Amplifying muon-to-positron conversion in nuclei with ultralight dark matter

This paper demonstrates that an ultralight scalar dark matter field coupling to neutrinos can significantly amplify the rate of lepton-flavour-violating muon-to-positron conversion in nuclei, thereby enabling upcoming experiments to establish stringent new constraints on neutrino-dark matter couplings that surpass current cosmological and terrestrial limits.

The Gist

The Big Picture: A Missing Puzzle Piece

Imagine the Standard Model of physics as a giant, mostly complete puzzle. It explains how the universe works very well, but there are a few glaring holes where pieces are missing. We know there is Dark Matter (invisible stuff holding galaxies together), but we don't know what it is. We also know neutrinos (ghostly particles) have mass, but we don't know why.

This paper proposes a clever way to find a specific type of dark matter by looking for a very rare, almost impossible event: a muon turning into a positron inside an atom.

The Problem: The "Ghost" That Won't Show Up

In the world of subatomic particles, a muon is like a heavy, unstable cousin of an electron. Usually, when a muon gets stuck inside an atom, it just turns into a regular electron and a neutrino.

However, physics allows for a "forbidden" magic trick: a muon could theoretically turn into a positron (the antimatter twin of an electron).

• The Catch: In our current understanding of the universe, this trick is so incredibly unlikely that it would take longer than the age of the universe to happen even once. It's like trying to win the lottery every day for a billion years and never winning.
• The Result: Because the rate is so low, our most sensitive detectors (like SINDRUM II, COMET, and Mu2e) can't see it yet. It's too quiet to hear.

The Solution: The "Cosmic Amplifier"

The authors suggest that the universe is filled with a special kind of dark matter called Ultralight Scalar Dark Matter (ULSDM).

• The Analogy: Imagine this dark matter isn't made of individual particles like tiny marbles, but is more like a gentle, invisible ocean wave that ripples through the entire universe. It's so light and spread out that it acts like a smooth, classical field rather than discrete particles.
• The Interaction: This "ocean" of dark matter interacts with neutrinos. The paper proposes that if this dark matter wave passes through an atom where a muon is trying to turn into a positron, it acts like a volume knob or a megaphone.

How the Magic Trick Gets Easier

Normally, the muon-to-positron conversion is suppressed because the "bridge" between the two particles is too weak.

1. Without Dark Matter: The muon tries to jump the gap, but the bridge is too flimsy. Nothing happens.
2. With Dark Matter: The ultralight dark matter field (the "ocean wave") couples with the neutrinos involved in the process. It effectively stiffens the bridge.
3. The Result: The "volume" of the event is turned up. The dark matter field adds a little extra energy and push to the process, making the impossible event happen frequently enough that our detectors might finally hear it.

The "Smoking Gun" vs. The "False Alarm"

Usually, if scientists saw a muon turn into a positron, they would say, "Aha! This proves that the universe violates a fundamental rule called Lepton Number Conservation (LNV)." It would be a "smoking gun" for new physics.

However, this paper points out a twist:

• Because this specific type of dark matter carries the "Lepton Number" itself, it can facilitate this conversion even if the fundamental rules of the universe aren't broken.
• The Analogy: Imagine a strict bouncer at a club (the law of physics) who won't let you in without a ticket (Lepton Number). Usually, you can't get in. But if the bouncer is actually a friend in disguise (the dark matter) who hands you a ticket, you get in. The club is full, but you didn't break the rules; your friend just helped you.
• Why this matters: If we see this event, it doesn't automatically prove the universe's laws are broken; it might just mean this specific dark matter exists. Conversely, if we don't see it, we can rule out certain ways this dark matter could exist.

What the Paper Actually Does

The authors did the math to see how much this "dark matter ocean" could boost the signal.

• They calculated that for very light dark matter (masses between $10^{-22}$ and $10^{-10}$ electron-volts), the boost could be huge.
• They looked at the limits set by current experiments (SINDRUM II) and future ones (COMET and Mu2e).
• The Finding: They drew a map (Figure 3 in the paper) showing which combinations of "dark matter mass" and "interaction strength" are now ruled out because we haven't seen the signal yet.
• The Conclusion: Future experiments like COMET and Mu2e are sensitive enough to detect this dark matter if it exists in a specific range. In fact, these particle experiments might be better at finding this specific type of dark matter than looking at the stars or the early universe (cosmology).

Summary

This paper suggests that a "sea" of ultralight dark matter could act as a cosmic amplifier, making a nearly impossible particle transformation (muon to positron) happen often enough to be detected. If we don't see it in upcoming experiments, we can draw a line in the sand and say, "This specific type of dark matter doesn't exist." It turns a particle physics experiment into a powerful telescope for hunting dark matter.

Technical Summary

1. Problem Statement

The paper addresses the extreme suppression of the lepton-number and lepton-flavour-violating (LNV/LFV) process of muon-to-positron conversion in nuclei ($\mu^- + N \to e^+ + N'$).

• Standard Model (SM) Context: In the SM extended with Majorana neutrino masses, this process is strictly forbidden for massless neutrinos. Even with massive Majorana neutrinos, the branching ratio is suppressed to $\sim 10^{-40}$ due to helicity suppression and the tiny scale of neutrino masses.
• Experimental Gap: Current and next-generation experiments (SINDRUM II, COMET, Mu2e) aim for sensitivities down to $10^{-17}$. However, the SM prediction is roughly 20–25 orders of magnitude below these detection thresholds.
• The Challenge: There is a need for Beyond-the-Standard-Model (BSM) mechanisms that can dynamically enhance this rate to observable levels without violating existing constraints from neutrinoless double-beta decay ($0\nu\beta\beta$) or cosmology.

2. Methodology

The authors propose a mechanism where an Ultralight Scalar Dark Matter (ULSDM) field, denoted as $\phi$, couples to neutrinos and amplifies the conversion rate.

• Theoretical Framework:
  • They introduce an effective dimension-six operator involving the ULSDM field $\phi$ (carrying two units of Lepton Number, $L$) and lepton doublets: $\mathcal{L} \supset \frac{y_{\mu e}}{\Lambda^2} (L_\mu H)(L_e H)\phi^* + \text{h.c.}$
  • After electroweak symmetry breaking, this yields an interaction term: $g_{\mu e} \nu_\mu \nu_e \phi^* + \text{h.c.}$, where $g_{\mu e}$ is the flavor-off-diagonal coupling.
• Coherent Field Approximation:
  • For ULSDM masses in the range $10^{-22} - O(1)$ eV, the field is treated as a coherent classical background: $\phi(t) = \phi_0 \cos(m_\phi t)$.
  • The local dark matter density $\rho_\phi \approx 0.3 \text{ GeV/cm}^3$ determines the amplitude $\phi_0 = \sqrt{2\rho_\phi/m_\phi}$.
• Mechanism of Amplification:
  • The ULSDM field induces a time-dependent contribution to the effective off-diagonal Majorana mass ($m_{\mu e}$) governing the conversion:
    $$m_{\mu e}(t) = (m_{\mu e})_{\text{vac}} + g_{\mu e}\phi_0 \cos(m_\phi t)$$
  • Since experimental timescales are much longer than the oscillation period (for most $m_\phi$), the rate is determined by the time-averaged squared mass:
    $$\langle |m_{\mu e}|^2 \rangle \approx \langle |(m_{\mu e})_{\text{vac}}|^2 \rangle + \frac{1}{2}(g_{\mu e}\phi_0)^2$$
  • Crucially, the ULSDM term allows the process to occur even if the vacuum Majorana mass is zero (i.e., even if total Lepton Number is conserved in the vacuum sector), provided the ULSDM carries the necessary lepton number.

3. Key Contributions

• Novel Probe of Flavor Structure: This is the first work to utilize $\mu^- \to e^+$ conversion to probe flavor-off-diagonal couplings ($g_{\mu e}$) of neutrinos to ULSDM. Previous works focused on flavor-diagonal couplings via $0\nu\beta\beta$.
• Cosmic Amplifier: The authors demonstrate that the coherent ULSDM background acts as a "cosmic amplifier," boosting the effective Majorana mass squared by a term proportional to $g_{\mu e}^2 \rho_\phi / m_\phi$.
• Decoupling from LNV: They show that the observation of $\mu^- \to e^+$ in this scenario does not necessarily imply vacuum Lepton Number Violation (LNV), as the ULSDM field itself carries the necessary quantum numbers.
• Complementarity: The study establishes $\mu^- \to e^+$ as a complementary probe to $0\nu\beta\beta$ (which constrains diagonal entries) and charged lepton flavor violation (CLFV) processes like $\mu \to e\gamma$ (which are highly suppressed in this specific model).

4. Results

• Branching Ratio Enhancement: The presence of ULSDM can raise the branching ratio $R_{\mu \to e^+}$ from $\sim 10^{-40}$ to the range of $10^{-17} - 10^{-16}$, making it accessible to upcoming experiments like COMET Phase-II and Mu2e.
• Exclusion Contours (Parameter Space):
  • The authors derived exclusion contours in the $g_{\mu e}$ vs. $m_\phi$ plane (Figure 3).
  • Current Bounds: SINDRUM II currently excludes $g_{\mu e} \gtrsim 10^{-5}$ for certain masses.
  • Future Sensitivity: COMET Phase-II and Mu2e can probe couplings as low as $g_{\mu e} \sim 10^{-10}$ for scalar masses $m_\phi \lesssim 10^{-11}$ eV.
• Comparison with Other Constraints:
  • Cosmology/Astrophysics: Constraints from Big Bang Nucleosynthesis (BBN), Cosmic Microwave Background (CMB) free-streaming, and Supernova 1987A cooling generally limit $g_{\mu e} \lesssim 10^{-5} - 10^{-7}$.
  • Superiority of Lab Experiments: The paper argues that for $m_\phi \gtrsim 10^{-19}$ eV, future $\mu^- \to e^+$ experiments will provide stronger, direct constraints than indirect cosmological bounds.
• Time-Dependent Signatures: For very light masses ($m_\phi \lesssim 10^{-22}$ eV), the oscillation period becomes comparable to experimental integration times, potentially leading to observable temporal modulations in the signal, offering a "smoking gun" signature for the ULSDM origin.

5. Significance

• Bridging Physics Frontiers: This work bridges the fields of neutrino physics, lepton flavor violation, and dark sector phenomenology. It suggests that rare nuclear processes can serve as sensitive detectors for ultralight dark matter.
• Model-Independent Constraints: The derived limits on $g_{\mu e}$ are largely model-independent regarding the UV completion, relying only on the effective field theory description of the interaction.
• Experimental Guidance: The results provide a strong motivation for $\mu^- \to e^+$ experiments (COMET, Mu2e) to consider ULSDM as a viable BSM target. It suggests that if these experiments reach their design sensitivities, they could either discover ULSDM or place the most stringent limits on flavor-off-diagonal neutrino-DM couplings to date.
• Reinterpretation of Null Results: If $\mu^- \to e^+$ is not observed, the non-observation translates directly into tight exclusion regions for the ULSDM parameter space, specifically ruling out the "neutrinophilic" scenario for specific mass-coupling combinations.

In summary, the paper proposes that ultralight scalar dark matter can dynamically enhance the rate of muon-to-positron conversion, transforming a theoretically unobservable process into a powerful tool for probing new physics and dark matter properties.