The Muon and Tau Electric Dipole Moments in the B-L Supersymmetric Standard Model

This paper investigates the electric dipole moments of muons and taus within the B-L supersymmetric standard model, demonstrating that both the traditional $\mu$-term and model-specific CP-violating parameters can produce significant contributions, with muon EDMs potentially falling within the sensitivity of upcoming Phase II experiments and tau EDMs reaching magnitudes around $10^{-21}e\cdot\text{cm}$.

The Gist

Imagine the universe as a giant, intricate clockwork machine. For a long time, scientists thought they understood how this machine ticked, but they noticed a tiny, unexplained wobble. This wobble is called CP-violation (Charge-Parity violation). It's a subtle asymmetry in how particles behave compared to their mirror images.

In our current best theory of physics (the Standard Model), this wobble is so tiny that it can't explain a massive mystery: why the universe is made of matter instead of being an empty void where matter and antimatter canceled each other out. Scientists suspect there must be a hidden "gear" or "spring" in the machine that creates a bigger wobble, but we haven't found it yet.

This paper is a detective story looking for that hidden gear, specifically focusing on two suspects: the Muon and the Tau. These are heavy cousins of the electron. The researchers are asking: If we look at these particles, can we find a bigger wobble (called an Electric Dipole Moment, or EDM) that points to new physics?

Here is the breakdown of their investigation using simple analogies:

1. The New Theory: The "B-L" Expansion

The authors are testing a specific theory called the B-L Supersymmetric Standard Model (B-LSSM).

• The Analogy: Think of the Standard Model as a standard house with a certain number of rooms. The B-LSSM is like adding a new, secret wing to that house. This new wing includes extra particles (like a new type of gauge boson called $Z'$) and new rules for how they interact.
• The Goal: They want to see if this "secret wing" creates a stronger wobble in the Muon and Tau particles than the standard house does.

2. The Search for the "Wobble" (EDM)

An Electric Dipole Moment (EDM) is like a tiny internal compass inside a particle.

• The Analogy: Imagine a spinning top. If it's perfectly balanced, it spins straight up. If it has an EDM, it's like the top is slightly lopsided, causing it to wobble as it spins.
• The Catch: In the old theory, this wobble is so small it's invisible. But if the "secret wing" of the B-LSSM exists, it might make the wobble much bigger—big enough that our new, super-sensitive microscopes (experiments) might finally see it.

3. The Investigation: Two Types of Suspects

The researchers looked at two different types of "suspects" (parameters) that could cause this wobble:

• The "Old" Suspects (General SUSY): These are variables like $\mu$ and $A_l$ that exist in almost all versions of this theory.

  • Finding: They found that these old suspects are the main drivers of the wobble. If you turn up the "volume" on these parameters (specifically the $\mu$ term), the wobble gets huge.
  • The Analogy: It's like turning up the volume on a radio. The louder you turn it, the clearer the signal becomes.

• The "New" Suspects (B-LSSM Specific): These are unique variables ($M_{BB'}$, $M_{BL}$, $\mu_\eta$) that only exist in this specific "secret wing" theory.

  • Finding: These new suspects also cause a wobble, but they are a bit more complicated. Sometimes they make the wobble bigger, but if they get too heavy (too massive), they stop contributing, a phenomenon the paper calls "decoupling."
  • The Analogy: Imagine these are new instruments in a band. They add a unique flavor to the music, but if they are too far away from the stage (too heavy), the audience can't hear them anymore.

4. The Results: What the Math Says

The team crunched the numbers to see what the wobble would look like in a real experiment.

• For the Muon ($d_\mu$):

  • The Result: The theory predicts a wobble that is right on the edge of what a new, upcoming experiment (called "Phase II") is designed to detect.
  • The Analogy: It's like a detective saying, "The suspect is hiding in the next room, and the new security camera we are installing next year will definitely catch them."
  • Significance: If the Phase II experiment sees this wobble, it proves the "secret wing" (B-LSSM) is real. If it doesn't, it means the "volume" on the main suspect ($\mu$) must be turned down very low.

• For the Tau ($d_\tau$):

  • The Result: The wobble here is predicted to be even larger (about $10^{-21}$), but the Tau particle is very short-lived and hard to study.
  • The Analogy: The signal is loud, but the messenger (the Tau particle) dies before it can deliver the message to our current detectors. It's a "loud whisper" that we can't quite hear yet with today's equipment.

5. The Conclusion

The paper concludes that the B-LSSM theory is a very strong candidate for explaining these missing pieces of physics.

• The "old" suspects (the $\mu$ term) are doing the heavy lifting.
• The "new" suspects (the B-LSSM specific parts) add interesting complexity but don't dominate the result.
• The Big Picture: We are on the verge of a breakthrough. The upcoming Phase II experiment for the Muon is sensitive enough that, if the B-LSSM theory is correct, we should see the "wobble" very soon. If we don't see it, we will have to rewrite the rules of the "secret wing" entirely.

In short, this paper is a roadmap telling experimentalists: "Look here, at the Muon, with your new Phase II microscope. If you see this specific wobble, you've found the hidden gear that explains why our universe exists."

Technical Summary

Technical Summary: The Muon and Tau Electric Dipole Moments in the B-L Supersymmetric Standard Model

Problem Statement
The Standard Model (SM) contains CP-violating (CPV) phases within the Cabibbo-Kobayashi-Maskawa (CKM) matrix, but these are insufficient to explain the observed baryon asymmetry of the universe. Consequently, new physics (NP) models introducing additional CPV sources are required. While the electric dipole moments (EDMs) of the electron, neutron, and mercury are tightly constrained, the experimental limits on the muon ($d_\mu$) and tau ($d_\tau$) EDMs remain significantly weaker due to the short lifetimes of these leptons. However, upcoming experiments, particularly those utilizing frozen-spin techniques, aim to improve sensitivity to $d_\mu$ by several orders of magnitude (reaching $6 \times 10^{-23} \, e \cdot \text{cm}$ in Phase II) and $d_\tau$ to $\sim 10^{-19} \, e \cdot \text{cm}$. This work addresses the theoretical predictions for $d_\mu$ and $d_\tau$ within the $U(1)_{B-L}$ extended Supersymmetric Standard Model (B-LSSM), a framework that extends the MSSM to include a local $U(1)_{B-L}$ gauge symmetry, new scalar singlets, and a $Z'$ gauge boson.

Methodology
The authors perform a comprehensive analytical and numerical study of charged lepton EDMs in the B-LSSM.

1. Analytical Framework: The study derives general analytical expressions for $d_\mu$ and $d_\tau$ by calculating contributions from both one-loop and two-loop diagrams.
   • One-loop contributions: Dominated by diagrams involving slepton-neutralino and chargino-sneutrino loops. The B-LSSM introduces three additional neutralinos and new CPV parameters ($M_{BB'}$, $M_{BL}$, $\mu_\eta$) that contribute at this level.
   • Two-loop contributions: The authors incorporate Barr-Zee type diagrams involving new scalars and the $Z'$ gauge boson. Corrections from top/stop quarks to the Higgs mass matrix are included. The analysis explicitly neglects two-loop contributions from squark-sector CPV parameters, as these are heavily suppressed by existing constraints on neutron and heavy quark EDMs.
2. Numerical Implementation: The authors utilize a benchmark parameter set consistent with LHC data, Higgs signal measurements, and flavor constraints (e.g., $B_s \to \mu^+\mu^-$). Key fixed parameters include $\tan\beta=10$, $M_{Z'}=5$ TeV, and specific mass scales for sleptons and squarks. A random scan is performed over the CPV phases ($\theta_\mu, \theta_{A_l}, \theta_{BB'}, \theta_{BL}, \theta_{\mu_\eta}$) to explore the parameter space.

Key Contributions

• Analytical Formulation: The paper presents the first complete analytical calculation of $d_\mu$ and $d_\tau$ in the B-LSSM, explicitly including the effects of the new $Z'$ boson and the three B-LSSM-specific CPV parameters ($M_{BB'}$, $M_{BL}$, $\mu_\eta$) at both one-loop and two-loop levels.
• Decoupling Analysis: The study systematically investigates the decoupling behavior of SUSY particles. It confirms that while heavy sleptons and Higgsinos suppress EDMs, the B-LSSM-specific parameters exhibit more complex asymptotic behaviors, with suppression only dominating at very large mass scales ($\gtrsim 20$ TeV).
• Flavor Dependence: The work highlights the strong flavor dependence of the EDMs, noting that $d_\tau$ is significantly enhanced relative to $d_\mu$ due to the proportionality of the chirality-flip terms to the Yukawa couplings ($d_\tau/d_\mu \sim m_\tau/m_\mu$).

Results

• Dominant Sources: The numerical results indicate that the traditional $\mu$-term (specifically its CPV phase $\theta_\mu$) provides the dominant contribution to both $d_\mu$ and $d_\tau$. The B-LSSM-specific parameters ($M_{BB'}$, $M_{BL}$, $\mu_\eta$) induce significant subleading effects, modulating the total prediction but not overriding the $\mu$-term dominance.
• Magnitude of Predictions:
  • Muon EDM ($d_\mu$): In regions with large $\tan\beta$ and relatively light Higgsinos ($|\mu| \lesssim 0.4$ TeV), the predicted $|d_\mu|$ reaches $\sim 1.0 \times 10^{-22} \, e \cdot \text{cm}$. A substantial portion of the parameter space with large CPV phases ($0.25\pi \lesssim |\theta_\mu| \lesssim 0.75\pi$) falls within the projected sensitivity of the Phase II experiment ($6 \times 10^{-23} \, e \cdot \text{cm}$).
  • Tau EDM ($d_\tau$): The model predicts $|d_\tau|$ can reach approximately $1.8 \times 10^{-21} \, e \cdot \text{cm}$. While this is a significant enhancement, the authors note that such values remain challenging to observe with near-future experimental capabilities.
• Parameter Interplay: The scatter plots reveal a sinusoidal dependence of the EDMs on $\theta_\mu$. The B-LSSM-specific parameters create a "band" of uncertainty around the central prediction, demonstrating their ability to significantly modulate the theoretical outcome even when $\theta_\mu$ is the primary driver.

Significance
The paper concludes that the B-LSSM offers a theoretically attractive framework where the predicted muon EDM is well within the reach of next-generation experiments. Specifically, the Phase II muon EDM experiment has the potential to either verify or rule out significant regions of the B-LSSM parameter space characterized by substantial CPV phases in the Higgsino mass parameter. Conversely, a null result would impose tight constraints on $\theta_\mu$, requiring either small phases or severe fine-tuning via destructive interference among multiple CPV sources. The study underscores the importance of extending EDM investigations beyond the electron to muons and taus to probe new CPV sources in extended supersymmetric models.