 γ1 0000 0  0 σ   (8) γ1 = x , (3)  00000 γ2 0 0 0 0    σx 0 !  000000 γ2 0 0 0  −    0000000 γ3 0 0  0 σ   2 = y  00000000 γ1 0  γ , (4)   σy 0 !  000000000 σ0  −     0 σ We see that, if we work with the coordinate vectors γ3 = z , (5) σz 0 ! ( √2 exp(ikxi) , √2 exp(iωt)), we have to multiply the − l.h.s of the modified Dirac equation (7) by the Jacobian where σν are the Pauli matrices. corresponding to these new coordinates. This Jacobian On a mathematical point of view, the Dirac matrices is equal to √2N where N is the dimension of the large are representative of infinitesimal rotations within the matrix M. wavefunction of a given elementary particle. The modified dirac equation (7) may be reduced to Let us make two hypotheses: first, the elementary the dirac equation (6). So, the eigenfunctions of the wave function corresponds to the eigenfunction of a modified Dirac equation may be found by making per- square potential with dimensions corresponding to the mutations of x, y and z in the eigenfunctions of the Dirac = Planck length (we take ~ 1 here). This first hypothesis equation: this means that x, y, z are equivalent. leads to elementary eigenfunctions which are equal to So the large matrix containing all combinations of γµ √2 exp(ikxi) if the eigenfunction corresponds to a space has a dimension of 4X3X3 + 2 = 38 :i.e. 38X38. The dimension or equal to √2 exp(iωt) if the eigenfunction eigenvalue of the Dirac equation is equal to 1, so the corresponds to a temporal dimension. eigenvalue m of the modified Dirac equation is equal to m = √238 = 219eV/c2. 2 2. Mass of the electron Finally, if one deals with eV/c units, the theoretical mass of the electron is equal to 219eV/c2 524keV ( the Following Olivi-Tran and Gottiniaux [4], the electron measured mass of the electron is 511keV≈). is fourdimensional. So, the wavefunction has four subcomponents which may be seen also as wavefunctions. In order to get the mass of the electron, let us analyze 3. Mass of the quark up the Dirac equation which may be rewritten: Following Olivi-Tran and Gottiniaux [4], the quark µ iγ ∂µψ = mψ (6) up is threedimensional. So, the wavefunction has three 2 subcomponents which may be seen also as wavefunc- as the electron neutrino has only one spatial dimension, tions. In order to get the mass of the quark up, let us there is no ’rotations’ between different spatial dimen- analyze the Dirac equation which may be rewritten: sions (as for the electron and the quark). In the propagation equation for neutrino, there would be no Dirac µ = iγ ∂µψ mcψ (9) matrices and only two subeigenfunctions would come A hint is to write equation (9) with the use of one unique into account: one for space the other for time. matrix M containing all Dirac matrices. Moreover, one A theoretical propagation function for the neu- √ has to take into account the fact that this large matrix has trino would then use 2 subfunctions 2 exp(iωt) and √ to contain all possible combinations of Dirac matrices 2 exp(ikx). So the mass of the electron neutrino would = √ 2 = 2 over x, y and z. Hence, we modify the Dirac equation be m 2 2eV/c (the measured mass of the electron neutrino is 2eV). by putting more constraints: all space dimensions have ≈ to be equivalent and time cannot go backwards. 5. Mass of the Higgs boson iM∂µψ = mψ (10)

Thus there are 3 possibilities of arranging γ1, γ2, γ3; all In a previous paper, Olivi-Tran [5] showed that the space dimensions are equivalent. Because time cannot Higgs field corresponds to the solution of the mass- go backwards, we take into account half of the matrix less Klein-Gordon propagation equation coupled to a γ0 which is equal to the Pauli matrix σ0, for each pair hypercubic fourdimensional potential. The first family of matrices γµ. The large matrix M containing all com- of fermions acquire mass by interacting with the Higgs binations of γµ matrices is then band diagonal with this field. band diagonal equal to: The Higgs field gives mass to the previous fermions and is fourdimensional. The Higgs potential has to be (γ1, γ2, γ3,σ0, γ2, γ3, γ1,σ0, γ3, γ1, γ2,σ0) (11) able to contain a combination of electron, quark up and electron neutrino. If all space dimensions are equivalent this leads to all In order to contain all the previous combinations, possible combinations of γµ,σ0. Equation (10) may be the resulting combination of the coordinate vectors reduced to the Dirac equation (9) if one takes into ac- √2 exp(ikx ) for spatial dimensions has a norm equal count that the solutions of the Dirac are equivalent by i to 218 (electron : combinations over x, y, z) multiplied making permutations of x, y and z. by 218 (quark up : combinations over (x, y);(y, z) and So the large matrix containing all combinations has a (x, z)) ; the electron neutrino is included in these last. dimension of 3X3X4 + 3X2 = 42. The resulting combination is 2 for the temporal dimen- We see that, if we work with the coordinate vectors sion (coordinate vector √2 exp(iωt) identical for all 3 ( √2 exp(ikx ) , √2 exp(iωt)), we have to multiply the i fermions) l.h.s of the modified Dirac equation by the Jacobian cor- So, if we make the hypothesis that the Higgs field is a responding to these new coordinates. This Jacobian is combination of the field corresponding to the electron, equal to √2N where N is the dimension of the large ma- the quark up and the electron neutrino, the mass of the trix M (i.e. 42). Higgs boson would be equal to 218.218.2 = 237eV/c2 The eigenvalue of the Dirac equation is equal to 1, so 137GeV (the measured mass of the Higgs boson is≈ the eigenvalue m of the modified Dirac equation is equal 126GeV). to m = √242 = 221eV/c2. Finally, if one deals with eV/c2 units, the mass of the 21 2 quark up is equal to 2 eV/c 2.1MeV (the measured 6. Conclusion mass of the quark up is 2.3MeV≈). The masses of the first family of fermions and of the 4. Mass of the electron neutrino Higgs boson are integer powers of 2 within experimen- tal errors. As what I wrote in the introduction [4], other Following Olivi-Tran [4], the electron neutrino is fermions are excited states of the first family and thus twodimensional: one temporal dimension and one spa- they do not follow the same rule regarding their masses. tial dimension. Up to now, there is no existing theoreti- Now, one has to analyze the masses of the W and Z cal propagation equation for the neutrinos. Let us make bosons which maybe can be found theoretically with the the hypothesis that all space dimensions are equivalent: same reasonings. 3 References

[1] N.Olivi-Tran and P.M.Gauthier, The FLRW cosmological model revisited: Relation on the local time with the local curvature and consequences on the Heisenberg uncertainty principle Adv. Studies Theor. Phys. vol.2 no 6 (2008) 267-270 [2] N.Olivi-Tran What if our three dimensional curved universe was embedded in four dimensional space? Consequences on the EPR paradox Adv. Studies Theor. Phys., Vol. 3, (2009), no. 12, 489 - 492 [3] N.Olivi-Tran Dimensional analysis of Einstein’s fields equations Adv. Studies Theor. Phys., Vol. 3, (2009), no. 1, 9 - 12 [4] N.Olivi-Tran and N.Gottiniaux A classification of elementary particles in d = 4 following a simple geometrical hypothesis in real space Adv. Studies Theor. Phys., Vol. 7, 2013, no. 18, 853-857 [5] N.Olivi-Tran Is it the Higgs scalar field? Advanced Studies in Theoretical Physics Vol. 4, 2010, no. 13, 633 - 636