τ-lepton as a Composition of Massless Preons: an Alternative to Higgs Mechanism

Abstract

Within the framework of the confinement mechanism proposed earlier by the author in QCD the problem of masses for fundamental fermions in particle physics is discussed by the example of τ-lepton τ ⁻. It is shown that the observed parameters of τ-lepton such as its mass and magnetic moment can be obtained in a preon model dynamically due to a preon gauge interaction. The radius of τ-lepton is also estimated. Under the circumstances preons might be massless in virtue of existence of the nonzero chiral limit for the preon interaction energy.

Appendix

In Subsection 2.3 we built the three-body leptonic wave functions as tensorial products from the two-body wave functions of premesons. The latter, however, are represented by triplets ψ of 4D-Dirac spinors [see, e.g., (2)], so ψ belongs to the Hilbert space ${\mathcal H}=L_{2}^{12}({\mathbb R}^{3})$. Under the situation we here recall some facts about tensorial products of Hilbert spaces necessary for building the lepton wave functions. For more information see, e.g., [20] and references therein.

For two Hilbert spaces ${\mathcal H}_{1}$ and ${\mathcal H}_{2}$, by definition, their tensorial product ${\mathcal H}_{1}\otimes {\mathcal H}_{2}$ consists from every possible linear combinations of elements of the form x ₁⊗x ₂, where $\mathrm {\mathbf {x}}_{1}\in {\mathcal H}_{1}$, $\mathrm {\mathbf {x}}_{2}\in {\mathcal H}_{2}$. The obvious relations take place

$$ (\alpha\mathrm{\mathbf{x}}_{1}+\beta\mathrm{\mathbf{x}}_{2})\otimes\mathrm{\mathbf{y}}=\alpha(\mathrm{\mathbf{x}}_{1}\otimes\mathrm{\mathbf{y}})+ \beta(\mathrm{\mathbf{x}}_{2}\otimes\mathrm{\mathbf{y}})\>, $$

(A.1)

$$ x\otimes(\alpha\mathrm{\mathbf{y}}_{1}+\beta\mathrm{\mathbf{y}}_{2})=\alpha(\mathrm{\mathbf{x}}\otimes\mathrm{\mathbf{y}}_{1})+ \beta(\mathrm{\mathbf{x}}\otimes\mathrm{\mathbf{y}}_{2}) $$

(A.2)

with arbitrary complex numbers α, β.

Scalar product in ${\mathcal H}_{1}\otimes {\mathcal H}_{2}$ is defined by the equality

$$ (\mathrm{\mathbf{x}}_{1}\otimes\mathrm{\mathbf{y}}_{1},\mathrm{\mathbf{x}}_{2}\otimes\mathrm{\mathbf{y}}_{2})= (x_{1},x_{2})_{1}(y_{1},y_{2})_{2}\>, $$

(A.3)

where (x ₁, x ₂)₁, (y ₁, y ₂)₂ are the scalar products, respectively, in ${\mathcal H}_{1}$ and ${\mathcal H}_{2}$. Under the situation, if {e _i } is an orthonormal basis in ${\mathcal H}_{1}$ and {f _j } is an orthonormal basis in ${\mathcal H}_{2}$ then h _ij = e _i ⊗f _j is an orthonormal basis in ${\mathcal H}_{1}\otimes {\mathcal H}_{2}$.

Further, if A is a linear operator in ${\mathcal H}_{1}$ and B is a linear operator in ${\mathcal H}_{2}$ then tensorial (Kronecker) product A ⊗ B acts in ${\mathcal H}_{1}\otimes {\mathcal H}_{2}$ and is defined by relation (A ⊗ B)(x ₁⊗x ₂) = A x ₁⊗B x ₂ with the properties

$$ A\otimes(\alpha B_{1}+\beta B_{2})=\alpha A\otimes B_{1}+ \beta A\otimes B_{2}\>, $$

(A.4)

$$ (\alpha A_{1}+\beta A_{2})\otimes B=\alpha A_{1}\otimes B+ \beta A_{2}\otimes B\>, $$

(A.5)

$$ A_{1}A_{2}\otimes B_{1}B_{2}=(A_{1}\otimes B_{1}) (A_{2}\otimes B_{2})\>. $$

(A.6)

In a similar way, the Kronecker sum of A and B is defined by A⊕B = A ⊗ I + I ⊗ B with identical operator I. Under the circumstances, let λ _i be the eigenvalues A and μ _j be those of B (listed according to multiplicity). Then the eigenvalues of A ⊗ B are λ _i μ _j while those of A⊕B will be λ _i + μ _j .

All the above is directly generalized to the arbitrary number of Hilbert spaces. For example, the operator A ₁⊗I ⊗ I + I ⊗ A ₂⊗I + I ⊗ I ⊗ A ₃ has the eigenvalues λ _i + μ _j + ν _k in space ${\mathcal H}_{1}\otimes {\mathcal H}_{2}\otimes {\mathcal H}_{3}$ if λ _i , μ _j , ν _k are the eigenvalues of A ₁, A ₂ and A ₃, respectively, in ${\mathcal H}_{1}$, ${\mathcal H}_{2}$, ${\mathcal H}_{3}$.