Revisiting $\mu$-$e$ conversion in $R$-parity violating SUSY

This paper revisits $\mu$-$e$ conversion in R-parity violating supersymmetry by incorporating renormalization group running effects to derive updated constraints on trilinear couplings, demonstrating that while these effects are often modest, they can significantly tighten limits in specific cases and highlighting the superior sensitivity of upcoming COMET and Mu2e experiments compared to current decay searches.

The Gist

Imagine the universe is a giant, bustling party where particles are the guests. In this party, there's a strict rule: Lepton Flavor. Think of this like a dress code. Electrons are wearing blue suits, muons are wearing red suits, and taus are wearing green suits.

In the Standard Model (our current best understanding of physics), these guests are polite. A muon (red suit) can turn into a neutrino, but it will never, ever change its suit to become an electron (blue suit) on its own. That would be "Lepton Flavor Violation" (LFV), and it's basically forbidden in our current rulebook.

However, physicists suspect there's a hidden "New Physics" guest list—maybe Supersymmetry (SUSY)—where the rules are a bit looser. If these new rules exist, a muon might sneakily swap its red suit for a blue one.

The Detective Work: The µ-e Conversion

This paper is about a specific, high-stakes detective game called µ-e conversion.

Imagine a muon (red suit) gets captured by an atom in a metal block (like Titanium or Gold). Usually, it just hangs out and eventually turns into a neutrino. But if the "New Physics" rules are real, the muon might skip the neutrino step entirely and instantly transform into an electron (blue suit) right there in the atom.

The paper asks: "How good are our new detectors at catching this suit-swapping trick?"

The Cast of Characters: R-Parity Violation

The authors are looking at a specific theory called R-parity violating Supersymmetry.

• The Villains: In this theory, there are invisible "coupling constants" (let's call them λ and λ') that act like secret handshakes. If these handshakes happen, the muon can change its suit.
• The Goal: The authors want to figure out exactly how strong these handshakes can be before we would have already seen them in our experiments.

The Plot Twist: The "Time Travel" Effect (RG Running)

Here is the clever part of the paper. The authors realized that these secret handshakes aren't static; they change depending on the "energy" of the universe.

Think of it like a zoom lens on a camera:

1. High Energy (The Zoom Out): At the very beginning of the universe (or at the highest energy scales where new particles exist), the rules for these handshakes are one way.
2. Low Energy (The Zoom In): As we look at the low-energy world where our experiments happen (like the muon in a metal block), the laws of physics "run" or evolve. The strength of the handshakes changes as they travel from the high-energy realm down to our low-energy lab.

The authors call this Renormalization Group (RG) running.

• The Analogy: Imagine you are trying to guess the weight of a suitcase at the airport. If you weigh it at the top of a mountain (high energy), it might feel light. But as you carry it down to sea level (low energy), the air pressure changes, and the scale might read differently.
• The Discovery: The authors found that for most cases, this "weight change" (RG running) is small—maybe a 10-30% difference. But for some specific combinations of handshakes, the effect is huge! It can change the predicted limits by up to 80%. If you ignore this "time travel" effect, you might think a rule is safe when it's actually broken, or vice versa.

The Showdown: Three Different Experiments

The paper compares three different ways to catch the muon changing its suit:

1. µ-e Conversion: The muon turns into an electron inside an atom. (The main focus of this paper).
2. µ → eγ: The muon turns into an electron and shoots out a flash of light (a photon).
3. µ → 3e: The muon turns into three electrons at once.

The Verdict:
The authors used a supercomputer to crunch the numbers and found that µ-e conversion is the ultimate detective.

• In many cases, the other two experiments (light flash or three electrons) are too weak to see the signal because of "GIM suppression" (a fancy way of saying the universe cancels out the signal for those specific handshakes).
• But the µ-e conversion experiment (like the upcoming COMET and Mu2e projects) is so sensitive that it can catch the muon changing suits even when the other experiments are blind.

Why This Matters

This paper is a "reality check" for future experiments.

• The Good News: The upcoming experiments (COMET and Mu2e) are going to be incredibly powerful. They will be able to test these "secret handshakes" much better than we ever could before.
• The Warning: If we want to interpret the results correctly, we must account for that "zoom lens" effect (RG running). If we ignore it, we might misjudge how strong the new physics is.

In a nutshell:
The authors took a complex theory about muons changing into electrons, applied a sophisticated mathematical "zoom lens" to see how the rules change from the high-energy universe down to our labs, and concluded that our new, super-sensitive detectors are going to be the best tools we have ever had to find out if the universe has a hidden "suit-swapping" rule. If they don't find anything, they will have proven that the universe is even more boring (and strict) than we thought!

Technical Summary

1. Problem Statement

The process of muon-to-electron conversion ($\mu$-$e$ conversion) in nuclei is a premier probe for Charged Lepton Flavor Violation (cLFV). While the Standard Model (SM) with massive neutrinos predicts a negligible rate due to $(m_\nu/m_W)^4$ suppression, many Beyond Standard Model (BSM) theories predict observable rates.

The paper focuses on the Supersymmetric (SUSY) model with R-parity violation (RPV). Specifically, it investigates the constraints on trilinear RPV couplings ($\lambda_{ijk}$ and $\lambda'_{ijk}$) using current and future experimental data. Previous studies often neglected the full impact of Renormalization Group (RG) running effects from the high new physics scale ($\Lambda_{NP} \sim \text{TeV}$) down to the low-energy scale ($\Lambda_{low} \sim 2 \text{ GeV}$), potentially underestimating or misestimating constraints on specific coupling combinations. The authors aim to provide a comprehensive update of these limits, incorporating RG evolution and comparing $\mu$-$e$ conversion against other cLFV processes like $\mu \to e\gamma$ and $\mu \to 3e$.

2. Methodology

The authors employ an Effective Field Theory (EFT) framework to connect the high-energy SUSY model to low-energy observables. The methodology proceeds in three main stages:

A. Matching to SMEFT (Standard Model Effective Field Theory)

• Scale: $\Lambda_{NP}$ (New Physics scale, assumed $\sim \text{TeV}$).
• Process: The RPV superpotential terms ($\lambda_{ijk} L_i L_j E^c_k$ and $\lambda'_{ijk} L_i Q_j D^c_k$) are integrated out to generate dimension-6 operators in the SMEFT basis.
  • $\lambda'$ terms: Contribute at the tree level via squark exchange, generating four-fermion operators of the type $(\ell \ell q q)$.
  • $\lambda$ terms: Contribute at the one-loop level (radiative), generating dipole operators ($O_{eW}, O_{eB}$) and four-fermion operators.
• Tools: The matching conditions are derived analytically and verified using the Matchete package.

B. RG Running (Renormalization Group Evolution)

• Scale: From $\Lambda_{NP}$ down to the Electroweak scale ($\Lambda_{EW}$), and then to the low-energy scale ($\Lambda_{low} \approx 2 \text{ GeV}$).
• Process:
  1. SMEFT to LEFT: Operators are evolved from $\Lambda_{NP}$ to $\Lambda_{EW}$, then matched to the Low-Energy Effective Field Theory (LEFT) at the electroweak scale.
  2. LEFT Evolution: The LEFT Wilson coefficients are evolved from $\Lambda_{EW}$ to $2 \text{ GeV}$ using QCD and QED anomalous dimension matrices.
• Tools: The authors utilize the Python package wilson to solve the RG equations automatically. This step is crucial as it accounts for operator mixing and scaling effects that can significantly alter the coefficients.

C. Calculation of Conversion Rates

• Formalism: The LEFT operators are matched to the nuclear physics formalism for $\mu$-$e$ conversion.
• Nuclear Physics: The conversion rate $\Gamma_{conv}$ is calculated for specific isotopes ($^{27}\text{Al}$, $^{48}\text{Ti}$, $^{197}\text{Au}$) using overlap integrals ($S, V, D$) derived from nuclear wavefunctions.
• Comparison: The theoretical rates are compared against:
  • Current limits: SINDRUM II ($\mu$-$e$ conversion), MEG-II ($\mu \to e\gamma$), SINDRUM ($\mu \to 3e$).
  • Future sensitivities: COMET (Phase I/II), Mu2e (Run I/II), Mu3e.

3. Key Contributions

1. Comprehensive RG Analysis: The paper provides a rigorous calculation of RG running effects for 15 combinations of $\lambda'$ couplings and 6 combinations of $\lambda$ couplings. It demonstrates that while RG effects are often small ($<30\%$), they can be substantial ($\sim 80\%$) for specific coupling combinations where cancellations occur at tree level.
2. Updated Constraints: The authors derive new upper limits on the product of couplings $|\lambda^{(\prime)}_{ijk} \lambda^{(\prime)*}_{prs}|$ (normalized by sparticle masses squared) using the most recent experimental data and projected future sensitivities.
3. Process Comparison: A detailed comparison is made between $\mu$-$e$ conversion, $\mu \to e\gamma$, and $\mu \to 3e$. The study highlights that while $\mu \to e\gamma$ currently provides the tightest constraints for dipole-dominated scenarios, $\mu$-$e$ conversion is superior for constraining tree-level four-fermion operators (dominated by $\lambda'$) and is the only process capable of constraining certain combinations suppressed by GIM mechanisms in other channels.
4. Identification of Neglected Channels: The authors identify specific coupling combinations (e.g., $\lambda'_{23k}\lambda'^*_{12k}$) that were previously underexplored or neglected because they are suppressed in $\mu \to e\gamma$ and $\mu \to 3e$ but dominate in $\mu$-$e$ conversion due to CKM matrix insertions.

4. Key Results

A. Impact of RG Running

• General Case: For most coupling combinations, RG running effects modify the constraints by less than 30%.
• Critical Cases: For specific combinations like $|\lambda'_{211}\lambda'^*_{111}|$, RG effects can weaken the constraint by 30–50% due to enhanced cancellations between vector charges $V(p)$ and $V(n)$.
• Enhancement: For combinations like $|\lambda'_{23k}\lambda'^*_{12k}|$ and $|\lambda'_{13k}\lambda'^*_{22k}|$, RG running improves the constraints (lowers the upper limit) by up to 80%, making the limits significantly tighter than previous tree-level estimates.

B. Constraints on $\lambda'$ Couplings

• Tree-Level Dominance: $\mu$-$e$ conversion provides the strongest constraints for $\lambda'$ couplings because they contribute at the tree level.
• Superiority over Decays: For many $\lambda'$ combinations, $\mu$-$e$ conversion limits are orders of magnitude stronger than those from $\mu \to e\gamma$ or $\mu \to 3e$.
• Future Sensitivity: The upcoming COMET Phase I and Mu2e experiments are projected to improve limits by 2–3 orders of magnitude, reaching sensitivities of $\sim 10^{-23} \text{ TeV}^{-2}$ for Wilson coefficients, far surpassing current decay experiments for these specific operators.

C. Constraints on $\lambda$ Couplings

• Loop Dominance: $\lambda$ couplings contribute primarily via loops. Consequently, $\mu \to 3e$ currently provides the strongest constraints for the first three combinations (which have tree-level contributions to $\mu \to 3e$).
• Future Outlook: Future $\mu$-$e$ conversion experiments are expected to surpass $\mu \to 3e$ and $\mu \to e\gamma$ in sensitivity for the remaining $\lambda$ combinations, which are suppressed in decay channels.

D. Neutrino Mass and Other Constraints

• The paper discusses constraints from neutrino mass generation (via one-loop diagrams). For certain combinations (e.g., $\lambda'_{133}\lambda'^*_{233}$), neutrino mass limits are currently stronger than cLFV limits. However, for many other combinations, $\mu$-$e$ conversion provides the most stringent bounds.
• Constraints from meson decays ($K \to \pi \nu \bar{\nu}$, $B \to X_s \nu \bar{\nu}$) are generally weaker than the projected $\mu$-$e$ conversion limits.

5. Significance

• Experimental Guidance: The paper serves as a critical guide for the COMET and Mu2e collaborations, demonstrating that these next-generation experiments will not only test new physics but will be uniquely capable of probing specific trilinear RPV coupling combinations that are invisible or poorly constrained by decay experiments.
• Theoretical Precision: By rigorously including RG running, the authors correct previous approximations, showing that neglecting these effects can lead to significant errors (up to 80%) in the derived bounds for specific parameter spaces.
• Model Discrimination: The comparative analysis of $\mu$-$e$ conversion, $\mu \to e\gamma$, and $\mu \to 3e$ offers a strategy for distinguishing between different new physics scenarios. For instance, a signal in $\mu$-$e$ conversion without a signal in $\mu \to 3e$ would strongly favor specific $\lambda'$-dominated RPV scenarios over others.

In conclusion, this work establishes that $\mu$-$e$ conversion is the most powerful tool for constraining trilinear R-parity violating SUSY in the near future, provided that Renormalization Group effects are properly accounted for.