New limits on the Pauli forbidden transitions in 12C nuclei obtained with the complete Borexino dataset

Using the complete Borexino dataset from 2007 to 2021, researchers established the most stringent experimental limits to date on the lifetime of $^{12}\text{C}$ nuclei against Pauli exclusion principle-forbidden transitions, setting lower bounds ranging from $10^{30}$ to $10^{32}$ years for various decay channels and deriving extremely tight upper limits on the relative strengths of such non-Paulian processes.

The Gist

The Great Cosmic "No-Double-Booking" Rule

Imagine a grand, cosmic hotel called the Atom Hotel. In this hotel, there is a very strict, unbreakable rule known as the Pauli Exclusion Principle. This rule, established by physicist Wolfgang Pauli in 1925, states that no two identical guests (like electrons or protons) can ever occupy the exact same room at the same time.

If a room is full, a new guest simply cannot move in. They must find an empty room on a higher floor. This rule is the reason matter has structure; without it, all atoms would collapse into a tiny, featureless blob.

The Big Question: What if the Rule is Broken?

For nearly a century, scientists have wondered: Is this rule 100% perfect, or is there a tiny, almost invisible chance that a guest could sneak into a full room?

If a proton or neutron (the "guests" inside the nucleus of an atom) were to break this rule and jump into a room that was already full, it would be a "forbidden transition." This wouldn't just be a quiet sneaking-in; it would be a chaotic event. The atom would suddenly become unstable and release a burst of energy—like a firework exploding inside the nucleus.

The Detective: Borexino

To catch this cosmic rule-breaker, scientists used a massive detector called Borexino.

• The Location: It's buried deep underground in a mountain in Italy (Gran Sasso). This is like a soundproof bunker; the mountain blocks out the noise of cosmic rays from space, leaving the detector in a state of "silence."
• The Size: The detector is a giant sphere filled with 278 tons of a special, ultra-pure liquid (scintillator). Think of it as a giant, glowing swimming pool.
• The Job: When a particle interacts with this liquid, it creates a tiny flash of light. The detector is so sensitive it can see a single photon (a particle of light) from a billion miles away.

The scientists looked specifically at Carbon-12 atoms (the stuff in your body and the air) inside this liquid. They waited for 14 years, watching billions of carbon atoms, hoping to see one of them break the Pauli rule.

The Search: What Were They Looking For?

The scientists imagined a scenario where a proton or neutron in a Carbon-12 atom tries to jump from a higher floor (the "P-shell") down to a lower floor (the "S-shell") that is already completely packed with other particles.

If this happened, the atom would have to get rid of the extra energy. It could do this in a few ways, like a magician pulling different tricks:

1. The Flash: It shoots out a high-energy gamma ray (a burst of light).
2. The Ejection: It kicks out a proton or a neutron.
3. The Transformation: It turns into a different particle (beta decay).

The Borexino team watched the "pool" for these specific flashes or ejected particles. They set up filters to ignore the "noise" (like background radiation or cosmic rays) so they could hear the faint "sneak" of a rule-breaking event.

The Results: The Rule Holds Firm

After analyzing data collected over 14 years, the result was a massive relief for the laws of physics: They found absolutely nothing.

Not a single Carbon-12 atom broke the rule.

Because they didn't find any rule-breakers, they could calculate how unlikely it is for this to happen. They set new, incredibly strict limits on how long an atom can exist before it might break the rule.

• The Old Limit: We knew it was rare.
• The New Limit: We now know it is astronomically rare.

To put their new numbers in perspective:

• The lifetime of a Carbon-12 atom before it might break the Pauli rule is at least 100,000,000,000,000,000,000,000,000,000,000 years (10³² years).
• To visualize this: The entire universe is only about 13.8 billion years old (10¹⁰ years). The time it would take for a Carbon atom to break this rule is 10 billion billion times longer than the age of the universe.

The "Relative Strength" Metaphor

The paper also calculated the "strength" of this forbidden transition compared to normal physics. They used a number called δ² (delta squared).

Imagine normal physics is a roaring waterfall. The forbidden transition is a single drop of water trying to flow upstream against that waterfall.

• For the electromagnetic force (gamma rays), the chance of a drop going upstream is 1 in 10⁵⁷.
• For the strong nuclear force (protons/neutrons), the chance is 1 in 10⁶¹.

These numbers are so small they are almost meaningless in human terms. It's like winning the lottery every second for a trillion years and still losing.

Why Does This Matter?

You might ask, "If the rule is so perfect, why bother checking?"

1. Testing the Foundation: The Pauli Exclusion Principle is a cornerstone of quantum mechanics. If we found even a tiny crack in it, our entire understanding of the universe (chemistry, stars, matter) would need to be rewritten.
2. New Physics: Some theories suggest that at the very beginning of the universe or at the smallest scales (Planck scale), the rules of quantum mechanics might get a little fuzzy. By pushing the limits of how perfect the rule is, we are testing if there is "new physics" hiding in the shadows.
3. The VIP Experiment: While other experiments (like VIP) test this with electrons in copper wires, Borexino is unique because it tests this with nucleons (protons and neutrons) inside the nucleus. It's a different kind of test, and Borexino has now set the world record for this specific test.

The Takeaway

The Borexino collaboration has acted as the ultimate cosmic bouncer. They stood guard for 14 years in a silent, deep-mountain bunker, watching billions of atoms. They confirmed that the "No-Double-Booking" rule of the universe is not just a suggestion; it is a law so rigid that it has held firm for longer than the universe has existed.

The universe, it seems, is very orderly indeed.

Technical Summary

1. Problem Statement and Motivation

The Pauli Exclusion Principle (PEP) is a fundamental postulate of quantum mechanics stating that no two identical fermions can occupy the same quantum state simultaneously. While PEP is a cornerstone of physics, its fundamental origin remains unexplained, and theoretical models (e.g., non-commutative quantum gravity, extra dimensions, or Lorentz invariance violation) suggest that minute violations might exist.

The paper addresses the search for PEP-violating transitions in nucleons within the Carbon-12 ($^{12}\text{C}$) nucleus. Specifically, it investigates hypothetical transitions where a nucleon (proton or neutron) jumps from the partially filled $1P_{3/2}$ shell to the fully occupied $1S_{1/2}$ shell. Such a transition would create an excited "non-Paulian" nucleus ($^{12e}\text{C}$) and release energy via the emission of:

• $\gamma$-rays (Electromagnetic interaction)
• Protons ($p$) or Neutrons ($n$) (Strong interaction)
• Electrons/Positrons ($e^{\mp}$) and neutrinos (Weak interaction, $\beta$-decay)

Previous experiments had set limits on these processes, but the sensitivity was limited by detector mass, background noise, or measurement duration. This study aims to provide the most stringent experimental constraints to date using the complete dataset from the Borexino experiment.

2. Methodology

Experimental Setup

The study utilizes data from the Borexino detector, located at the Laboratori Nazionali del Gran Sasso (Italy).

• Detector: A large-volume liquid scintillator detector containing 278 tons of pseudocumene ($C_9H_{12}$) doped with PPO.
• Environment: Encased in a stainless steel sphere and a water tank acting as a Cherenkov muon veto to shield against cosmic rays and external radiation.
• Dataset: Data collected over 14 years (2007–2021), providing an exposure of 576.2 kton-days.
• Key Advantages: Extremely low background levels, large target mass, and long integration time.

Signal Identification and Selection

The analysis searches for specific signatures of non-Paulian transitions in the energy spectrum:

1. $\gamma$-emission ($^{12}\text{C} \to ^{12e}\text{C} + \gamma$): Searches for high-energy gamma rays ($\sim$16–19 MeV).
2. Proton emission ($^{12}\text{C} \to ^{11e}\text{B} + p$): Protons are identified via their scintillation light (quenching factor applied) and pulse shape discrimination (using a Multi-Layer Perceptron, MLP, to distinguish protons from electrons/alphas).
3. Neutron emission ($^{12}\text{C} \to ^{11e}\text{C} + n$): Identified via delayed coincidence. The prompt signal is the recoil proton from neutron scattering; the delayed signal is the 2.22 MeV $\gamma$-ray from neutron capture on hydrogen ($n + p \to d + \gamma$).
4. $\beta$-decay ($^{12}\text{C} \to ^{12e}\text{N/B} + e^{\mp} + \nu$): Searches for high-energy electrons/positrons ($>12.5$ MeV) resulting from weak transitions.

Data Analysis Strategy

• Monte Carlo (MC) Simulations: Used to model detector response, energy resolution, and efficiency for all reaction channels. Simulations were tuned against calibration data from $^{241}\text{Am-}^9\text{Be}$ sources.
• Background Estimation: Backgrounds from solar neutrinos ($^8\text{B}$), cosmogenic isotopes, and natural radioactivity ($^{208}\text{Tl}$) were modeled. In low-energy regions, a conservative approach assumed zero expected background ($N_{exp}=0$) where signal-to-background ratios were favorable.
• Statistical Treatment: Upper limits on event counts ($S_{lim}$) were calculated using the Feldman-Cousins procedure at 90% Confidence Level (C.L.).

3. Key Contributions

• Comprehensive Dataset: This is the first analysis using the complete 14-year Borexino dataset, significantly increasing statistical power compared to previous partial analyses.
• Multi-Channel Analysis: The study simultaneously sets limits on electromagnetic, strong, and weak interaction channels for the same nucleus ($^{12}\text{C}$), allowing for a direct comparison of the relative strengths of PEP violation across different fundamental forces.
• Advanced Discrimination: The application of MLP-based pulse shape analysis significantly improved the rejection of background events in the proton emission channel.
• Theoretical Framework: The results are translated not only into lifetime limits but also into limits on the relative strength parameter ($\delta^2$), representing the mixing probability of non-fermionic symmetric components.

4. Key Results

The study established the following lower limits on the mean lifetime ($\tau$) for PEP-forbidden transitions in $^{12}\text{C}$ (all at 90% C.L.):

Transition Channel	Process	Lower Limit on Lifetime ($\tau$)	Previous Best Limit	Improvement Factor
Electromagnetic	$^{12}\text{C} \to ^{12e}\text{C} + \gamma$	$\ge 1.1 \times 10^{32}$ y	$4.2 \times 10^{24}$ y	$\sim 10^7$
Strong (Proton)	$^{12}\text{C} \to ^{11e}\text{B} + p$	$\ge 1.0 \times 10^{31}$ y	$8.9 \times 10^{29}$ y	$\sim 100$
Strong (Neutron)	$^{12}\text{C} \to ^{11e}\text{C} + n$	$\ge 2.0 \times 10^{31}$ y	$2.1 \times 10^{22}$ y	$\sim 10^9$
Weak ($\beta^-$)	$^{12}\text{C} \to ^{12e}\text{N} + e^- + \bar{\nu}_e$	$\ge 6.4 \times 10^{30}$ y	$3.1 \times 10^{24}$ y	$\sim 10^6$
Weak ($\beta^+$)	$^{12}\text{C} \to ^{12e}\text{B} + e^+ + \nu_e$	$\ge 6.6 \times 10^{30}$ y	$2.6 \times 10^{24}$ y	$\sim 10^6$

Relative Strength Limits ($\delta^2 = \lambda_{forbidden}/\lambda_{normal}$):

• Electromagnetic ($\delta^2_\gamma$): $\le 1.0 \times 10^{-57}$
• Strong ($\delta^2_N$ for $n, p$): $\le 7.0 \times 10^{-61}$
• Weak ($\delta^2_\beta$): $\le 9.6 \times 10^{-36}$

Note: The strong interaction limits are the most stringent, while the weak interaction limits are less restrictive due to the inherently slower rate of normal weak decays.

5. Significance

• Stringency: These results represent the most stringent experimental constraints to date on PEP violation in nuclei. The limits on the neutron and proton emission channels improve upon previous results by up to 9 orders of magnitude.
• Theoretical Implications: The derived limits on $\delta^2$ severely constrain theoretical models that predict PEP violation, such as those involving non-commutative geometry or extra dimensions. The fact that no signal was observed reinforces the validity of the PEP and the standard model of particle physics.
• Methodological Benchmark: The paper demonstrates the capability of large-volume liquid scintillators to perform precision nuclear physics searches, utilizing their low background and large mass to probe rare processes that are inaccessible to smaller detectors.
• Comparison with Other Experiments: The limits on $\gamma$-emission are comparable to those from the Kamiokande experiment (using $^{16}\text{O}$), while the limits on nucleon emission are significantly superior to those from DAMA/LIBRA and NEMO-2.

In conclusion, the Borexino collaboration has successfully utilized a decade-and-a-half of data to push the boundaries of PEP violation searches, confirming the principle's robustness in the nuclear sector with unprecedented precision.