SU Symmetry Research Articles

Projective Spin Adaptation for the Exact Diagonalization of Isotropic Spin Clusters

Spin Hamiltonians, like the Heisenberg model, are used to describe the magnetic properties of exchange-coupled molecules and solids. For finite clusters, physical quantities, such as heat capacities, magnetic susceptibilities or neutron-scattering spectra, can be calculated based on energies and eigenstates obtained by exact diagonalization (ED). Utilizing spin-rotational symmetry SU(2) to factor the Hamiltonian with respect to total spin S facilitates ED, but the conventional approach to spin-adapting the basis is more intricate than selecting states with a given magnetic quantum number M (the spin z-component), as it relies on irreducible tensor-operator techniques and spin-coupling coefficients. Here, we present a simpler technique based on applying a spin projector to uncoupled basis states. As an alternative to Löwdin’s projection operator, we consider a group-theoretical formulation of the projector, which can be evaluated either exactly or approximately using an integration grid. An important aspect is the choice of uncoupled basis states. We present an extension of Löwdin’s theorem for s=12 to arbitrary local spin quantum numbers s, which allows for the direct selection of configurations that span a complete, linearly independent basis in an S sector upon the spin projection. We illustrate the procedure with a few examples.

Quantum inspired kernel matrices: Exploring symmetry in machine learning

Quantum machine learning, an emerging field at the intersection of quantum computing and classical machine learning, has shown great promise in enhancing computational capabilities beyond classical bounds. A key element in this area of research is the utilization of Quantum Kernel Estimators, traditionally grounded in the symmetries of SU(2) groups associated with qubits. This study extends the conceptual framework of Quantum Kernel Estimators to incorporate a broader spectrum of symmetry groups. By harnessing the various structures of Lie groups, we develop novel quantum-inspired feature maps that offer more flexible and potentially powerful ways to encode and compress classical data into quantum states. We present a comprehensive theoretical introduction for this approach, followed by a methodology that integrates the developed feature maps into quantum-inspired kernel classifiers. Our results, derived from a series of computational experiments across various datasets, demonstrate the efficacy of this approach in comparison to traditional quantum and classical machine learning models. The findings not only underline the versatility of Lie-group theory in potentially enhancing quantum machine learning algorithms but also open new avenues for exploring complex symmetries in quantum information processing. This research bridges a gap between the study of symmetries and machine learning, paving the way for more sophisticated quantum algorithms capable of tackling complex, high-dimensional data in ways previously unattainable.

A realistic theory of E6 unification through novel intermediate symmetries

We propose a non-supersymmetric E6 GUT with the scalar sector consisting of 650 ⨁ 351′ ⨁ 27. Making use of the first representation for the initial symmetry breaking to an intermediate stage, and the latter two representations for second-stage breaking to the Standard Model and a realistic Yukawa sector, this theory represents the minimal E6 GUT that proceeds through one of the intermediate stages that are novel compared to SU(5) or SO(10) GUT: trinification SU(3)C × SU(3)L × SU(3)R, SU(6) × SU(2) and flipped SO(10) × U(1). We analyze these possibilities under the choice of vacuum that preserves a ℤ2 “spinorial parity”, which disentangles the chiral and vector-like fermions of E6 and provides a dark matter candidate in the form of a (scalar) inert doublet. Three cases are shown to consistently unify under the extended survival hypothesis (with minimal fine-tuning): trinification symmetry SU(3)C × SU(3)L × SU(3)R with either LR or CR parity, and SU(6)CR × SU(2)L. Although the successful cases give a large range for proton lifetime estimates, all of them include regions consistent with current experimental bounds and within reach of forthcoming experiments. The scenario investigated in this paper essentially represents the unique (potentially) viable choice in the class of E6 GUTs proceeding through a novel-symmetry intermediate stage, since non-minimal alternatives seem to be intrinsically non-perturbative.

Proxy-SU(4) symmetry in A = 60–90 region

Applications of the proxy-SU(3) model of Bonatsos and collaborators to nuclei in A = 60–90 region introduces proxy-SU(4) symmetry. Shell model spaces with single particle orbits 1 p 3/2, 1 p 1/2, 0 f 5/2 and 0 g 9/2 are essential for these nuclei and also protons and neutrons in this region occupy the same single particle orbits. With this and applying the ‘proxy scheme’, the 0 g 9/2 changes to 0 f 7/2 giving the spectrum generating algebra U(40) ⊃ [U(10) ⊃ G ⊃ SO(3)] ⨂ [SU ST (4) ⊃ SU S (2) ⨂ SU T (2)]. With G = SU(3), we have the proxy-SU(3) model. It is easy to see that proxy-SU(3) symmetry implies goodness of the SU(4) symmetry appearing above, i.e. proxy-SU(4) symmetry. Shell model calculations pointing out the need for 0 g 9/2 orbit, ground state masses, shape changes and shape co-existence in A = 60–90 region and GT distributions clearly show the importance of proxy-SU(4) in this mass region. Besides presenting this evidence, new proxy schemes with G = SU(5), SO(6) and SO(10) that are generated by good proxy-SU(4) symmetry are described in some detail. An important feature is that the four proxy symmetries SU(3), SO(6), SU(5) and SO(10) appear twice.

Predictive Dirac neutrino spectrum with strong CP solution in SU(5)L × SU(5)R unification

We develop a grand unified theory of matter and forces based on the gauge symmetry SU(5)L × SU(5)R with parity interchanging the two factor groups. Our main motivation for such a construction is to realize a minimal GUT embedding of left-right symmetric models that provide a parity solution to the strong CP problem without the axion. We show how the gauge couplings unify with an intermediate gauge symmetry SU(3)cL × SU(2)2L × U(1)L × SU(5)R, and establish its consistency with proton decay constraints. The model correctly reproduces the observed fermion masses and mixings and leads to naturally light Dirac neutrinos with their Yukawa couplings suppressed by a factor MI/MG, the ratio of the intermediate scale to the GUT scale. We call this mechanism type II-Dirac seesaw. Furthermore, the model predicts δCP = ±(130.4±1.2)° and \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${m}_{{\ u }_{1}}$$\\end{document} = (4.8 – 8.4) meV for the Dirac CP phase and the lightest neutrino mass. We demonstrate how the model solves the strong CP problem via parity symmetry.

On the symmetries of elementary fermions

This paper deals with the symmetries of elementary fermions and their derivation from fundamental physical principles such as the Lorentz invariance and from the spinor-helicity formalism employed to three-vectors. The generators of the Lorentz group are discussed, and the physics of the associated chiral spins is used to establish the chiral symmetry SU(2). The helicity of spin is defined which, when being applied to the hadronic isospin, yields the symmetry group SU(4). The Kronecker product of these two basic symmetries defines a unified symmetry SU(2)⊗SU(4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$SU(2) \\otimes SU(4)$$\\end{document} of the basic light fermions, namely the chiral doublets of the single lepton and triple quarks of the first family. Breaking this symmetry, according to mechanisms that are used in the electroweak interactions of the Standard Model (SM) of quantum field theory, yields quantum electrodynamics (QED) with symmetry group U(1), weak interactions with symmetry group SU(2), and what we like to name quantum hadrodynamics (QHD, akin to quantum chromodynamics QCD of the SM) with symmetry group SU(3). The gauge bosons associated with QED and QHD remain massless, but the weak bosons and the V bosons, related to the transformation of the quarks into a lepton and vice versa, become massive by the Higgs mechanism. Their masses are defined by the Higgs vacuum and the two coupling constants involved in the unified model. The V-boson mass is predicted to be 35.4 GeV. Furthermore, a possible explanation of hadron confinement is given in terms of the two hadronic charge operators. Moreover, the concept of hypercharge as used in the SM is not needed. Various versions of the extended Dirac equation that include the above symmetry groups are derived.

Non-Abelian chiral soliton lattice in rotating QCD matter: Nambu-Goldstone and excited modes

The ground state of QCD with two flavors at a finite baryon chemical potential under rapid rotation is a chiral soliton lattice (CSL) of the η meson, consisting of a stack of sine-Gordon solitons carrying a baryon number, due to the anomalous coupling of the η meson to the rotation. In a large parameter region, the ground state becomes a non-Abelian CSL, in which due to the neutral pion condensation each η soliton decays into a pair of non-Abelian sine-Gordon solitons carrying S2 moduli originated from Nambu-Goldstone (NG) modes localized around it, corresponding to the spontaneously broken vector symmetry SU(2)V. There, the S2 modes of neighboring solitons are anti-aligned, and these modes should propagate in the transverse direction of the lattice due to the interaction between the S2 modes of neighboring solitons. In this paper, we calculate excitations including gapless NG modes and excited modes around non-Abelian and Abelian (η) CSLs, and find three gapless NG modes with linear dispersion relations (type-A NG modes): two isospinons (S2 modes) and a phonon corresponding to the spontaneously broken vector SU(2)V and translational symmetries around the non-Abelian CSL, respectively, and only a phonon for the Abelian CSL because of the recovering SU(2)V. We also find in the deconfined phase that the dispersion relation of the isospinons becomes of the Dirac type, i.e. linear even at large momentum.

Domain-wall Skyrmion phase in a rapidly rotating QCD matter

Based on the chiral perturbation theory at the leading order, we show a signal of the presence of a new phase in rapidly rotating QCD matter with two flavors, that is a domain-wall Skyrmion phase. Based on the chiral Lagrangian with a Wess-Zumino-Witten (WZW) term responsible for the chiral anomaly and chiral vortical effect, it was shown that the ground state is a chiral soliton lattice (CSL) consisting of a stack of η-solitons in a high density region under rapid rotation. In a large parameter region, a single η-soliton decays into a pair of non-Abelian solitons, each of which carries SU(2)V/U(1) ≃ ℂP1 ≃ S2 moduli as a consequence of the spontaneously broken vector symmetry SU(2)V. In such a non-Abelian CSL, we construct the effective world-volume theory of a single non-Abelian soliton to obtain a d = 2 + 1 dimensional ℂP1 model with a topological term originated from the WZW term. We show that when the chemical potential is larger than a critical value, a topological lump supported by the second homotopy group π2(S2) ≃ ℤ has negative energy and is spontaneously created, implying the domain-wall Skyrmion phase. This lump corresponds in the bulk to a Skyrmion supported by the third homotopy group π3[SU(2)] ≃ ℤ carrying a baryon number. This composite state is called a domain-wall Skyrmion, and is stable even in the absence of the Skyrme term. An analytic formula for the effective nucleon mass in this medium can be written only in terms of the meson’s constants as 42πfπfη/mπ~1.21\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ 4\\sqrt{2}\\pi {f}_{\\pi }{f}_{\\eta }/{m}_{\\pi}\\sim 1.21 $$\\end{document} GeV with the decay constants fπ and fη of the pions and η meson, respectively, and the pion mass mπ. This is reasonably heavier than the nucleon mass in the QCD vacuum.

Description of intruder levels in the 162,164,166Dy nuclei by two different algebraic approaches

In this paper, we tried to describe both normal and intruder levels of the ground and first excited beta and gamma rotational bands of 162,164,166Dy deformed nuclei in the framework of the interacting boson model. A normal Hamiltonian of SU(3) dynamical symmetry limit is extended by adding a 2p-2 h excitation term. Also, the results are compared with the predictions of the partial dynamical symmetry for these levels of considered nuclei. The results show that, this extension removes the degeneracy suggested by the pure SU(3) symmetry and makes satisfactory agreement with the experimental counterparts for the energy levels, too. Also, a comparison between the results of this extended formalism and partial dynamical symmetry shows the advantages of the first one in the description of intruder levels, whereas the latter, makes exact results for the normal energy levels of beta and gamma energy bands. The predictions of pure SU(3), partial dynamical symmetry SU(3) and mixed formalism for different quadrupole transition rates between normal and intruder levels of this nucleus are compared with the predictions of pseudo-SU(3) model and the advantages of each model have explained in detail.

Mass Generation of the Scalars Bosons in a Left-Right Symmetric Model with two Bidoublets

The search for new Physics justified by new discoverys, motivate among others, the realization of new models called Extensions of the Standard Model (ESM) which must include at low energy the Standar Model. (SM) The electroweak symmetry group SU(2) L ⊗ U(1) Y to the gauge group with left-right symmetry SU(2) L ⊗ SU(2) R ⊗ U(1) B − L ⊗ 𝒫, represents one of the minimal extensions of the SM, in which 𝒫 is a parity discrete symmetry such that left-right coupling constants satisfied gL = gR , and according to the hierarchy in the symmetry breaking (conditions that must be met by the vacuum expectation values introduced in the model) allow to obtain the SM. The aim of this work is to identify the Higgs boson of the SM, considering a more general scalar potential that must respect all the symmetries (local gauge invariant, discrete symmetries and Lorentz invariant) established in the model. We will take into account those previous works about the hierarchy conditions and certain approximations that must satisfy the vacuum expectation values for obtain greater simplicity in the caculations of scalar masses of the model.

Symmetries of the squeeze-driven Kerr oscillator

We study the symmetries of the static effective Hamiltonian of a driven superconducting nonlinear oscillator, the so-called squeeze-driven Kerr Hamiltonian, and discover a remarkable quasi-spin symmetry su(2) at integer values of the ratio of the detuning parameter Δ to the Kerr coefficient K. We investigate the stability of this newly discovered symmetry to high-order perturbations arising from the static effective expansion of the driven Hamiltonian. Our finding may find applications in the generation and stabilization of states useful for quantum computing. Finally, we discuss other Hamiltonians with similar properties and within reach of current technologies.

The production of charm pentaquark from B meson within SU(3) analysis

We study the masses and production modes of pentaquark with the quark constituent cq¯qqq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c\\bar{q}qqq$$\\end{document}, by the tridiquark-diquark model and systematical light flavor quark symmetry SU(3) analysis. The mass spectrums show that the S-wave singly charm pentaquark cn¯nnn,cs¯ssn,cn¯ssn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c{\\bar{n}}nnn,~c{\\bar{s}}ssn,~c{\\bar{n}}ssn$$\\end{document} and cn¯snn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c{\\bar{n}}snn$$\\end{document}(n=u,d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n=u,d)$$\\end{document} are above their strong decay thresholds, while cs¯nnn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c{\\bar{s}} nnn$$\\end{document} and cs¯snn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c\\bar{s} snn$$\\end{document} with parity 12-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{2}^-$$\\end{document} are below their strong decay thresholds, which imply the possibility of stable states. Furthermore, we discuss the production of the concerned 15 states from B meson, the analysis within SU(3) symmetry yields the golden channels of production process, B¯s0→Fs¯udd+p¯,B-→Fu¯dds-Λ¯0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\overline{B}}^0_s\\rightarrow F_{\\bar{s}udd}^{+} {\\overline{p}},\\ B^-\\rightarrow F_{\\bar{u}dds}^{-} {\\overline{\\Lambda }}^0$$\\end{document}, which expected to be work worthily in b-factory.

Sterile neutrinos in a left-right mirror model

In this work we investigate the possibility that one neutrino mirror plays the role of a sterile neutrinos with mass on the scale of a few eV ′ s, that is, mˆ 1 ∼ O(1 eV ). We consider the extension of the Standard Model with gauge symmetry SU(2)L ⊗ SU(2)R ⊗ U(1)Y ′ and additional exotic fermions known as mirror fermions. A double application of the seesaw type I approximation to the most general Majorana-type neutrino mass matrix and diagonalization is performed to study neutrino masses and mixing.

Physics implication from higher weak isospin decomposition

The SU(3)_Lotimes U(1)_X symmetry actually studied is directly broken to the electroweak symmetry SU(2)_Lotimes U(1)_Y by a Higgs triplet, predicting a relevant new physics at TeV scale. This work argues, by contrast, that the higher weak isospin SU(3)_L might be broken at a high energy scale, much beyond 1 TeV, by a Higgs octet to an intermediate symmetry SU(2)_Lotimes U(1)_{T_8} at TeV, before the latter U(1)_{T_8} recombined with U(1)_X defines (i.e., broken to) U(1)_Y by a Higgs singlet. The new physics coupled to SU(3)_L breaking phase is decoupled, whereas what remains is a novel family-nonuniversal abelian model, U(1)_{T_8}otimes U(1)_X, significantly overhauling the standard model as well as yielding consistent results for neutrino mass, dark matter, W-mass anomaly, and FCNC, differently from the usual 3-3-1 model.

Unitarity bounds on extensions of Higgs sector

It is widely believed that extensions of the minimal Higgs sector is one of the promising directions for resolving many puzzles beyond the Standard Model (SM). In this work, we study the unitarity bounds on the models by extending the two-Higgs-doublet model with an additional real or complex Higgs triplet scalar. By noting that the SM gauge symmetries SU(2)L × U(1)Y are recovered at high energies, we can classify the two-body scattering states by decomposing the direct product of two scalar multiplets into their direct sum of irreducible representations of electroweak gauge groups. In such state bases, the s-wave amplitudes of two-body scalar scatterings can be written in the form of block-diagonalized scattering matrices. Then the application of the perturbative unitarity conditions on the eigenvalues of scattering matrices leads to the analytic constraints on the model parameters. Finally, we numerically investigate the complex triplet scalar extension of the two-Higgs-doublet model, finding that the perturbative unitarity places useful stringent bounds on the model parameter space.

Describe of Nuclear Structure for Germanium (_32^66)Ge_34) Nucleus under frame (IBM-1, GVMI and VMI) Models

In this paper, the interacting bosons model-1 (IBM-1), a variable moment of inertia (VMI) and generalized moment of inertia (GVMI) models were used to calculate the energy levels of the positive parity and its gamma transitions as a function of the angular momentum of even-even (_32^66Ge_34) nucleus. To determine the dynamic symmetry of this nucleus, the ratios of the energy levels (E_4^+)⁄(E_2^+ ), (E_6^+)⁄(E_2^+), and (E_8^+)⁄(E_2^+) were evaluated and compared with experimental energy values and the ideal scheme of the three dynamic symmetries SU(5), SU(3), and O(6). The current study showed that the dynamic symmetry of this nucleus is determined to be O(6) - SU(5). The intersection of the energy band and the phenomenon of back bending were also studied using the (VMI) and (GVMI) models. These consequences were compared with the experimental results, and the results obtained have been in good agreement.

Algebra of the spinor invariants and the relativistic hydrogen atom

It is shown that the Dirac equation with the Coulomb potential can be solved using the algebra of the three spinor invariants of the Dirac equation without the involvement of the methods of supersymmetric quantum mechanics. The Dirac Hamiltonian is invariant with respect to the rotation transformation, which indicates the dynamical (hidden) symmetry SU(2) of the Dirac equation. The total symmetry of the Dirac equation is the symmetry SO(3)⊗SU(2). The generator of the SO(3) symmetry group is given by the total momentum operator, and the generator of SU(2) group is given by the rotation of the vector-states in the spinor space, determined by the Dirac, Johnson–Lippmann, and the new spinor invariants. It is shown that using algebraic approach to the Dirac problem allows one to calculate the eigenstates and eigenenergies of the relativistic hydrogen atom and reveals the fundamental role of the principal quantum number as an independent number, even though it is represented as the combination of other quantum numbers.

Axial-vector transition form factors of the singly heavy baryons

We investigate the axial-vector transition form factors of the lowest-lying singly heavy baryons within the framework of the chiral quark-soliton model. We consider the linear ${m}_{\mathrm{s}}$ corrections, dealing with the strange current quark mass ${m}_{\mathrm{s}}$ as a small perturbation. Since we have various relations between different transitions because of isospin symmetry and flavor SU(3) symmetry breaking, only two axial-vector transition form factors are independent. We present the numerical results for these form factors. The effects of the flavor SU(3) symmetry breaking turn out tiny, so we neglect them. We also compute the decay rates for several strong decays of the singly heavy baryons and compare the results with the experimental data and those from other models. While the results for the ${\mathrm{\ensuremath{\Sigma}}}_{c}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Lambda}}}_{c}^{+}+\ensuremath{\pi}$ and ${\mathrm{\ensuremath{\Sigma}}}_{c}^{*}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Lambda}}}_{c}^{+}+\ensuremath{\pi}$ decays are slightly overestimated in comparison with the corresponding experimental data, those for the ${\mathrm{\ensuremath{\Xi}}}_{c}^{*}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Xi}}}_{c}+\ensuremath{\pi}$ are in remarkable agreement with the data.

Features of Fermion Dynamics Revealed by SU2 Symmetry

We deal here with the notion of statistical order and apply it to a system of interacting fermions endowed with an SU2 × SU2 symmetry. The discussion takes place in a thermal quantum statistical scenario. Two distinct fermion–fermion interactions are at play. One of them is a well-known spin–flip interaction. The other is the pairing interaction responsible for nuclear superconductivity. We used novel statistical quantifiers that yield insights regarding changes in the statistical order produced when the values of the pertinent coupling constants vary. In particular, we show that judicious manipulation of the energy cost associated with statistical order variations with the fermion number is the key to understanding important details of the associated dynamics.

Leptons and other forces of nature

Assuming that neutrinos are not Majorana fermions and the right handed Dirac neutrino does not exist, we propose a model in which the second and the third generations of the leptons are composites, while the first generation is fundamental. The composite states are formed by the fundamental leptons and two new fundamental hidden scalar particles. In addition, there exist two hidden forces besides the standard model interactions. The gauge symmetry SU(2) L ⊗ U(1) Y ⊗ U(1) h ⊗ SU(2) h of the electroweak and the hidden forces breaks down to U(1) Y ⊗ U(1) h after the spontaneous symmetry breaking. We explain the neutrino masses in terms of the binding dynamics of a hidden force. The phenomenon of neutrino oscillation can also be explained by our model in a dynamical framework of the hidden forces.