Abstract:
We introduce an alternative mechanism for generating neutrino masses by mapping the traditional left-handed leptonic chiral doublet to a modified framework where the neutrino and charged lepton fields act as a single entity. By exploiting the Gell-Mann-Nishijima relation, we construct this left-handed configuration to share identical weak isospin and weak hypercharge quantum numbers with the right-handed charged lepton field. We demonstrate that this shared quantum framework allows the Higgs-field vector to accommodate non-zero mass eigenvalues for both neutrinos and anti-neutrinos. This structural alignment eliminates the need for independent heavy sterile fields or additional unobserved gauge sectors while fully preserving the verified phenomenological predictions of standard electroweak theory. \end{abstract}
Introduction
Neutrino masses are not direct observables associated with individual flavor states; instead, they are modeled via a superposition of distinct mass eigenstates. The action of the mass operator $\hat{M}$ on a generic neutrino flavor field $\vert{}\nu_\ell\rangle$ ($\ell = e, \mu, \tau$) is expressed as follows:
\[ \hat{M}|\nu_\ell\rangle = U_{\ell1}m_{\nu 1}|\nu_1\rangle + U_{\ell2}m_{\nu 2}|\nu_2\rangle + U_{\ell3}m_{\nu 3}|\nu_3\rangle \]Where $m_{\nu i}$ $(i = 1, 2, 3$) represent the eigenvalues of the neutrino mass hierarchy, and $U_{\ell i}$ denote the mixing coefficients of the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix that dictate the flavor-mass composition.
While Equation (1) phenomenologically describes the mixing mechanism, it does not specify the underlying dynamic gauge mechanism through which neutrinos acquire mass. Within the standard electroweak framework, left-handed charged leptons acquire mass via a Yukawa coupling to the Higgs doublet. Conversely, the Standard Model predicts massless neutrinos, a conclusion that fundamentally conflicts with modern oscillation observations. In this paper, we resolve this discrepancy by introducing a alternative gauge assignment that accommodates neutrino mass generation without disrupting the established predictions of electroweak theory.
The Field Vectors
To establish the gauge framework, we define the scalar and leptonic fields. We begin with the Higgs doublet, expressed in the unitary gauge to ensure a massless photon and massive electroweak gauge bosons:
\begin{equation} \varphi^A = 0 \end{equation} \begin{equation} \varphi^B = \varphi_0 + \frac{h(x)}{\sqrt{2}} \end{equation} \begin{equation} \varphi = \begin{bmatrix} \varphi^A \\ \varphi^B \end{bmatrix} = \begin{bmatrix} 0 \\ \varphi^B \end{bmatrix} \end{equation}Where $\varphi_0$ represents the vacuum expectation value (VEV) of the scalar field, and $h(x)$ denotes the physical Higgs boson field.
\vspace{.5 cm}Next, we define the leptonic field representations. To facilitate the unified gauge transformation proposed in this framework, both the left-handed and right-handed fields are expressed as column vectors. For a given lepton flavor $\ell \in \{e, \mu, \tau\}$, the left-handed and right-handed field vectors are defined as:
\begin{equation} \psi_L = \begin{bmatrix} \nu_{\ell}\\ \ell_L \end{bmatrix} \end{equation} \begin{equation} \psi_R = \begin{bmatrix} 0\\ \ell_R \end{bmatrix} \end{equation}The corresponding Dirac adjoint (anti-lepton) field vectors are defined as row vectors to ensure proper matrix multiplication:
\begin{equation} \bar{\psi}_L = \begin{bmatrix} \bar{\nu}_{\ell} & \bar{\ell}_L \end{bmatrix} \end{equation} \begin{equation} \bar{\psi}_R = \begin{bmatrix} 0 & \bar{\ell}_R \end{bmatrix} \end{equation}The New Approach
We begin by examining the standard interaction Lagrangian for a given charged lepton flavor $\ell$, governed by the generic Yukawa coupling $G_\ell$:
\begin{equation} L_{int} = -G_\ell(\bar{\psi}_L\varphi\psi_R + \bar{\psi}_R\varphi^\top\psi_L) \end{equation}To accommodate non-zero mass terms for both neutrinos and anti-neutrinos, we introduce the following structural modifications:
\begin{equation} \phi= \bar{\psi}_L\varphi\psi_R + \bar{\psi}_R\varphi^\top\psi_L \end{equation} \begin{equation} %23 \bar{\psi}_L \varphi = \begin{bmatrix} 0 & \bar{\nu}_\ell + \bar{\ell}_L \end{bmatrix} \begin{bmatrix} 0 \\ \varphi^B \end{bmatrix} = (\bar{\nu}_\ell + \bar{\ell}_L) \varphi^B \end{equation} \begin{equation} \bar{\psi}_L\varphi\psi_R = (\bar{\nu}_\ell + \bar{\ell}_L) \varphi^B \begin{bmatrix} 0 \\ \ell_R \end{bmatrix} = \begin{bmatrix} 0 \\ (\bar{\nu}_\ell + \bar{\ell}_L) \varphi^B \ell_R \end{bmatrix} \end{equation} \begin{equation} \bar{\psi}_R\varphi^\top\psi_L = \begin{bmatrix} 0 & \bar{\ell}_R \end{bmatrix} \begin{bmatrix} 0 & \varphi^B \end{bmatrix} \begin{bmatrix} 0 \\ \nu_\ell + \ell_L \end{bmatrix} \end{equation} \begin{equation} \bar{\psi}_R\varphi^\top\psi_L = \begin{bmatrix} 0 & \bar{\ell}_R\varphi^B (\nu_\ell + \ell_L) \end{bmatrix} \end{equation} \begin{equation} (\bar{\psi}_R\varphi^\top\psi_L)^\top = \begin{bmatrix} 0 \\ \bar{\ell}_R\varphi^B (\nu_\ell + \ell_L) \end{bmatrix} \end{equation} \begin{equation} \phi' = \begin{bmatrix} 0 \\ (\bar{\nu}_\ell + \bar{\ell}_L) \varphi^B e_R + \bar{\ell}_R\varphi^B (\nu_\ell + \ell_L) \end{bmatrix} \end{equation} Why change $ \begin{bmatrix} \nu_\ell \\ \ell_L \end{bmatrix} $ to $ \begin{bmatrix} 0 \\ \nu_\ell + \ell_L \end{bmatrix} $? Consider the Gell-Mann-Nishijima relation and Table 1: \begin{equation} Q = I^3 + \frac{Y}{2} \end{equation}
Where $\rho$ designates the particle species, $I$ represents the weak isospin, $Y$ denotes the weak hypercharge, and $Q$ is the net electric charge. Crucially, the composite quantum state $\nu_\ell + \ell_L$ exhibits the exact same weak isospin and weak hypercharge as the right-handed charged lepton field $\ell_R$. This allows the state $\nu_\ell + \ell_L$ to be treated as a single left-handed physical entity interacting with $\varphi^B$ and $\bar{\ell}_R$. Mechanistically, its Dirac adjoint counterpart $\bar{\nu}_\ell + \bar{\ell}_L$ functions as a single left-handed anti-particle state coupled directly to $\varphi^B$ and $\ell_R$. Because the coupling strength governing neutrinos differs from that of charged leptons, we represent the neutrino interaction scale using $\sqrt{G_{\nu_\ell i}}\sqrt{G_\ell}$. Since observations constrain neutrinos to three distinct physical mass eigenstates, this coupling parameter is explicitly indexed by $i$, where $i = 1, 2, 3$. The standard Yukawa coupling $G_\ell$ is retained for the charged lepton field components:
\begin{equation} \phi'' = \begin{bmatrix} 0 \\ (\sqrt{G_{\nu_\ell i}}\bar{\nu}_\ell + \sqrt{G_\ell}\bar{\ell}_L) \varphi^B \sqrt{G_\ell}\ell_R + \sqrt{G_\ell}\bar{\ell}_R\varphi^B (\sqrt{G_{\nu_\ell i}}\nu_\ell + \sqrt{G_\ell}\ell_L) \end{bmatrix} \end{equation} \begin{equation} \phi'' = \begin{bmatrix} 0 \\ \sqrt{G_{\nu_\ell i}}\sqrt{G_\ell}\bar{\nu}_\ell\varphi^B \ell_R + G_\ell\bar{\ell}_L\varphi^B \ell_R + \sqrt{G_{\nu_\ell i}}\sqrt{G_\ell}\bar{\ell}_R\varphi^B \nu_\ell + G_\ell\bar{\ell}_R\varphi^B \ell_L \end{bmatrix} \end{equation} \begin{equation} \phi'' = \begin{bmatrix} 0 \\ \sqrt{G_{\nu_\ell i}}\sqrt{G_\ell}(\bar{\nu}_\ell\varphi^B \ell_R + \bar{\ell}_R\varphi^B \nu_\ell) + G_\ell(\bar{\ell}_L\varphi^B \ell_R + \bar{\ell}_R\varphi^B \ell_L) \end{bmatrix} \end{equation} \begin{equation} \phi'' = \sqrt{G_{\nu_\ell i}}\sqrt{G_\ell} \begin{bmatrix} 0 \\ \bar{\nu}_\ell\varphi^B \ell_R + \bar{\ell}_R\varphi^B \nu_\ell \end{bmatrix} + G_\ell \begin{bmatrix} 0 \\ \bar{\ell}_L\varphi^B \ell_R + \bar{\ell}_R\varphi^B \ell_L \end{bmatrix} \end{equation} \begin{equation} \phi'' = \sqrt{G_{\nu_\ell i}}\sqrt{G_\ell} \begin{bmatrix} 0 \\ \bar{\nu}_\ell \ell_R + \bar{\ell}_R\nu_\ell \end{bmatrix}\varphi^B + G_\ell \begin{bmatrix} 0 \\ \bar{\ell}_L \ell_R + \bar{\ell}_R \ell_L \end{bmatrix} \varphi^B \end{equation}We map these configurations directly to the effective mass framework of the Dirac Lagrangian. Here, the total mass parameter $M_i$ encapsulates both the static charged lepton mass and the probabilistically varying neutrino mass eigenstate, requiring the total mass term to be explicitly indexed by the eigenstate index $i$:
\begin{equation} L_{mass} = -M_i(\bar{\psi}_L\psi_R + \bar{\psi}_R\psi_L) \end{equation} We calculate the total mass $M_i$: \begin{equation} M_i =\frac{-L_{mass}}{\bar{\psi}_L\psi_R + \bar{\psi}_R\psi_L} \end{equation}Next, we calculate the charged lepton (anti-lepton) mass $m_\ell$. To ensure that $m_\ell$ precisely reproduces empirical experimental measurements within this modified architecture, a minor numerical calibration can be applied to the coupling parameter $G_\ell$:
\begin{equation} %32 m_\ell = \frac{G_\ell(\bar{\ell}_L \ell_R + \bar{\ell}_R \ell_L)\varphi^B}{\bar{\psi}_L\psi_R + \bar{\psi}_R\psi_L} \end{equation} The mass of the left-handed neutrino (anti-neutrino) is the difference between the total mass and the charged lepton (anti-lepton) mass: \begin{equation} m_{\nu i} = M_i - m_\ell \end{equation} The probability of $m_{\nu i}$ is $|U_{\ell i}|^2$.Discussion
The mechanism introduced in this paper offers a distinct conceptual advantage over conventional extensions of the Standard Model. The prevailing paradigm for generating non-zero neutrino masses relies on the addition of gauge-singlet, right-handed sterile neutrino fields. In extensions such as the Type-I Seesaw Mechanism, these sterile fields are assigned an explicit, unconstrained Majorana mass scale ($M_R$) that is typically assumed to reside near the Grand Unification Theory (GUT) scale ($10^{14}$ to $10^{15}$~GeV). While mathematically elegant in suppressing the light active neutrino masses to the sub-electronvolt range through the relation $m_\nu \approx m_D^2 / M_R$ ($m_D = $ Dirac mass), this approach introduces a severe hierarchy problem and relies on a colossally heavy, unobservable sector that remains entirely decoupled from modern high-energy collider physics.
In direct contrast, our framework achieves non-zero neutrino masses without introducing independent heavy sterile fields, additional gauge symmetries, or fine-tuned mass scales. Rather than expanding the particle spectrum, the mechanism modifies the underlying arrangement of the existing lepton content. By recognizing that the composite state $\nu_\ell + \ell_L$ shares identical weak isospin and weak hypercharge with the right-handed charged lepton field $\ell_R$, the mass terms emerge naturally via the standard Higgs-field vector $\varphi^B$. This eliminates the need for a decoupled, high-energy sterile mass altogether. By mapping the flavor-mass variations directly to the mixing elements of the PMNS matrix, our model provides a minimalist alternative that handles mass generation entirely within the observable electroweak sector while leaving the verified low-energy phenomenological predictions intact.
Conclusion
In this paper, we have presented an alternative framework for generating non-zero neutrino masses without introducing heavy sterile fields or modifying the standard field content of the electroweak theory. By treating the left-handed neutrino and left-handed charged lepton fields as a unified entity under a modified gauge assignment, we demonstrated that a valid mass term can be naturally accommodated via the Higgs-field vector. This construction successfully accounts for the tiny mass eigenstates required by neutrino oscillation data while preserving the verified predictions and mass relations of the standard electroweak gauge sectors.
References
1. Maki, M. Nakagawa, and S. Sakata, ``Remarks on the Unified Model of Elementary Particles,'' Progress of Theoretical Physics 28}(5), 870 (1962).
2. W. F. Valle, ``Neutrino Physics Overview,'' Journal of Physics: Conference Series 53(1), 473--505 (2006), arXiv:hep-ph/0608101.
3. McMahon, Quantum Field Theory Demystified--A Self-Teaching Guide. McGraw-Hill Companies, Inc., 2008.
© 2026 G.M. Jackson
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