Rare B meson decays in the Minimal R-symmetric Supersymmetric Standard Model

This paper analyzes lepton flavor violating B meson decays within the Minimal R-symmetric Supersymmetric Standard Model, finding that while experimental constraints on slepton and squark mass matrices significantly influence predictions, the branching ratio for $B^0_d \rightarrow \mu \tau$ remains four orders of magnitude below future experimental sensitivity, making the $B^0_d \rightarrow \mu \tau$ decay more likely to be observed.

The Gist

The Big Picture: Hunting for Ghosts in the Machine

Imagine the Standard Model of physics as a massive, incredibly complex rulebook for how the universe works. It's been tested for decades and works perfectly... mostly. But physicists suspect there are "ghosts" in the machine—hidden particles and forces we haven't seen yet. This is what we call New Physics.

One of the best ways to find these ghosts is to look for "forbidden" moves. In the Standard Model, certain particle swaps are so rare they are basically impossible. It's like trying to win the lottery every single day for a year; the odds are so low that if it happens, something outside the rules must be helping you.

This paper focuses on a specific "forbidden move": Lepton Flavor Violation (LFV).

• The Players: Imagine B-mesons (heavy, unstable particles made of a bottom quark) as a busy train station.
• The Rule: Usually, when a B-meson decays (breaks apart), it turns into a specific type of passenger (a lepton, like an electron or a muon).
• The Forbidden Move: The paper looks at cases where a B-meson decays into two different types of passengers at the same time (e.g., a muon and a tau particle). In our current rulebook, this is like a train arriving with a ticket for a "Red Car" but the passengers getting off are wearing "Blue" and "Green" uniforms. It shouldn't happen.

The New Theory: The "R-Symmetric" Upgrade

The authors are testing a specific theory called the Minimal R-symmetric Supersymmetric Standard Model (MRSSM).

• Supersymmetry (SUSY): Think of this as a "Shadow World." For every particle we know (like an electron), there is a heavier, invisible "shadow twin" (a selectron).
• The "R-Symmetry" Twist: Most versions of this Shadow World theory have a problem: the shadows interact in messy, chaotic ways that break the rules of the universe (causing CP violation and flavor issues).
• The Fix: The MRSSM introduces a special rule called R-symmetry. Imagine this as a strict bouncer at a club. The bouncer says, "No messy interactions allowed!" This forces the shadow particles to behave in a very specific, orderly way. This makes the theory cleaner and more predictive.

The Investigation: How They Did It

The authors didn't just guess; they ran a massive simulation, like a video game where they tweak the settings to see what happens.

1. Setting the Stage: They had to make sure their "Shadow World" looked like our real world. They adjusted 14 different knobs (parameters) to ensure the model predicted a Higgs boson (the particle that gives mass to others) with the correct weight (125 GeV) and matched other known measurements like the W-boson mass.
2. The Constraints (The Speed Bumps): They had to make sure their model didn't break other known laws. For example, if the shadow particles were too heavy or interacted too strongly, they would have caused other rare decays (like a muon turning into an electron and a photon) that we have already seen limits on. They had to steer clear of these "speed bumps."
3. The Prediction: Once the model was tuned to be realistic, they asked: "Okay, given these rules, how often should our 'forbidden' B-meson train swaps happen?"

The Results: What Did They Find?

The results are a mix of "good news" for theorists and "bad news" for experimentalists (in a way).

• The "Off-Diagonal" Secret Sauce: The paper found that the likelihood of these rare decays depends heavily on something called "off-diagonal entries" in the mass matrices.
  • Analogy: Imagine the shadow particles have a family tree. Usually, a "muon-shadow" only talks to other "muon-shadows." But in this model, there are secret back-channels (off-diagonal entries) that let a "muon-shadow" talk to a "tau-shadow." The strength of this back-channel determines how often the forbidden B-meson decay happens.
• The Numbers:
  • Electron-Muon swaps ($B \to e\mu$): These are predicted to be incredibly rare—so rare that even future, super-powerful telescopes probably won't see them. They are about 100 million times less likely than what we might hope to detect.
  • Muon-Tau swaps ($B \to \mu\tau$): This is the most promising candidate. The model predicts these happen more often than the electron swaps, but they are still 10,000 times less likely than what the next generation of experiments (like the High-Luminosity LHC) can currently detect.

The Conclusion: A Long Shot, But Worth Watching

The paper concludes that while the Minimal R-symmetric Supersymmetric Standard Model is a beautiful, orderly theory that solves many problems, it predicts that these specific "forbidden" B-meson decays are extremely rare.

• The Verdict: If we look for these decays in the near future, we probably won't find them. The "ghosts" in this specific Shadow World are very shy.
• The Silver Lining: The decay $B^0_d \to \mu\tau$ (Muon to Tau) has the highest chance of being seen, even if it's still a long shot. It's like looking for a needle in a haystack, but this specific needle is slightly less hidden than the others.

In short: The authors built a very strict, orderly version of a "Shadow World" theory. They crunched the numbers and found that while this theory is mathematically beautiful, it predicts that the universe is even more boring (regarding these specific rare swaps) than we hoped. We will need to build even bigger, more sensitive particle colliders to catch a glimpse of these elusive events.

Technical Summary

1. Problem Statement

Lepton Flavor Violating (LFV) decays of B mesons (specifically $B^0_q \to l_1 l_2$, where $q=d,s$ and $l_1, l_2 \in \{e, \mu, \tau\}$ with $l_1 \neq l_2$) are highly suppressed in the Standard Model (SM) due to tiny neutrino masses, rendering their branching ratios (BR) far below current experimental sensitivity ($\sim 10^{-52}$). However, these processes are powerful probes for New Physics (NP).

The paper addresses the prediction of these rare decays within the Minimal R-symmetric Supersymmetric Standard Model (MRSSM). Unlike the standard MSSM, the MRSSM incorporates a global $U(1)_R$ symmetry which forbids trilinear $A$-terms and Majorana gaugino masses, replacing them with Dirac masses. The authors aim to determine the upper limits of LFV B-meson decays in this specific framework, constrained by:

• The observed 125 GeV Higgs boson mass.
• Electroweak precision observables (W mass, oblique parameters $S, T, U$).
• Existing experimental limits on radiative lepton decays ($l_2 \to l_1 \gamma$) and other B-physics observables ($B \to X_s \gamma$, $B_s \to \mu\mu$).

2. Methodology

Theoretical Framework

• Model: The MRSSM extends the MSSM by adding chiral adjoint superfields ($\hat{O}, \hat{T}, \hat{S}$) and R-Higgs doublets ($\hat{R}_u, \hat{R}_d$). This structure enforces Dirac gaugino masses and eliminates $A$-terms, altering the flavor structure compared to the MSSM.
• Flavor Violation Source: In the MRSSM, LFV arises primarily from off-diagonal entries in the slepton ($m^2_{\tilde{l}}, m^2_{\tilde{r}}$) and squark ($m^2_{\tilde{q}}, m^2_{\tilde{u}}, m^2_{\tilde{d}}$) mass matrices. Notably, the R-symmetry forbids left-right slepton mixing (no $A$-terms), a key difference from the MSSM.
• Calculation Formalism:
  • The process $B^0_q \to l_1 l_2$ occurs at the one-loop box level involving charginos/neutralinos and sfermions.
  • The authors derived analytic expressions for the effective Lagrangian coefficients ($B^\alpha_{\beta\delta}$) and form factors ($F_S, F_P, F_V, F_A$).
  • The branching ratio is calculated using the squared amplitude derived from these form factors.

Numerical Analysis Strategy

• Tools: Calculations were performed using BSMArts, SARAH, HiggsTools, and MultiNest.
• Parameter Scanning:
  • A scan was performed over 14 model parameters (including $\tan\beta$, Dirac gaugino masses, Higgsino masses, and trilinear couplings $\lambda, \Lambda$) to find the region compatible with a 125 GeV Higgs mass at the $1\sigma$ level.
  • Mass Insertion Approximation: Off-diagonal elements in sfermion mass matrices were parameterized as $\delta_{ij}^X = (m^2_X)_{ij} / \sqrt{(m^2_X)_{ii}(m^2_X)_{jj}}$.
  • Constraints: The parameter space was filtered against:
    • Higgs mass ($125.2 \pm 0.11$ GeV).
    • Electroweak precision data ($S, T, U$ parameters, $m_W$).
    • $B \to X_s \gamma$ and $B_{d,s} \to \mu\mu$ limits.
    • Radiative lepton decays ($\mu \to e\gamma$, $\tau \to e\gamma$, $\tau \to \mu\gamma$) to constrain slepton mixing ($\delta_{ij}^L$).

3. Key Contributions

1. First Comprehensive MRSSM Study on LFV B-Decays: This work provides the first detailed analysis of $B^0_q \to l_1 l_2$ specifically within the R-symmetric framework, highlighting the distinct phenomenological consequences of Dirac gaugino masses and the absence of $A$-terms.
2. Constraint Propagation: The authors rigorously propagate constraints from low-energy lepton flavor violation ($l_2 \to l_1 \gamma$) to the B-meson sector, establishing strict upper bounds on the relevant mass insertion parameters ($\delta_{ij}^L$).
3. Parameter Sensitivity Analysis: The study isolates the dependence of branching ratios on specific parameters, particularly $\tan\beta$, squark mixing ($\delta_{ij}^Q$), and slepton mixing ($\delta_{ij}^L$), demonstrating that $\tan\beta$ plays a critical role in enhancing these rates.

4. Key Results

• Allowed Parameter Space: The authors identified a viable parameter space at the $1\sigma$ level that reproduces the SM-like Higgs mass. Key values include $\tan\beta \approx 17.4$, Dirac gaugino masses $M_D \approx 600-630$ GeV, and specific ranges for trilinear couplings.
• Constraints on Mixing Parameters:
  • Radiative lepton decays constrain the slepton mixing parameters to:
    • $\log_{10} \delta_{12}^L \leq -2.6$
    • $\log_{10} \delta_{13}^L \leq -0.2$
    • $\log_{10} \delta_{23}^L \leq -0.1$
• Branching Ratio Predictions:
  • $B^0_q \to e\mu$: The predicted BR is extremely suppressed, around $O(10^{-18})$, which is 8 orders of magnitude below future experimental sensitivity.
  • $B^0_q \to e\tau$ and $B^0_q \to \mu\tau$: These channels are enhanced by large $\tan\beta$ and squark mixing. The upper predictions reach $O(10^{-10})$.
  • Specific Case ($B^0_d \to \mu\tau$): The predicted BR is approximately four orders of magnitude below the future experimental limit ($1.3 \times 10^{-6}$).
• Dependence on $\tan\beta$: The branching ratios for $B^0_d \to l_1 l_2$ increase significantly with $\tan\beta$. At $\tan\beta \sim 50$, the rates for $B^0_s \to e\tau$ and $B^0_s \to \mu\tau$ can be enhanced up to $O(10^{-10})$.
• Role of Squark Mixing: The parameters $\delta_{13}^Q$ and $\delta_{23}^Q$ have significant impacts. Interestingly, $B^0_d$ and $B^0_s$ decays respond oppositely to changes in these parameters due to the specific quark flavor composition.

5. Significance and Conclusion

The paper concludes that while the MRSSM can accommodate the observed Higgs mass and satisfy all current electroweak and flavor constraints, the predicted rates for LFV B-meson decays remain challenging to detect.

• Observability: The decay $B^0_d \to \mu\tau$ is identified as the most promising channel, with a predicted rate only four orders of magnitude below future experimental sensitivities. This suggests that while difficult, it is not impossible to observe in future high-luminosity experiments (e.g., LHCb Upgrade II or Belle II).
• Model Discrimination: The distinct suppression of $B^0_q \to e\mu$ compared to $\tau$-involving channels, driven by the specific constraints on $\delta_{ij}^L$ and the absence of $A$-terms in the MRSSM, offers a potential signature to distinguish this model from the MSSM or other New Physics scenarios.
• Future Outlook: The authors suggest that further efforts should focus on the $B^0_d \to \mu\tau$ channel, as it represents the most viable window for testing the MRSSM via rare B decays in the near future.