First measurement of the absolute branching fractions of $\Sigma^+$ nonleptonic decays and test of the $\Delta I = 1/2$ rule

Using a large sample of $J/\psi$ events collected by the BESIII detector, this study presents the first absolute measurements of the $\Sigma^+ \to p \pi^0$ and $\Sigma^+ \to n \pi^+$ branching fractions, which significantly deviate from previous PDG values and provide strong evidence ($>5\sigma$) for the presence of a $\Delta I = 3/2$ transition amplitude in $\Sigma$ hyperon decays, challenging the strict validity of the $\Delta I = 1/2$ rule.

The Gist

Imagine the universe as a giant, bustling construction site. Most of the time, the workers (particles) follow strict blueprints (the laws of physics). But sometimes, they pull a "magic trick" where they vanish and reappear as something else. This is called decay.

This paper is like a team of master auditors (the BESIII Collaboration) who have spent years watching these magic tricks to see if the workers are following the rules or if they're sneaking in some extra moves.

Here is the story of what they found, explained simply:

1. The Mystery of the "Sigma Plus" Worker

The main character in this story is a particle called the Sigma Plus ($\Sigma^+$). Think of it as a very unstable Lego brick. It doesn't like to stay alone for long; it wants to break apart into smaller pieces.

For decades, scientists had a "rulebook" (called the PDG, or Particle Data Group) that told them exactly how often this brick breaks into two specific combinations:

• Combo A: A proton and a neutral pion (like a proton and a ghost).
• Combo B: A neutron and a charged pion (like a neutron and a charged spark).

The rulebook said: "About 51.5% of the time it does Combo A, and 48.4% of the time it does Combo B."

2. The "Double-Tag" Detective Trick

The problem was that the old rulebook numbers were based on rough estimates and old experiments. It was like trying to guess the weight of a watermelon by looking at a blurry photo from 1980.

The BESIII team at the Beijing Electron Positron Collider decided to do a direct, high-precision measurement. They used a clever trick called the "Double-Tag" method.

• The Analogy: Imagine you are at a party where couples always arrive together. You want to count how many people are wearing red hats.
  • The Single Tag: You spot one person in a couple wearing a red hat. You know their partner is there, even if you can't see them clearly.
  • The Double Tag: You spot the partner (the "tag") clearly first. Because you know they are a perfect pair, you know exactly where the other person (the "signal") must be. You can then look specifically at that spot to see what they are wearing.

By using this method on billions of particle collisions (specifically from a particle called the $J/\psi$), they could count the Sigma Plus decays with incredible accuracy, eliminating the guesswork.

3. The Big Surprise: The Rulebook Was Wrong!

When the team finally counted the results, they found something shocking. The Sigma Plus wasn't following the old rulebook at all.

• The New Reality:
  • Combo A: Happens 49.79% of the time.
  • Combo B: Happens 49.87% of the time.

The Metaphor: It's like if a coin that everyone thought was weighted to land on Heads 51% of the time actually lands on Heads and Tails almost exactly 50/50.

The difference between their new numbers and the old rulebook is huge in the world of physics. It's a 4.4-sigma and 3.4-sigma difference. In "everyday" terms, if you flipped a coin 10,000 times and got a result this different from what you expected, you would be 99.99% sure that your original expectation was wrong.

4. The "Isospin" Rule Breaker

The paper also tests a famous law of physics called the $\Delta I = 1/2$ rule.

• The Analogy: Imagine a dance floor where the music (the weak force) tells dancers to change partners. The rule says, "You can only change partners by swapping 1/2 of a step."
• The Expectation: For a long time, physicists thought Sigma particles only did this "1/2 step" dance. They thought the "3/2 step" dance didn't exist.
• The Discovery: The new, precise measurements show that the Sigma particles are doing the "3/2 step" dance!

The math shows that the "3/2 step" is happening with a significance of more than 5-sigma. This is the "gold standard" in physics for a discovery. It means the old rule isn't just slightly off; it's incomplete. There is a hidden "extra move" in the dance that we didn't know about.

Why Does This Matter?

You might ask, "Who cares if a tiny particle breaks apart 1% differently than we thought?"

1. Fixing the Blueprint: These Sigma particles are the "parents" of many other, heavier particles. If we get the Sigma's numbers wrong, our calculations for the "children" particles are also wrong. This paper fixes the foundation for future studies.
2. Understanding the Universe: The fact that the "3/2 step" exists helps us understand how the fundamental forces of nature (specifically the Weak Force) interact with the strong force that holds atoms together. It's a clue to solving the mystery of why the universe is made of matter and not just energy.
3. New Physics: It suggests that our current models of how particles behave are missing a piece of the puzzle. Just like finding a new continent changes a map, finding this "extra step" changes our map of the subatomic world.

The Bottom Line

The BESIII team acted like the ultimate auditors. They took a long-standing assumption about how a specific particle decays, measured it with laser precision, and proved that the old textbooks were wrong. They also found a "secret move" in the particle's dance that breaks an old rule, opening the door to a deeper understanding of how the universe works at its smallest scale.

Technical Summary

1. Problem Statement

The paper addresses two critical issues in hyperon physics:

• Inconsistency in Branching Fractions (BFs): Previous values for the absolute branching fractions of the dominant nonleptonic decays of the $\Sigma^+$ hyperon ($\Sigma^+ \to p\pi^0$ and $\Sigma^+ \to n\pi^+$) listed in the Particle Data Group (PDG) were derived from relative measurements or indirect inferences (e.g., inferring $\Sigma^+ \to p\pi^0$ from $\Sigma^+ \to p\gamma$). These values lacked direct absolute measurements and had missing systematic uncertainties. Recent BESIII measurements of $\Sigma^+ \to p\gamma$ revealed a $4.2\sigma$ discrepancy from the PDG value, casting doubt on the reliability of the existing $\Sigma^+ \to p\pi^0$ data.
• Testing the $\Delta I = 1/2$ Rule: While the $\Delta I = 1/2$ rule (the dominance of isospin-changing transitions by $1/2$ in nonleptonic weak decays) is well-established in kaon systems, recent observations in other hyperon decays suggest potential violations. A precise test requires high-precision absolute branching fractions and decay parameters to extract the S-wave and P-wave amplitudes. Previous data were insufficient to perform a stringent test for $\Sigma$ hyperons.

2. Methodology

The study utilizes data collected by the BESIII detector at the BEPCII collider.

• Dataset: $(10087 \pm 44) \times 10^6$ $J/\psi$ events collected at a center-of-mass energy of $\sqrt{s} = 3.097$ GeV.
• Production Mechanism: The analysis exploits the production of quantum-entangled $\Sigma^+ \bar{\Sigma}^-$ pairs from $J/\psi$ decays ($J/\psi \to \Sigma^+ \bar{\Sigma}^-$).
• Double-Tag (DT) Technique:
  • Single Tag (ST): The $\bar{\Sigma}^-$ is reconstructed via its decay $\bar{\Sigma}^- \to \bar{p}\pi^0$. The recoil mass against this tag is used to identify the event.
  • Double Tag (DT): The signal $\Sigma^+$ decays are reconstructed from the remaining particles in the event recoiling against the ST $\bar{\Sigma}^-$.
  • Signal Reconstruction:
    • For $\Sigma^+ \to p\pi^0$: Only the proton ($p$) is reconstructed on the signal side; the $\pi^0$ is inferred via recoil mass.
    • For $\Sigma^+ \to n\pi^+$: Only the pion ($\pi^+$) is reconstructed; the neutron ($n$) is inferred via recoil mass.
  • Advantage: This method cancels many systematic uncertainties associated with the ST selection and does not require knowledge of the total number of $J/\psi$ events or the $\Sigma^+$ production cross-section.
• Analysis Steps:
  1. Event Selection: Charged tracks are identified using the MDC and Time-of-Flight system. Photons are identified in the EMC.
  2. ST Yield: An unbinned maximum likelihood fit is performed on the recoil mass spectrum ($RM_{\bar{\Sigma}^-}$) to determine the number of ST events ($N_{ST}$).
  3. DT Yield: A similar fit is performed on the recoil mass against the reconstructed $\bar{\Sigma}^- p$ or $\bar{\Sigma}^- \pi^+$ system to determine the number of DT events ($N_{DT}$).
  4. Branching Fraction Calculation: The absolute BF is calculated using the formula:
     $$\mathcal{B}_{sig} = \frac{N_{DT}}{N_{ST} \times \epsilon_{sig}}$$
     where $\epsilon_{sig}$ is the signal efficiency derived from Monte Carlo (MC) simulations.
  5. Systematic Uncertainties: Sources include tracking/PID efficiencies, signal/background shape modeling, MC generator modeling, and mass window requirements.

3. Key Contributions

• First Absolute Measurement: This is the first direct, absolute measurement of the branching fractions for $\Sigma^+ \to p\pi^0$ and $\Sigma^+ \to n\pi^+$.
• High Precision: The statistical precision achieved is significantly higher than previous measurements, with total uncertainties (statistical + systematic) of approximately 0.5%.
• Amplitude Extraction: Using the new BFs combined with previously measured decay parameters ($\alpha$), the authors extracted the S-wave ($A$) and P-wave ($B$) decay amplitudes with unprecedented precision.
• Rigorous $\Delta I = 1/2$ Test: The paper provides the first quantitative test of the $\Delta I = 1/2$ rule in $\Sigma$ hyperon decays using these new amplitudes.

4. Results

Branching Fractions:
The measured absolute branching fractions are:

• $\mathcal{B}(\Sigma^+ \to p\pi^0) = (49.79 \pm 0.06_{stat} \pm 0.22_{syst})\%$
• $\mathcal{B}(\Sigma^+ \to n\pi^+) = (49.87 \pm 0.05_{stat} \pm 0.29_{syst})\%$

Comparison with PDG:

• The result for $\Sigma^+ \to p\pi^0$ deviates from the PDG value by 4.4$\sigma$.
• The result for $\Sigma^+ \to n\pi^+$ deviates from the PDG value by 3.4$\sigma$.
• The ratio $\mathcal{B}(\Sigma^+ \to p\pi^0) / \mathcal{B}(\Sigma^+ \to n\pi^+) = 0.9984 \pm 0.0017 \pm 0.0069$ deviates from the PDG ratio by 5.5$\sigma$.

$\Delta I = 1/2$ Rule Test:
The authors tested the relation derived from isospin symmetry:
$$M_{\Sigma^- \to n\pi^-} - M_{\Sigma^+ \to n\pi^+} + \sqrt{2}M_{\Sigma^+ \to p\pi^0} = 0$$
where $M$ represents the $A$ (S-wave) or $B$ (P-wave) amplitudes.

• S-wave ($A$) test: The deviation from zero is $-0.127 \pm 0.014$ (in units of $G_F m_{\pi^+}^2$), a >5$\sigma$ deviation.
• P-wave ($B$) test: The deviation from zero is $-2.78 \pm 0.16$, also a >5$\sigma$ deviation.

5. Significance

• Resolution of Discrepancies: The new measurements resolve the tension between the BESIII $\Sigma^+ \to p\gamma$ measurement and the PDG values, suggesting the PDG values for $\Sigma^+$ nonleptonic decays were inaccurate due to reliance on indirect relative measurements.
• Evidence for $\Delta I = 3/2$ Amplitudes: The >5$\sigma$ violation of the $\Delta I = 1/2$ rule in $\Sigma$ decays provides compelling evidence for the presence of non-negligible $\Delta I = 3/2$ transition amplitudes. This challenges the universality of the rule and offers new constraints for theoretical models of isospin symmetry breaking in weak interactions.
• Theoretical Impact: The precise S-wave and P-wave amplitudes serve as critical inputs for:
  • Understanding non-perturbative QCD dynamics.
  • Predicting CP violation in hyperon decays.
  • Calculating rates for rare decays like $\Sigma^+ \to p l^+ l^-$.
  • Benchmarking theoretical frameworks (e.g., Chiral Perturbation Theory, Skyrme models, QCD sum rules), none of which currently fully reproduce the observed amplitude pattern.
• Future Applications: These precise BFs are fundamental inputs for studying heavier baryons (e.g., $\Lambda_c$, $\Xi_c$, $\Lambda(1405)$) that decay into final states containing $\Sigma^+$.

In conclusion, this work establishes a new standard for hyperon decay measurements, corrects long-standing PDG values, and provides the first definitive experimental evidence of significant $\Delta I = 3/2$ contributions in $\Sigma$ hyperon nonleptonic decays.