Scalarful double beta decay

This paper re-evaluates scalar emissions in double beta decay using modern effective field theory methods, assessing experimental sensitivity through spectral shape analysis while extending the study to massive scalars, sterile neutrino couplings, and exotic right-handed interactions.

The Gist

Imagine the atomic nucleus as a tiny, bustling factory. Inside, particles are constantly rearranging themselves. Usually, when a specific type of factory (a nucleus) tries to fix a problem by turning two neutrons into two protons, it has to release two tiny messengers called neutrinos to balance the books. This is a known process called "two-neutrino double beta decay" ($2\nu\beta\beta$). It's like a factory paying its debts by sending out two very light, ghostly envelopes.

However, physicists have a wilder theory: What if the factory could pay its debts without sending those two neutrino envelopes? This is called neutrinoless double beta decay ($0\nu\beta\beta$). If we find this, it proves that neutrinos are their own antiparticles (like a coin that is heads on both sides) and that the universe breaks a fundamental rule called "lepton number conservation." This would be a Nobel Prize-level discovery.

But here is the twist in this paper: What if the factory doesn't just skip the neutrinos, but instead throws out a secret package?

The "Ghost Package" (The Scalar)

The authors of this paper are investigating a scenario where, instead of just two electrons flying out, the nucleus also emits a mysterious, invisible particle called a scalar (often called a "Majoron" in physics jargon). Think of this scalar as a ghostly balloon that floats away with some of the energy.

Because this balloon takes some energy with it, the two electrons that do come out don't have their full energy. Instead of having a sharp, distinct energy (like a laser pointer), they have a smeared-out, continuous range of energies (like a flashlight beam).

The Detective Work: Finding the Needle in the Haystack

The problem is that the "normal" process (sending two neutrinos) also produces a smeared-out energy spectrum. It's like trying to find a specific type of cloud in a sky full of clouds.

The authors act as cosmic detectives. They ask: "Can we tell the difference between the 'ghost balloon' signal and the normal 'two-neutrino' background just by looking at the shape of the energy clouds?"

They used advanced mathematics (Effective Field Theory) to build a super-precise map of what these energy clouds should look like if a ghost balloon were present. They looked at three main clues:

1. Total Energy: How much energy the two electrons have combined.
2. Individual Energy: How the energy is split between the two electrons.
3. Angle: The angle at which the two electrons fly apart. Do they fly back-to-back (like a recoil from a gun) or in the same direction?

The Findings: What the Paper Says

The paper is essentially a "Sensitivity Report" for future experiments. Here are the key takeaways in plain English:

• The Best Clue is Total Energy: If you want to catch this ghost balloon, the most powerful tool is measuring the total energy of the two electrons. The "shape" of the energy curve changes significantly if a scalar is involved.
• The "Xenon" Champion: Among the different types of atoms (isotopes) scientists use to test this, Xenon-136 is currently the best detective. It gives the tightest constraints on how strong the "ghost balloon" interaction can be.
• The "Funnel" Mystery: When the scalar interacts with a special type of "sterile" neutrino (a heavy, invisible cousin of the normal neutrino), the math gets weird. Depending on the mass of the sterile neutrino, the signal can suddenly vanish. It's like a radio station that works perfectly, then hits a dead zone where the signal drops to zero, and then comes back. This creates a "funnel" in the data where we can't set any limits at all.
• Heavy Balloons: What if the ghost balloon is too heavy to be created directly? The paper also looks at "off-shell" scenarios where the balloon is virtual and decays into neutrinos later. This distorts the background in a different way, but it's much harder to detect.
• Right-Handed Currents: The paper also checks a weird scenario where the electrons are "right-handed" (a specific spin orientation). In this case, the electrons prefer to fly in the same direction, whereas normal electrons fly apart. This is a huge clue! If we see electrons flying together, we know something exotic is happening.

Why Does This Matter?

We are building massive detectors deep underground (like the LEGEND or nEXO experiments) to catch these rare events. They will collect millions of data points.

This paper tells the experimentalists: "Don't just look for a spike in the data. Look at the shape of the curve. If the curve looks a little bit like this specific mathematical function, you might have found a new particle!"

It also warns them: "Be careful with your theoretical calculations." The biggest uncertainty isn't the detector; it's our current understanding of how the nucleus behaves (the Nuclear Matrix Elements). If our map of the nucleus is slightly off, we might miss the signal or think we found it when we didn't.

The Bottom Line

This paper is a user manual for the next generation of particle physics. It refines the theoretical tools needed to distinguish between the "boring" background noise of the universe and the "exciting" signal of new physics. It tells us that if we look closely at the energy and angles of electrons in double beta decay, we might just find a ghostly scalar particle that could explain dark matter or the origin of neutrino mass.

Technical Summary

1. Problem Statement

Neutrinoless double beta decay ($0\nu\beta\beta$) is a primary probe for physics beyond the Standard Model (BSM), specifically testing lepton number violation and the Majorana nature of neutrinos. The canonical $0\nu\beta\beta$ mode involves two neutrons decaying into two protons and two electrons without neutrino emission. However, many BSM models (e.g., Majoron models, axion-like particles, or sterile neutrino scenarios) predict alternative decay modes where a light scalar boson ($\phi$) is emitted alongside the electrons ($0\nu\beta\beta\phi$).

The challenge lies in distinguishing these scalar-emitting signals from the irreducible Standard Model background: two-neutrino double beta decay ($2\nu\beta\beta$). While $0\nu\beta\beta\phi$ produces a continuous energy spectrum distinct from the mono-energetic peak of canonical $0\nu\beta\beta$, it overlaps significantly with the $2\nu\beta\beta$ spectrum. Previous analyses often relied on simplified approximations or rescaling massless limits to massive cases, potentially missing spectral shape features. Furthermore, theoretical uncertainties in Nuclear Matrix Elements (NMEs) and phase space factors complicate the extraction of robust limits on scalar-neutrino couplings.

2. Methodology

The authors employ an Effective Field Theory (EFT) approach to revisit the theoretical underpinnings of scalar-emitting $0\nu\beta\beta$ decays. Their methodology involves:

• Theoretical Framework:

  • Deriving decay amplitudes for $X \to Y + e^- + e^- + \phi$ considering general scalar-neutrino couplings ($g_{ij}, \bar{g}_{ij}$) for active, sterile, and mixed neutrino sectors.
  • Calculating the neutrino potential $V(q^2)$, distinguishing between long-range (potential) and short-range (hard) neutrino exchanges.
  • Extending the state-of-the-art framework for canonical $0\nu\beta\beta$ to include scalar emission, incorporating sterile neutrinos and massive scalars.
  • Deriving expressions for the phase space factor ($\Omega_\phi$) and angular correlations between the emitted electrons.

• Statistical Analysis:

  • Utilizing a frequentist approach with a likelihood ratio test statistic ($q_{\theta_\phi}$) to set exclusion limits.
  • Comparing the BSM signal ($0\nu\beta\beta\phi$) against the SM background ($2\nu\beta\beta$) using Asimov datasets (simulated data representing the background-only hypothesis).
  • Analyzing multiple observables: summed electron energy ($\epsilon$), individual electron energies ($E_{e1}, E_{e2}$), and the angular correlation ($\cos\theta$).
  • Systematically varying theoretical parameters (e.g., $\xi_{31}$, $\Delta_{0,2}$) and NMEs to quantify uncertainties.

• Scenarios Investigated:

  1. Scalar coupling to active neutrinos.
  2. Scalar coupling to sterile neutrinos (massive BSM neutrinos).
  3. Mixed couplings (active-sterile).
  4. Off-shell heavy scalars ($m_\phi > Q$-value) decaying into neutrinos.
  5. Exotic right-handed lepton currents.

3. Key Contributions

• Improved Spectral Description: The paper provides a more rigorous derivation of the $0\nu\beta\beta\phi$ spectrum compared to seminal earlier works. It explicitly accounts for the dependence of the amplitude on neutrino masses and scalar masses, rather than relying on simple phase-space rescaling.
• Sterile Neutrino "Funnel": The authors identify a critical phenomenon in scalar couplings to sterile neutrinos. Due to a sign flip in the amplitude as the sterile neutrino mass crosses a specific threshold (related to the interpolation of NMEs around 2 GeV), the decay rate can vanish. This creates a "funnel" in the exclusion limits where constraints on the coupling disappear entirely for specific mass ranges.
• Multi-Observable Analysis: The study evaluates the utility of combining energy and angular observables. While the summed energy is generally the most powerful discriminator, the paper demonstrates that angular correlations become crucial for specific BSM scenarios, such as those involving right-handed currents, where the angular distribution differs significantly from the $2\nu\beta\beta$ background.
• Off-Shell Heavy Scalars: The analysis extends to scalars heavier than the decay Q-value, treating them as off-shell particles that decay into neutrino pairs, thereby distorting the $2\nu\beta\beta$ spectrum.

4. Results

• Sensitivity to Active Neutrinos:

  • The summed electron energy is the most sensitive observable for constraining scalar-active neutrino couplings ($g_{ee}$).
  • Among isotopes studied ($^{76}\text{Ge}$, $^{82}\text{Se}$, $^{100}\text{Mo}$, $^{136}\text{Xe}$), $^{136}\text{Xe}$ provides the most stringent limits up to its Q-value.
  • Theoretical uncertainties in NMEs (specifically the interpolation parameters for heavy neutrinos) dominate over uncertainties in the $2\nu\beta\beta$ background parameters (like $\xi_{31}$).
  • Limits on massive scalars are weaker than massless ones due to phase space suppression, but the spectral shape analysis provides tighter constraints than simple rescaling methods used in previous literature.

• Sterile Neutrino Couplings:

  • For pseudoscalar couplings to sterile neutrinos, limits are robust.
  • For scalar couplings, a funnel region appears where the limit drops to zero (no constraint possible) due to the amplitude crossing zero. The exact location of this funnel depends on NME fit parameters but is a robust feature of the theory.

• Right-Handed Currents:

  • In scenarios involving right-handed lepton currents, the angular correlation of electrons exhibits a distinct trend (preferring collinear emission) compared to the back-to-back preference of $2\nu\beta\beta$.
  • A multi-observable analysis (energy + angle) improves limits on these couplings, particularly for scalar masses around 1 MeV, though the improvement is marginal compared to NME uncertainties.

• Off-Shell Scalars:

  • For scalars with $m_\phi > Q$, the distortion of the $2\nu\beta\beta$ spectrum is suppressed by the heavy scalar propagator ($1/m_\phi^4$), leading to significantly weaker limits compared to on-shell production.

5. Significance

This work significantly refines the theoretical toolkit for searching for scalar particles in double beta decay experiments. By moving beyond simplified approximations and incorporating state-of-the-art EFT descriptions and NME uncertainties, the authors provide:

1. Realistic Sensitivity Projections: More accurate forecasts for upcoming experiments (e.g., LEGEND, nEXO, CUPID) regarding their ability to probe Majoron-like models and sterile neutrinos.
2. Methodological Guidance: Highlighting that while summed energy is generally optimal, angular correlations are essential for specific exotic scenarios (like right-handed currents) and that multi-observable analyses are necessary to avoid missing signals in "funnel" regions or specific kinematic configurations.
3. Theoretical Robustness: The identification of the sterile neutrino "funnel" warns experimentalists that null results in specific mass ranges do not necessarily rule out scalar couplings, necessitating a careful interpretation of exclusion plots.

The paper concludes that while NME uncertainties currently limit the precision of coupling constraints, the qualitative picture remains stable, and the proposed spectral shape analyses offer a powerful avenue to distinguish BSM physics from the Standard Model background.