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The proton has no lifetime, only a cross-section: Triality-locked baryon number and vortex-catalyzed nucleon decay

This paper proposes that if baryon number is an exact discrete gauge symmetry locked to the Z3 center of color, protons possess no intrinsic lifetime but instead decay via a vortex-catalyzed process with a cross-section dependent on local cosmic defect density, resulting in a unique signature of environment-dependent, clustered ∆B=3 events rather than standard ∆B=1 decays.

Original authors: Mohammad Hannan

Published 2026-07-29

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohammad Hannan

Original paper licensed under CC BY 4.0 (https://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

The Great Proton Puzzle: Why Everything Might Be Stable (Until It Isn't)

Imagine the universe is built out of tiny, unbreakable Lego bricks. For decades, physicists have been obsessed with one specific brick: the proton. It's the core of every atom in your body, the stars, and the coffee in your mug. The big question has always been: "How long does a proton last?" The standard answer is that protons are almost eternal, but they might eventually fall apart into lighter particles, like a Lego brick slowly crumbling into dust over billions of years. If protons do decay, it would be a massive discovery, proving that the laws of physics we know today are just a slice of a much bigger, stranger picture.

To understand this new idea, you need to know two things. First, there's a rule called "Baryon Number." Think of it as a cosmic accounting system where protons and neutrons (baryons) have a value of +1, and their opposites have -1. In the old story, this number could change by 1, meaning a single proton could vanish. Second, there are "topological defects." Imagine a knotted rope or a twist in a fabric that can't be untied without cutting it. In the early universe, as things cooled down, the fabric of reality might have developed these permanent knots or strings. Usually, we think of these as just heavy, invisible objects floating around. But what if they aren't just floating? What if they are the only reason protons ever disappear?

The Paper's Big Twist: Protons Don't Have a "Lifetime"

In this research article, independent researcher Mohammad Hannan proposes a radical shift in how we view the proton. He argues that the proton doesn't have a "lifetime" at all. Instead of being a ticking clock that eventually runs out of time, the proton is perfectly stable forever—unless it bumps into something specific.

The Main Idea: The "Catalysis" Theory
Hannan suggests that protons are absolutely safe in empty space. They will never decay on their own. However, the universe is filled with invisible, cosmic "vortex strings" (the knots mentioned earlier). These strings are like cosmic speed bumps or catalysts. When a proton happens to collide with one of these strings, the string acts like a magic wand, temporarily breaking the rules that keep the proton safe. In that split second of collision, the proton can change, but only in a very specific way.

What This Paper Rules Out
The paper explicitly argues against the most popular theories of proton decay.

No Single Proton Decay: The standard idea that a single proton turns into a positron and a pion (a process called $\Delta B = 1$ ) is strictly forbidden in this model. If you see a single proton vanish on its own, this theory is wrong.
No Neutron Oscillation: The idea that a neutron can spontaneously turn into an antineutron ( $\Delta B = 2$ ) is also impossible here.
No "Mass Scale" Decay: The paper rejects the idea that there is a universal, constant rate of decay that happens everywhere in the universe at the same time.

The New Rule: Triality and the "Group of Three"
The paper builds a mathematical model (using a group called $SU(4)_{cB}$ ) where baryon number is treated like a "fourth color" of charge. This creates a strict lock: you can only break a proton if you break three of them at the same time.

The "Three-for-One" Rule: Instead of one proton disappearing, the theory predicts that three nucleons (protons or neutrons) must vanish together when they hit a string. This is a $\Delta B = 3$ event.
The Cosmic String Core: Inside the core of these invisible strings, the rules of physics change. The "lock" is temporarily opened, allowing three baryons to interact and transform into other particles (like pions and neutrinos).

How It Works: The "Cross-Section" vs. "Lifetime"
The paper replaces the concept of "lifetime" with "cross-section."

Old View: A proton has a timer. It might last $10^{34}$ years.
New View: A proton has a target size. It only "decays" if it hits a string. The rate of decay depends entirely on how many strings are floating around in your neighborhood.
The Formula: The paper gives a simple equation: $\Gamma_p = n_{def} \sigma v$ $Γ_{p} = n_{d e f} σ v$ .
- $\Gamma_p$ is the rate of decay.
- $n_{def}$ is the local density of these cosmic strings (how crowded they are).
- $\sigma$ is the size of the target (the cross-section).
- $v$ is the speed of the proton.

This means the "lifetime" of a proton isn't a constant number of nature. It changes depending on where you are. If you were inside a neutron star where these strings might be packed tightly together, protons would seem to decay much faster. If you are in empty space with no strings nearby, they are perfectly immortal.

What the Paper Predicts (and How to Test It)
The authors are very clear: this idea is "sharply falsifiable," meaning it's easy to prove wrong.

The "Clustered" Signal: If this theory is right, we shouldn't see random, isolated proton decays. Instead, we should see a "train" of events. As a cosmic string sweeps through a detector (like a giant water tank), it would leave a trail of three-baryon disappearances in a straight line, all happening at the exact same time.
The "Three-Particle" Event: The decay products would be three nucleons disappearing at once, releasing about 2.7 GeV of energy, rather than the single-particle decay we usually look for.
Gravitational Waves: These cosmic strings would wiggle and create ripples in spacetime called gravitational waves. The paper suggests that if we detect a specific pattern of these waves (using instruments like LISA or pulsar timing arrays) at the same time as we see these clustered proton decays, it would be "smoking-gun" evidence.

The Bottom Line
This paper doesn't claim to have found the answer yet. It offers a new framework to explain why we haven't seen protons decay yet: maybe they aren't decaying on their own. It suggests that the "proton lifetime" we measure in experiments is actually just a measure of how many cosmic strings are passing through our detector. If we ever see a single proton decay on its own, this theory is dead. But if we see a line of three-baryon events appearing in a straight line, correlated with gravitational waves, we might have just found the cosmic strings that hold the universe together.

Technical Summary: "The proton has no lifetime, only a cross-section: Triality-locked baryon number and vortex-catalyzed nucleon decay"

Problem Statement
Current experimental limits on proton decay (e.g., $\tau(p \to e^+\pi^0) > 2.4 \times 10^{34}$ yr) are universally interpreted as constraints on an intrinsic, constant proton lifetime determined by a high-energy mass scale (e.g., Grand Unified Theory scales). This interpretation assumes a spontaneous, environment-independent first-order decay process. The paper challenges this paradigm, arguing that the proton may possess no intrinsic lifetime constant. Instead, it proposes that baryon number ( $B$ ) is exactly conserved in the vacuum, and apparent decay is actually a catalysis event triggered by the interaction of nucleons with topological defects.

Methodology and Theoretical Framework
The author constructs a specific field-theoretic framework to realize this scenario:

$SU(4)_{cB}$ Embedding: Baryon number is embedded as a "fourth colour" within a unified $SU(4)_{cB}$ gauge group. The symmetry breaks in two stages:
- $SU(4)_{cB} \xrightarrow{M_*} SU(3)_c \times U(1)_B$
- $U(1)_B \xrightarrow{v_\Phi} SU(3)_c \times Z^{(B)}_3$
  The second stage is driven by a scalar field $\Phi$ with baryon number $B_\Phi = 3$ .
Triality-Locking and Selection Rules: The breaking of $U(1)_B$ leaves an exact discrete gauge symmetry $Z^{(B)}_3$ . Under this symmetry, quarks carry charge $+1$ and leptons charge $0$. This enforces a strict selection rule: any gauge-invariant operator must satisfy $\Delta B \equiv 0 \pmod 3$ . Consequently, single-nucleon decay ( $\Delta B = 1$ ) and neutron-antineutron oscillation ( $\Delta B = 2$ ) are exactly forbidden at all scales, including by quantum gravity, as they violate this discrete gauge symmetry.
Topological Defects: The breaking $U(1)_B \to Z^{(B)}_3$ generates stable $Z_3$ baryonic vortices (cosmic strings) due to the non-trivial first homotopy group $\pi_1(U(1)/Z_3) = Z_3$ .
- Core Physics: Inside the vortex core, the scalar condensate $\langle \Phi \rangle$ vanishes, locally restoring the full $U(1)_B$ symmetry. This suspension of the $Z^{(B)}_3$ selection rule allows $\Delta B = 3$ operators to become unsuppressed within the core.
- Effective Operator: A dimension-15 operator (in the bulk) becomes effective as a six-fermion operator ( $\sim (udd)^3$ ) inside the core, mediating processes like $p + n + n \to \text{mesons} + \bar{\nu}\bar{\nu}\bar{\nu}$ .
Catalysis Mechanism: The paper applies the Callan-Rubakov and Alford-March-Russell-Wilczek (AMW) mechanisms. Nucleons scattering off the vortex core experience wavefunction amplification (Aharonov-Bohm effect), leading to a catalysis cross-section ( $\sigma_{cat}$ ) that is not suppressed by the high unification scale $M_*$ . The cross-section is estimated to range from geometric overlap values ( $\sim 10^{-32} \text{ cm}^2$ ) to the strong-interaction unitarity bound ( $\sim 10^{-26} \text{ cm}^2$ ).

Key Results and Derivations

The Master Relation: The paper derives that the observable "decay rate" is not a constant but a function of the local defect density:
$\Gamma_p = n_{def} \sigma_{cat} v$
where $n_{def}$ is the local density of baryonic-defect encounters, $\sigma_{cat}$ is the catalysis cross-section, and $v$ is the relative velocity. The apparent lifetime is $\tau_p^{eff} = \Gamma_p^{-1}$ .
Anomaly Cancellation: The author proves that the $Z^{(B)}_3$ symmetry is free of gauge and gravitational anomalies within the Standard Model embedding, requiring no additional spectator fermions.
Cosmological Evolution: The defect network is modeled as a scaling solution of local cosmic strings. The present-day density is suppressed by the expansion of the universe, but the network radiates a stochastic gravitational-wave background (SGWB).
Reinterpretation of Limits: Existing experimental null results (e.g., Super-Kamiokande) are recast not as lower bounds on a mass scale, but as upper bounds on the local cosmic defect flux ( $n_{def}$ ).

Significance and Claims
The paper claims to offer a "maximally falsifiable" alternative to mass-scale theories of nucleon decay. Its significance lies in reorganizing the concept of proton stability around a measurable quantity (cosmic defect flux) rather than an unobservable high-energy mass scale.

The paper asserts that this framework makes four decisive, unique predictions that distinguish it from standard GUTs or hidden portal models:

Absolute Stability of $\Delta B = 1$ and $\Delta B = 2$ : No single-nucleon decay ( $p \to e^+\pi^0$ ) or neutron-antineutron oscillation will ever be observed.
Dominant $\Delta B = 3$ Channel: Any observed signal must be a correlated, simultaneous disappearance of three nucleons (e.g., $pnn$) with $\Delta(B-L)=0$ .
Clustered, Collinear Events: Unlike the uniform, Poisson-distributed events of spontaneous decay, vortex catalysis produces events clustered along the linear path of a passing string, occurring within a narrow time window ( $\sim 0.25$ ms for a detector of Hyper-K size).
Environment Dependence and GW Correlation: The apparent decay rate varies with the local density of defects (e.g., enhanced in neutron stars or the solar core). Furthermore, a detection of $\Delta B = 3$ events should be correlated with a stochastic gravitational-wave background from the string network, detectable by Pulsar Timing Arrays (PTA) or LISA.

The author concludes that while the constituent elements (colour centre locking, sphaleron rules, defect catalysis) are established physics, the synthesis proposing that the proton has "no lifetime, only a cross-section" constitutes a new, testable paradigm.

The Great Proton Puzzle: Why Everything Might Be Stable (Until It Isn't)

The Paper's Big Twist: Protons Don't Have a "Lifetime"

Technical Summary: "The proton has no lifetime, only a cross-section: Triality-locked baryon number and vortex-catalyzed nucleon decay"

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