⚛️ high-energy theory

Chiral phases for massive fermions and spinor classifications

This paper explores the Lounesto classification of regular spinors by introducing two chiral phases—one arising from gamma matrix choices and another from unconstrained massive field degrees of freedom—to demonstrate that varying these phases generates all regular spinor classes and suggests that the Standard Model's fermion chiral phases constitute new physical parameters that are unmeasurable within current weak interactions.

Original authors: Cheng-Yang Lee

Published 2026-08-04

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Cheng-Yang Lee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe is built from a giant, invisible Lego set. The smallest, most fundamental pieces of this set are tiny particles called fermions, which include the electrons that power our lights and the quarks that make up the atoms in our bodies. For decades, physicists have used a very specific instruction manual, called the Standard Model, to understand how these pieces fit together. This manual relies on a mathematical tool called "spinors" to describe how these particles spin and move. Think of a spinor not just as a particle, but as a tiny, spinning top that carries a secret code—a hidden "phase" or setting—that tells it how to interact with the rest of the universe.

Usually, scientists have assumed there is only one correct way to set this code for every particle, much like assuming every Lego brick has only one specific color and shape. However, a new perspective suggests that these spinning tops might have a hidden dial that we haven't been turning. This dial could change the particle's "chiral phase," a fancy way of saying it can twist its internal structure in different ways without breaking the fundamental laws of physics. The big question is: do these hidden dials actually exist in nature, and if they do, could they explain mysteries like why neutrinos have mass or what dark matter is made of?

This paper, written by Cheng-Yang Lee, dives deep into this mystery by re-examining the "instruction manual" for massive fermions. The author explores a classification system known as the Lounesto classification, which sorts all possible types of spinors into six different "families." While the Standard Model mostly uses the second family (Dirac spinors), this paper asks what happens if we tweak the hidden dials (the chiral phases) to create particles from the first and third families, or even just different versions of the second family.

The author discovers that these hidden dials are real mathematical possibilities that aren't forbidden by the laws of physics. By turning these dials, you can create new types of particles that behave differently than the ones we currently know. For instance, the paper shows that if you have two different particles (like an electron and a muon) and they have different settings on their hidden dials, they would interact with each other in strange, measurable ways. However, when the author looks at the particles we actually see in our daily lives, the story gets more subtle.

The paper finds that for the particles in the "lepton" family (like electrons and neutrinos), these hidden dials are essentially invisible to the weak nuclear force, the force responsible for radioactive decay. It's as if the dials are locked in a position that makes them impossible to detect with current experiments. But for the "quark" family (the building blocks of protons and neutrons), the story is different. The author calculates that if protons and neutrons had different dial settings, it would change how long a free neutron lives before it decays. The math shows that a difference in these settings would cause a massive change—up to 25%—in the neutron's lifetime. Since experiments tell us the neutron's lifetime is very precise and matches the standard prediction, the author concludes that protons and neutrons must have the exact same dial setting.

In short, the paper suggests that while these "chiral phases" are a new set of physical parameters that could exist, nature seems to have locked them into a specific configuration for the quarks we know. The author argues that these phases are a new kind of physical property for the Standard Model, but measuring them would likely require interactions beyond our current understanding of physics. It's a bit like finding a secret compartment in a familiar toy; the compartment exists, but you need a special, yet-to-be-invented key to open it and see what's inside.

Technical Summary: Chiral Phases for Massive Fermions and Spinor Classifications

Problem Statement
The Standard Model (SM) successfully describes massive and massless fermions using Dirac and Weyl fields, constructed from unitary irreducible representations of the Poincaré group. However, the Lounesto spinor classification reveals six distinct classes of spinors in four-dimensional Minkowski space-time: three regular (1st–3rd) and three singular (4th–6th). While Dirac spinors correspond to the 2nd class and Weyl/ELKO spinors to the 6th and 4th–5th classes respectively, the physical nature of the 1st and 3rd class regular spinors remains underexplored. Furthermore, it is unclear whether the construction of massive spin-half fields allows for additional degrees of freedom beyond the standard Dirac formulation, specifically regarding chiral phases that are unconstrained by Poincaré symmetry and locality.

Methodology
The author re-examines the construction of massive spin-half fields from first principles, focusing on the Lounesto classification and the role of chiral transformations. The methodology involves:

Identifying Two Chiral Phases:
- Phase $\alpha$ : Arises from the freedom in choosing the $\gamma^\mu$ matrices via a chiral transformation ( $\gamma^\mu_\alpha = e^{-i\gamma_5\alpha/2}\gamma^\mu e^{i\gamma_5\alpha/2}$ ). This transformation leaves the Lorentz generators ( $J_{\mu\nu}$ ) invariant.
- Phase $\beta$ : A degree of freedom inherent to the massive spin-half field solution, unconstrained by Poincaré symmetry and locality. It appears in the coefficients of the spinor solutions ( $u$ and $v$ ) when satisfying rotation invariance and locality constraints.
Deriving Spinor Solutions: By constructing the spinor field $\psi(x)$ with general coefficients $c_\pm$ and $d_\pm$ parameterized by phases $\beta_u$ and $\beta_v$ , and imposing locality via equal-time anti-commutators, the author derives the relationship $\beta_v = \beta_u + (2n+1)\pi$ . Setting $\beta_u = \beta$ , the spin-sums and bilinear covariants are calculated as functions of the difference $(\alpha - \beta)$ .
Classifying Regular Spinors: The author demonstrates that the Lounesto classes (1st, 2nd, and 3rd) are determined by the value of $(\alpha - \beta)$ $(α - β)$ :
- 1st Class: $(\alpha - \beta) \neq 0, \pi, \pi/2, 3\pi/2$ .
- 2nd Class (Standard Dirac): $(\alpha - \beta) = 0, \pi$ .
- 3rd Class: $(\alpha - \beta) = \pi/2, 3\pi/2$ .
Analyzing Physical Observables: The paper investigates the physical consequences of these phases by computing observables in specific interaction scenarios:
- Single Fermion Interactions: Analyzing Yukawa interactions to determine phase dependence in decay rates.
- Multiple Fermion Interactions: Examining interactions between different fermion species (e.g., $\psi$ and $\lambda$ ) where relative phases $(\beta_\psi - \beta_\lambda)$ may exist.
- Standard Model Applications: Applying these concepts to SM weak interactions, specifically charged-current lepton interactions and neutron $\beta$ -decay (quark sector).

Key Contributions and Results

Generalization of Dirac Spinors: The paper establishes that standard Dirac spinors are a specific case ( $\alpha = \beta = 0$ ) of a more general framework where $\alpha = \beta$ but not necessarily zero. This generalization allows for the existence of 1st and 3rd class regular spinors as physically distinct fields.
Phase-Dependent Observables:
- For single fermions, certain Yukawa interactions (e.g., $H'_Y = g'\bar{\psi}\psi\phi$ ) yield decay rates dependent on $(\alpha - \beta)$ . Specifically, the mass-squared contribution to the decay rate vanishes when $\beta = \alpha + \pi/4$ .
- For multiple fermions, interactions involving different species (e.g., flavor-violating Yukawa couplings) depend on the relative phase $(\beta_\psi - \beta_\lambda)$ . The decay rate is proportional to $[1 + \cos(\beta_\lambda - \beta_\psi)]$ , which can vanish for specific phase differences.
Standard Model Implications:
- Leptons: In SM weak charged-current interactions, the relative phases between leptons and neutrinos are global and cancel out. Consequently, lepton chiral phases ( $\beta_\ell, \beta_i$ ) are not measurable within the weak interaction and can theoretically differ between generations.
- Quarks: In neutron $\beta$ -decay, the relative phase between up and down quarks ( $\beta_u - \beta_d$ ) introduces a term proportional to $(f^2 - g^2)[1 - \cos(\beta_n - \beta_p)]$ in the decay rate. Since $f \neq g$ in the SM, a non-zero phase difference would cause significant deviations (up to 25%) from the SM prediction for the neutron lifetime.
- Constraint on Quark Phases: Given the tight experimental constraints on the neutron lifetime, the paper concludes that the chiral phases for up and down quarks must be identical ( $\beta_u = \beta_d$ ). Extending this to include quark mixing (CKM matrix), the author conjectures that all quarks in the SM share the same chiral phase.

Significance and Claims
The paper argues that the chiral phase is a new set of physical parameters for SM fermions, unconstrained by current Poincaré symmetry and locality requirements.

Distinct Physics: Fermionic fields constructed from different classes of regular spinors (1st, 2nd, 3rd) are physically distinct due to phase-dependent observables.
Intra-Class Variability: Even within the same class (e.g., the 2nd class/Dirac fermions), different fermion species may possess different chiral phases. While this is unobservable in single-species interactions or specific SM weak processes for leptons, it becomes a measurable parameter in interactions between different fermion generations or in processes beyond the SM.
Beyond the Standard Model: Measuring these phases requires interactions beyond the SM, such as flavor-violating processes (e.g., the Yukawa interaction in Eq. 38) or models like Two-Higgs-Doublet Models. The existence of these phases suggests that the SM fermions possess a hidden degree of freedom that could be a target for future searches for new physics.

The author concludes that while the SM fermions are currently consistent with a single chiral phase for quarks and potentially different phases for leptons, the theoretical framework necessitates treating these phases as fundamental parameters that may be species-dependent.

Technical Summary: Chiral Phases for Massive Fermions and Spinor Classifications

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