CP violation in $\Sigma^+\to p\ell^+\ell^-$ within the standard model and beyond

This paper explores the potential for observing significant $CP$ violation in the rare hyperon decays $\Sigma^+\to p\ell^+\ell^-$ and $\Sigma^+\to p\gamma$ at LHCb, noting that large Standard Model long-distance effects could amplify $CP$-violating signals arising from new physics.

The Gist

The Cosmic Mirror Test: A Simple Guide to the $\Sigma^+$ Decay Paper

Imagine you are standing in front of a mirror. In a perfect world, everything you see in the reflection should be a perfect, symmetrical copy of you. If you raise your right hand, your reflection raises its left. If you smile, it smiles back. This "sameness" is what physicists call Symmetry.

For a long time, scientists thought the universe worked this way: that the laws of physics were perfectly symmetrical. But there is a tiny, mysterious glitch in the cosmic mirror. This glitch is called CP Violation. It is the reason why, after the Big Bang, the universe didn't just result in a boring soup of light, but actually created the matter (atoms, stars, and people) that we see today.

This paper is about using a specific, rare "glitch" in a tiny particle to hunt for even deeper secrets of the universe.

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1. The Protagonist: The $\Sigma^+$ Particle

The star of this paper is a particle called the Sigma-plus ($\Sigma^+$). Think of the $\Sigma^+$ as a tiny, unstable spinning top. Because it is unstable, it doesn't stay a $\Sigma^+$ for long; it "decays" (breaks apart) into other, lighter particles.

Specifically, the researchers are looking at a rare way it breaks apart: it turns into a proton and two leptons (tiny particles like muons or electrons).

2. The Experiment: The Mirror Test

To find "CP Violation," scientists perform a mirror test. They watch the $\Sigma^+$ particle decay, and then they watch its "anti-particle" twin (the $\Sigma^-$) decay.

• In a symmetrical world: The two decays should happen at almost exactly the same rate. It’s like two identical twins performing the exact same dance routine.
• In a "glitchy" (CP-violating) world: One twin dances slightly faster or differently than the other.

If scientists see a significant difference between the $\Sigma^+$ and its twin, they have found a "glitch."

3. The "Long-Distance" Interference (The Secret Sauce)

The paper mentions something called "long-distance contributions." This sounds technical, but think of it like this:

Imagine you are trying to hear a specific person whispering in a crowded room. The "short-distance" part is the whisper itself. The "long-distance" part is the echo bouncing off the walls.

In this decay, the "echoes" (the Standard Model's internal physics) are actually quite loud and messy. While that sounds like a problem, the authors argue it’s actually a benefit. These loud "echoes" create a perfect background that can amplify any tiny, new signal from "New Physics." It’s like using a megaphone to make a tiny whisper much easier to hear.

4. The Mystery: "New Physics"

The "Standard Model" is our current rulebook for how the universe works. It’s a great rulebook, but it’s incomplete—it can’t explain everything (like Dark Matter or why we exist).

The authors are looking for "New Physics"—rules that aren't in our current book. They test several "What If?" scenarios:

• Supersymmetry: What if every particle has a heavy, "super" twin?
• Leptoquarks: What if there are strange "bridge" particles that allow quarks and leptons to talk to each other more easily?

5. The Conclusion: A Window of Opportunity

The most exciting part of the paper is the "Window of Opportunity."

The authors calculated that if certain "New Physics" exists, the difference between the $\Sigma^+$ and its twin could be huge—up to tens of percent! In the world of particle physics, a "tens of percent" difference is like a giant neon sign flashing in a dark room.

The Bottom Line:
The researchers are telling the scientific community: "Hey, look at this specific decay! We've checked the math, and if there's something new and exciting happening in the universe, this is exactly where we are going to see it first."

They are pointing the way for massive experiments (like the LHCb at CERN) to go out and find the "glitch" that explains how our universe was built.

Technical Summary

This technical summary outlines the research presented in the paper "CP violation in $\Sigma^+ \to p\ell^+\ell^-$ within the Standard Model and beyond" by He, Tandean, and Valencia.

1. Problem Statement

The recent observation of the rare hyperon decay $\Sigma^+ \to p\mu^+\mu^-$ by the LHCb collaboration has opened a new window into flavor physics. While the measured branching fraction aligns with Standard Model (SM) predictions, the potential for discovering CP violation (CPV) remains high.

The central problem addressed is determining whether CP-violating asymmetries in the dimuon ($\mu^+\mu^-$) and dielectron ($e^+e^-$) channels, as well as the radiative channel ($\Sigma^+ \to p\gamma$), can serve as sensitive probes for New Physics (NP). Specifically, the authors investigate how the large "absorptive phases" (long-distance contributions) inherent in the SM can interfere with NP amplitudes to produce observable CPV signals.

2. Methodology

The authors employ a theoretical framework based on Low-Energy Effective Field Theory (LEFT) to parameterize both SM and NP contributions.

• Amplitude Modeling: They construct a general decay amplitude $M$ for $\Sigma^+ \to p\ell^+\ell^-$ using a set of complex form factors ($\tilde{a}$ through $\tilde{k}$) that account for short-distance (quark-level) and long-distance (photon-exchange) effects.
• SM Baseline: The SM contribution is calculated using effective Hamiltonians involving photon- and $Z$-penguin/box diagrams. They utilize the "relativistic solution 4" for long-distance form factors, which best matches current LHCb data.
• NP Parameterization: NP is introduced via Wilson coefficients ($C_r$) associated with various quark-level operators (dipole, vector, axial-vector, scalar, and pseudoscalar).
• Observables: The study focuses on:
  • $\hat{\Delta}_\ell$: The decay rate asymmetry between the hyperon and antihyperon.
  • $\hat{\Delta}_\gamma$: The branching fraction asymmetry in the radiative decay.
  • $\hat{A}_\gamma$: The asymmetry in the decay asymmetry parameter $\alpha_\gamma$.
• Model Testing: They apply a model-independent approach to find maximum allowed NP effects and then test specific scenarios: Supersymmetry (SUSY), $ds\nu\nu$ interactions (linked to neutrino channels), and Scalar Leptoquarks (LQs).

3. Key Contributions

• Interference Mechanism: The paper identifies that the SM long-distance contributions provide significant strong phases. This is crucial because CP violation requires both a weak phase (from NP) and a strong phase (from SM) to manifest in rate asymmetries.
• Comprehensive Mapping: The authors provide a detailed mapping of how specific Wilson coefficients ($C_7^\pm$ and $C_9^\pm$) impact different CP-violating observables across three different decay modes.
• Constraint Integration: They synthesize constraints from various sectors—including Kaon decays ($K \to \pi \ell \ell$, $K \to \pi \nu \bar{\nu}$) and existing hyperon data—to define the "allowed" parameter space for NP.

4. Results

The study yields several critical quantitative findings:

• Large Potential Asymmetries: In a model-independent approach, the CP-violating asymmetry $\hat{\Delta}_\mu$ can be as high as tens of percent if the coefficients $C_7^-$ or $C_9^-$ are large. This is orders of magnitude larger than the SM expectation ($\hat{\Delta}_\mu^{SM} \approx 1.1 \times 10^{-4}$).
• Sensitivity of Channels:
  • $\hat{\Delta}_\mu$ (Dimuon): Highly sensitive to $C_7^-$ and $C_9^-$.
  • $\hat{\Delta}_e$ (Dielectron): Less sensitive to NP than the muon channel due to different scaling, but still a viable probe.
  • $\Sigma^+ \to p\gamma$ (Radiative): $\hat{A}_\gamma$ and $\hat{\Delta}_\gamma$ are primarily sensitive to the dipole operators $C_7^\pm$.
• Model-Specific Insights:
  • SUSY: Constraints from Kaon physics significantly restrict the possible CPV in hyperons, making the predicted asymmetries smaller than the model-independent maximums.
  • Leptoquarks: While LQs can contribute to these decays, current constraints from $K \to \pi \nu \bar{\nu}$ and $K \to \pi \ell \ell$ mean that most LQ models cannot fully saturate the largest possible model-independent bounds for $C_9^+$, though $C_9^-$ remains a significant "window" for NP.

5. Significance

This paper establishes $\Sigma^+ \to p\ell^+\ell^-$ as a high-priority channel for searching for New Physics.

The significance lies in the "wide window" identified for $C_7^-$ and $C_9^-$. Because these coefficients are not yet tightly constrained by other flavor observables, the hyperon sector offers a unique opportunity for the LHCb and future experiments (like BESIII, PANDA, or the Super Tau Charm Facility) to discover CP violation that would be invisible in the SM. It provides a roadmap for experimentalists to target specific asymmetries that could signal the presence of new particles or interactions.