Comprehensive analyses of rare $\Lambda_b \rightarrow \Lambda \ell^+ \ell^-$, $\Sigma_b \rightarrow \Sigma \ell^+ \ell^-$ and $\Xi_b \rightarrow \Xi \ell^+ \ell^-$ decays in 2HDM

This paper investigates rare dileptonic decays of $\Lambda_b$, $\Sigma_b$, and $\Xi_b$ baryons within the Type III Two-Higgs-Doublet Model, calculating key observables like branching ratios and forward-backward asymmetries using light-cone QCD form factors to compare predictions against Standard Model results and assess the model's viability for future experimental verification at LHCb and Belle II.

The Gist

Imagine the universe as a giant, incredibly complex machine. For decades, scientists have had a user manual for this machine called the Standard Model (SM). It's a brilliant manual that explains how most of the particles (the tiny building blocks of everything) behave. But, like any old manual, it has some pages missing. It doesn't explain things like dark matter or why there is more matter than antimatter.

This paper is like a team of mechanics (the authors) trying out a new, expanded version of the manual called the Two-Higgs-Doublet Model (2HDM). Specifically, they are testing a "Type III" version, which is a bit more flexible and allows for some interactions that the original manual strictly forbids.

Here is a breakdown of what they did, using simple analogies:

1. The Experiment: The "Heavy Baryon" Race

The scientists focused on three specific, heavy particles called baryons (named $\Lambda_b$, $\Sigma_b$, and $\Xi_b$). Think of these as heavy, three-wheeled delivery trucks made of quarks (the fundamental particles).

They wanted to see what happens when these trucks spontaneously break down into a lighter truck and a pair of "leptons" (like muons or tau particles, which are cousins of the electron). This is a very rare event, like a truck driving down the highway and suddenly turning into a bicycle and a pair of butterflies.

2. The Two Scenarios: The "Standard" vs. The "New"

The researchers calculated what this breakdown should look like under two different rulesets:

• The Standard Model (The Old Manual): This is the baseline. It predicts exactly how often this happens and how the particles fly apart.
• The 2HDM Type III (The New Manual): This model introduces a new character: a charged Higgs boson. Imagine the Standard Model has one "force carrier" (like a messenger) that delivers the decay. The 2HDM says, "Wait, there's actually two messengers, and a third, heavier, secret messenger (the charged Higgs) that we haven't fully accounted for yet."

3. The Investigation: Checking the "Fingerprints"

The scientists didn't just guess; they did the math to see how the presence of this "secret messenger" (the charged Higgs) would change the outcome of the crash. They looked at three main things:

• The Branching Ratio (How often it happens):

  • Analogy: If you flip a coin 1,000 times, how many times does it land on heads?
  • Finding: In the Standard Model, the coin lands on heads a specific number of times. In the 2HDM, if the "secret messenger" is light (low mass) and interacts strongly, the coin might land on heads much more often. The paper found that if this new particle exists and is relatively light (around 175 GeV), the decay rate could be significantly higher than the Standard Model predicts.

• The Forward-Backward Asymmetry (Which way they fly):

  • Analogy: When the truck breaks down, do the butterflies fly mostly forward, mostly backward, or evenly in both directions?
  • Finding: The Standard Model predicts a specific "lean" (asymmetry) in how the particles fly. The 2HDM predicts that this lean gets "flattened out." It's like the butterflies are less likely to choose a side and more likely to just drift. This is a huge clue. If experiments see the butterflies drifting evenly instead of leaning, it's a sign of the new physics.

• The "Long-Distance" Noise:

  • Analogy: Imagine trying to hear a whisper in a room where a loud band is playing. The band is the "long-distance" effect (particles like the J/$\psi$ that interfere with the signal).
  • Finding: The scientists carefully calculated how to filter out this noise to hear the true signal of the new physics. They found that even with the noise, the signal of the new model is still visible, especially in the "high-energy" parts of the crash.

4. The Results: What Did They Find?

• The "Light" Higgs is the Key: The new model only really changes the outcome if the "secret messenger" (the charged Higgs) is relatively light. If it's very heavy (like 1000 GeV), it's too heavy to interfere, and the universe looks just like the Standard Model again.
• Agreement with Data: When they compared their "New Manual" predictions with actual data from the LHCb and CDF experiments (which have already seen some of these rare crashes), they found something interesting. The data actually fits the "New Manual" (specifically with a light Higgs) quite well, sometimes even better than the old one in certain high-energy zones.
• The Tau Problem: They also looked at a heavier version of the lepton called the "tau." The math says the new model affects these too, but we don't have enough data yet to confirm it. It's like having a theory about a new species of bird, but we haven't spotted one in the wild yet.

5. The Conclusion: Why Does This Matter?

This paper is a roadmap for future experiments. It tells the scientists at the LHC (Large Hadron Collider) and Belle II detectors:

"Hey, keep your eyes peeled! If you see these heavy trucks breaking down more often than expected, or if the particles fly in a flatter pattern than we thought, it might be the first proof that our 'Standard Manual' is incomplete and that this 'Type III' model is real."

In a nutshell: The authors are testing a new theory of physics by simulating rare particle crashes. They found that if a specific new particle exists and is light, it would leave a distinct "fingerprint" on these crashes. Current data hints that this fingerprint might be there, and future, more powerful detectors will be the ones to confirm if we are truly looking at a new chapter in the story of the universe.

Technical Summary

1. Problem Statement

The Standard Model (SM) of particle physics, while highly successful, fails to address fundamental questions such as the nature of dark matter, the hierarchy problem, and the origin of neutrino masses. Furthermore, the discovery of the Higgs boson at 125 GeV has not yet confirmed whether it is the sole Higgs boson or part of an extended sector.

• Specific Gap: Rare flavor-changing neutral current (FCNC) decays of heavy baryons ($\Lambda_b, \Sigma_b, \Xi_b$) into a lighter baryon and a dilepton pair ($\ell^+ \ell^-$) are sensitive probes for New Physics (NP). While these decays are forbidden at the tree level in the SM (occurring only via loop diagrams), extensions like the Two-Higgs-Doublet Model (2HDM) can significantly alter their rates and kinematic distributions.
• Focus: This study specifically investigates the Type-III 2HDM, which allows for tree-level FCNCs via non-diagonal Yukawa couplings, a feature often suppressed in Type-I and Type-II models. The goal is to quantify how Type-III 2HDM parameters affect the observables of $\Lambda_b \to \Lambda \ell^+ \ell^-$, $\Sigma_b \to \Sigma \ell^+ \ell^-$, and $\Xi_b \to \Xi \ell^+ \ell^-$ decays.

2. Methodology

The authors employ a rigorous theoretical framework combining effective field theory with non-perturbative QCD calculations.

• Theoretical Framework (2HDM Type-III):

  • The model introduces five physical Higgs bosons: $h^0, H^0$ (CP-even), $A^0$ (CP-odd), and $H^\pm$ (charged).
  • Unlike Type-I/II, Type-III allows both Higgs doublets to couple to all fermion types, introducing non-diagonal Yukawa couplings ($\lambda_{tt}, \lambda_{bb}$). These couplings are treated as free complex parameters, potentially introducing CP-violating phases.
  • The study focuses on the $b \to s \ell^+ \ell^-$ transition, governed by an effective Hamiltonian.

• Effective Hamiltonian & Wilson Coefficients:

  • The transition is described by an effective Hamiltonian containing operators $O_7, O_9, O_{10}$ and new scalar/pseudoscalar operators $Q_1, Q_2$ arising from neutral Higgs exchange.
  • The authors calculate the modifications to the Wilson coefficients ($C_7, C_9, C_{10}$) induced by charged Higgs ($H^\pm$) loops and the new scalar operators induced by neutral Higgs ($h^0, H^0, A^0$) exchanges.
  • Renormalization group evolution is applied to run coefficients from the electroweak scale ($\mu = m_W$) down to the $b$-quark mass scale ($\mu = m_b$).

• Hadronic Matrix Elements:

  • To handle the non-perturbative QCD effects of the baryon transitions, the authors utilize Light-Cone QCD Sum Rules (LCSR).
  • They employ 12 independent form factors ($f_i, g_i, f^T_i, g^T_i$) to parameterize the transition matrix elements between the initial heavy baryon and the final light baryon. This approach avoids approximations based on heavy quark expansion or large energy limits, providing a "full theory" calculation.

• Observables Calculated:

  • Differential Decay Width ($d\Gamma/dq^2$): Derived from the squared amplitude.
  • Differential Branching Ratio ($d\mathcal{B}/dq^2$): Calculated as a function of the dilepton invariant mass squared ($q^2$).
  • Total Branching Ratio ($\mathcal{B}$): Integrated over the kinematically allowed $q^2$ range.
  • Lepton Forward-Backward Asymmetry ($A_{FB}$): A key angular observable sensitive to the interference between vector and axial-vector currents.

• Long-Distance (LD) Effects:

  • The analysis includes LD contributions from charmonium resonances ($J/\psi, \psi'$) using a Breit-Wigner parameterization, which significantly affects the low-$q^2$ region. Both LD-included and LD-subtracted results are presented.

3. Key Contributions

1. Comprehensive Baryonic Analysis: Unlike many studies focusing solely on $\Lambda_b$, this paper provides a unified, comparative analysis of three distinct baryonic systems ($\Lambda_b, \Sigma_b, \Xi_b$) within the same 2HDM Type-III framework.
2. Full QCD Form Factors: The use of LCSR-derived form factors for all three baryon transitions ensures a consistent treatment of hadronic uncertainties across the different channels.
3. Parameter Space Exploration: The study systematically varies the charged Higgs mass ($m_{H^\pm}$) from 175 GeV to 1000 GeV and the top-quark Yukawa coupling ($\lambda_{tt}$) from 0.05 to 0.30, mapping the sensitivity of observables to these BSM parameters.
4. Lepton Flavor Comparison: The paper explicitly compares muon ($\mu$) and tau ($\tau$) channels, highlighting the different sensitivities to New Physics due to the lepton mass dependence.

4. Results

• Branching Ratios:

  • SM Baseline: The SM predictions for $\Lambda_b \to \Lambda \mu^+ \mu^-$ are consistent with existing LHCb and CDF experimental data within uncertainties.
  • 2HDM Enhancement: In the Type-III 2HDM, the branching ratios can be significantly enhanced, particularly for low charged Higgs masses ($m_{H^\pm} \approx 175$ GeV) and large couplings ($\lambda_{tt} \approx 0.30$).
  • Decoupling: As $m_{H^\pm}$ increases (e.g., to 1000 GeV), the 2HDM predictions converge toward the SM values, demonstrating the decoupling behavior of the model.
  • Channel Sensitivity: The $\mu^+ \mu^-$ channels are generally more sensitive to NP variations than the $\tau^+ \tau^-$ channels. The $\Xi_b \to \Xi \mu^+ \mu^-$ channel shows particularly large relative enhancements in the high-$q^2$ region for light Higgs masses.

• Forward-Backward Asymmetry ($A_{FB}$):

  • SM Prediction: The SM predicts a distinct zero-crossing point and negative values at high $q^2$.
  • 2HDM Deviation: The Type-III 2HDM, especially with $m_{H^\pm} = 175$ GeV, tends to flatten the $A_{FB}$ curve or shift the zero-crossing point. In the $\tau$ channels, the asymmetry is suppressed and pulled closer to zero compared to the SM.
  • Experimental Agreement: For the $\Lambda_b \to \Lambda \mu^+ \mu^-$ channel, the experimental data shows the strongest agreement with the 2HDM prediction for $m_{H^\pm} = 175$ GeV, whereas higher masses align more closely with the SM.

• Long-Distance Effects:

  • LD contributions create prominent resonance peaks in the $q^2$ spectrum around 9.5 and 13.5 GeV$^2$ (charmonium region).
  • The qualitative trends of 2HDM deviations (enhancement at high $q^2$) remain robust even when LD effects are subtracted, making the high-$q^2$ region a cleaner probe for NP.

5. Significance

• Probing BSM Physics: The study demonstrates that rare baryonic decays are powerful tools for constraining the parameter space of the Type-III 2HDM. The specific "fingerprints" of low-mass charged Higgs bosons (enhanced branching ratios and flattened $A_{FB}$) provide clear targets for experimental verification.
• Experimental Guidance: The results suggest that future high-luminosity runs at LHCb and Belle II are critical. Specifically, precise measurements of the differential branching ratio and $A_{FB}$ in the high-$q^2$ region for $\Lambda_b, \Sigma_b,$ and $\Xi_b$ decays could distinguish between the SM and Type-III 2HDM scenarios.
• Theoretical Benchmark: By providing detailed numerical tables and graphical representations for all three baryon types and both lepton flavors, this work serves as a comprehensive benchmark for future experimental analyses and theoretical refinements in heavy baryon phenomenology.

In conclusion, the paper establishes that the Type-III 2HDM can induce significant deviations in rare baryonic dileptonic decays, particularly for light charged Higgs bosons, and that the lepton forward-backward asymmetry is a highly sensitive observable for detecting these effects.