Probing invisible particles with charm

This paper highlights the potential of rare charm hadron decays as clean probes for invisible particles like neutrinos, axion-like particles, and dark photons, demonstrating that existing and future high-luminosity experiments can significantly constrain or discover new physics beyond the Standard Model through branching ratios that are currently weakly limited.

The Gist

Imagine the universe as a giant, bustling city where particles are the citizens. Most of the time, these citizens interact in predictable ways, following the strict traffic laws of the "Standard Model" (our current best map of physics). But sometimes, a citizen might vanish into thin air, leaving behind only a gap in the traffic flow. This is called "missing energy."

This paper is like a team of detectives (physicists Gudrun Hiller and Dominik Suelmann) who are looking for these vanishing acts, specifically in the neighborhood of Charm particles (a type of subatomic particle). They are asking: "Could these missing particles be ghosts from a hidden dimension, or perhaps new types of neutrinos we haven't seen yet?"

Here is a breakdown of their investigation using simple analogies:

1. The Crime Scene: Charm Decays

In the world of particle physics, heavy particles (like Charm hadrons) are unstable. They naturally decay, or break apart, into lighter particles. Usually, we can see all the pieces.

• The Mystery: Sometimes, a Charm particle decays into a visible piece (like a pion or a proton) and something else that the detectors cannot see. It's like a magician pulling a rabbit out of a hat, but the rabbit disappears the moment it leaves the hat.
• The Goal: The authors want to know if this "invisible rabbit" is a standard ghost (a known neutrino) or a new, exotic creature (like a Dark Photon or an Axion).

2. The Suspects (The Invisible Particles)

The paper investigates four main types of "invisible" suspects that could be hiding in these decays:

• Neutrinos (Left and Right-handed): The standard "ghosts" that barely interact with anything. The paper also looks for "sterile" neutrinos, which are like ghosts that don't even talk to the standard ones.
• ALPs (Axion-Like Particles): Imagine these as tiny, wobbly ripples in a fabric of space. They are very light and could be a candidate for Dark Matter (the invisible stuff holding galaxies together).
• Dark Photons ($Z'$): Think of these as "shadow photons." Regular light (photons) interacts with us; these shadow photons only interact with the dark sector. They are like a secret radio frequency that only certain hidden devices can hear.

3. The Investigation Method: "The Clean Test"

The authors explain that in the Standard Model, Charm particles should not decay into invisibles very often. It's like a locked door that is supposed to stay shut.

• The Null Test: If they find any of these decays, it's a "smoking gun." It means the door was forced open by new physics. Because the expected background is so low, even a tiny signal would be a huge discovery.
• The Challenge: So far, no one has seen this happen. Experiments like BESIII and Belle II have set "speed limits" (upper limits) on how often this can happen, but those limits are still quite loose. It's like saying, "We haven't seen a car drive through the wall, but we only checked for 5 minutes."

4. The Tools: EFT and Recasting

To make sense of the data, the authors use a toolkit called Effective Field Theory (EFT).

• The Analogy: Imagine you are trying to figure out what a machine does by looking at the input and output, without seeing the gears inside. EFT is a mathematical way to describe all the possible "gears" (new physics) that could be turning, even if we don't know the exact blueprint of the machine.
• Recasting: The authors took old data from experiments and re-analyzed it with their new "glasses." They asked, "If the invisible particle was an ALP instead of a neutrino, would the old data still look the same?" They found that by re-interpreting the data, they could set much tighter rules on what these new particles could be.

5. The Findings: What's Possible?

The paper calculates how often these decays could happen if new physics exists:

• The "Big" Possibilities: If the new physics involves "chirality-flipping" (a specific way particles twist), the decay rate could be as high as 1 in 1,000 ($10^{-3}$) or 1 in 10,000 ($10^{-4}$). This is huge in particle physics!
• The "Strict" Possibilities: If the new physics is "heavy" and follows stricter rules (like the Standard Model's heavy partners), the rate is much lower, around 1 in 100,000 ($10^{-5}$).
• The "Weak" Constraints: For some specific types of invisible particles (like sterile neutrinos), the current rules are very weak. The decay could happen quite often, and we just haven't looked hard enough in the right places yet.

6. The Future: Where to Look Next

The authors point out that different types of invisible particles leave different "fingerprints" in the data.

• The Shape of the Signal: Just as different musical instruments sound different, different invisible particles create different patterns in the energy distribution of the decay.
• The Next Steps: They urge current and future experiments (like the Super Tau-Charm Factory or FCC-ee) to look at specific decay channels, such as a Charm baryon turning into a proton and an invisible particle ($\Lambda_c \to p + \text{invisible}$). This specific channel is a "golden mode" because it can tell us exactly which type of invisible particle is involved.

Summary

This paper is a roadmap for hunting invisible particles in the "Charm" sector of physics. It argues that:

1. Charm decays are a clean playground because the Standard Model predicts almost nothing should happen there.
2. New physics could be hiding in plain sight, potentially making these decays happen thousands of times more often than we thought.
3. By re-analyzing old data and looking at specific decay patterns, we can distinguish between different types of invisible particles (neutrinos, axions, dark photons).
4. Future experiments have the potential to either find these new particles or rule out large chunks of theoretical possibilities, effectively closing the case on these "missing energy" mysteries.

Technical Summary

Technical Summary: Probing Invisible Particles with Charm

Problem and Motivation
The paper addresses the search for invisible particles in rare flavor-changing neutral current (FCNC) decays of charm hadrons. While missing energy signatures are powerful probes for physics beyond the Standard Model (SM)—including sterile neutrinos, lepton number violation (LNV), axion-like particles (ALPs), and dark photons ($Z'$)—charm physics offers a unique window distinct from $b$-hadron and kaon programs. In the SM, $c \to u \nu \bar{\nu}$ transitions are heavily suppressed by the Glashow-Iliopoulos-Maiani (GIM) mechanism and CKM factors, rendering them unobservable in the foreseeable future. Consequently, any observation of charm decays into invisible final states constitutes a clean null test of the SM. Current experimental upper limits from BESIII and Belle are at the level of $10^{-5} - 10^{-4}$, leaving significant room for new physics (NP) contributions, particularly if the NP involves light particles or $SU(2)_L$ singlets that evade strong down-type quark constraints.

Methodology
The authors employ a systematic approach combining Effective Field Theory (EFT) and specific light NP models to analyze two-, three-, and four-body decays of charmed mesons ($D^0, D^+, D_s^+$) and baryons ($\Lambda_c, \Xi_c$).

1. Theoretical Frameworks:

   • SMEFT and $\nu$SMEFT: The study considers heavy new physics via the Standard Model Effective Field Theory (SMEFT) at dimension six (chirality-preserving) and dimension seven (chirality-flipping, including LNV). It also incorporates the $\nu$SMEFT, which includes light right-handed (sterile) neutrinos.
   • Weak Effective Theory (WET): Operators are matched from the high-scale SMEFT/$\nu$SMEFT down to the low-energy WET. This includes four-fermion operators for left- and right-handed neutrinos, as well as scalar, pseudoscalar, and tensor operators.
   • Light Mediators: The analysis extends to Axion-Like Particles (ALPs) and light $Z'$ bosons decaying into dark sector fermions. For ALPs and $Z'$, the authors distinguish between on-shell (long-lived) and off-shell (promptly decaying) scenarios, calculating the fraction of particles escaping the detector based on kinematic distributions and detector geometry (specifically BESIII).

2. Recasting Experimental Data:
   A critical methodological component is the "recast" of existing experimental searches. Since experimental collaborations often provide limits based on specific signal shapes (e.g., specific $q^2$ distributions for neutrinos), the authors re-evaluate these bounds for different NP models (ALPs, $Z'$, different chirality structures). They utilize likelihood functions incorporating Poisson statistics for signal events and Gaussian distributions for nuisance parameters (backgrounds, efficiencies) to derive model-independent or model-specific upper limits on branching ratios.

3. Hadronic Inputs:
   The calculation of branching ratios relies on form factors for transitions such as $D \to \pi$, $D \to V$ (vector mesons), $\Lambda_c \to p$, and $D \to \pi\pi$. The authors utilize lattice QCD results (Fermilab/MILC and ETM collaborations) and Light-Cone Sum Rules (LCSR), explicitly addressing uncertainties arising from discrepancies between different lattice calculations in the high-$q^2$ region.

Key Contributions and Results

• Branching Ratio Predictions: The paper establishes that NP-induced branching ratios can reach $10^{-3}$ for $Z'$ models and $10^{-4}$ for ALPs, and up to few $\times 10^{-5}$ for chirality-preserving SMEFT operators. Chirality-flipping operators (dimension seven SMEFT or $\nu$SMEFT) allow for branching ratios up to few $\times 10^{-4}$.
• Experimental Constraints:
  • Wilson Coefficients: The authors derive upper limits on combinations of Wilson coefficients ($x_{SP}, x_{LR}, x_T$) for light neutrino models. They find that limits from $D^0 \to \text{invisible}$ are significantly stronger (by nearly two orders of magnitude) than those from $D^0 \to \pi^0 \nu \bar{\nu}$ for scalar/pseudoscalar couplings.
  • ALP and $Z'$ Parameters: New constraints are placed on ALP couplings ($k_{12}^{V,A}/f$) and $Z'$ couplings ($x_{Z'}$) across different mass windows. For instance, the vector coupling of ALPs is constrained down to $|k_{12}^V|/f \sim 0.22 \times 10^{-7} \text{ GeV}^{-1}$ for $m_a=0$ via $D^+ \to \pi^+ + \text{invisible}$.
  • Recast Limits: The recasting of the $D^0 \to \pi^0 \nu \bar{\nu}$ search (BESIII) provides improved constraints on ALP models, particularly for masses $m_a > 1 \text{ GeV}$, where previous works had not accounted for the different signal shape.
• Model Discrimination: The study demonstrates that different NP models can be distinguished not only by the total branching ratio but by the shape of the differential distributions ($d\mathcal{B}/dq^2$).
  • Chirality: Scalar and pseudoscalar operators (allowed with RH neutrinos) exhibit unique $q^2$ behaviors, vanishing at the low-$q^2$ endpoint, whereas vector/axial-vector and tensor contributions remain finite.
  • Resonance vs. Contact: Long-lived $Z'$ or ALPs produce two-body decay kinematics (delta functions in $q^2$), while off-shell decays or contact interactions produce three-body distributions.
  • Mode Correlations: Ratios of branching fractions, such as $\mathcal{B}(D \to \pi X) / \mathcal{B}(\Lambda_c \to p X)$, provide distinct signatures for ALPs, $Z'$, and neutrino models, aiding in the identification of the underlying NP structure.

Significance and Claims
The authors claim that rare charm decays into invisible final states are "clean null tests" of the SM that are currently "essentially unconstrained" for specific light NP parameters, limited only by weak lifetime constraints ($O(10^{-1})$). The paper highlights that:

1. Sensitivity: Current and future experiments (BESIII, Belle II, STCF, FCC-ee, CEPC) have the potential to probe these decays with high sensitivity, potentially reaching scales of $\Lambda_{LNV} \gtrsim 1.5 \text{ TeV}$ and $\Lambda_{\nu\text{SMEFT}} \gtrsim 2.1 \text{ TeV}$.
2. Complementarity: Charm physics complements $b$-physics and kaon programs, particularly in probing $SU(2)_L$ singlets and light mediators that might evade constraints from down-type quark transitions.
3. Necessity of Global Analysis: The paper emphasizes that distinguishing models requires a global analysis of multiple decay modes ($D \to \pi, D \to V, \Lambda_c \to p, D \to \pi\pi$) and differential distributions, as single-mode searches may not be sufficient to disentangle couplings (e.g., vector vs. axial-vector ALP couplings).
4. Future Needs: The authors modestly note that sharper interpretations depend on improved hadronic form factors, particularly for $D \to V$ and $D \to \pi\pi$ transitions, and encourage experimental collaborations to broaden signal regions and present data in a reinterpretable format.

In conclusion, the paper provides a comprehensive roadmap for using charm hadron decays to probe invisible sectors, offering updated theoretical predictions, recast experimental limits, and strategies for model discrimination that are currently accessible to existing and upcoming high-luminosity charm factories.