⚛️ phenomenology

The lepton flavor universality including the $b\rightarrow c l \nu$ process in the $U(1)_X$ SSM

This paper investigates lepton flavor universality in $b \rightarrow c \ell \nu$ processes within the $U(1)_X$ SSM framework, demonstrating that while the model cannot fully match experimental data, specific parameter combinations yield improved numerical results over the Standard Model.

Original authors: Meng-Zi Cao, Shu-Min Zhao, Yue-Tong Liu, Shuang Di, Rong-Zhi Sun, Xing-Xing Dong

Published 2026-09-25

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CC BY 4.0

Original authors: Meng-Zi Cao, Shu-Min Zhao, Yue-Tong Liu, Shuang Di, Rong-Zhi Sun, Xing-Xing Dong

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast catalog of the universe's building blocks, there exists a rule so fundamental that physicists have long treated it as an unbreakable law: nature does not play favorites based on the weight of a particle. This principle, known as lepton flavor universality, suggests that the forces governing the interactions of electrons, muons, and tau particles are identical, differing only in how their mass affects their motion. For decades, the Standard Model of physics has relied on this symmetry to describe how matter behaves. However, in recent years, experiments have begun to uncover subtle cracks in this foundation. When heavy particles containing a bottom quark decay into lighter particles, the rate at which they produce the heaviest lepton, the tau, seems to differ from what the established laws predict. These discrepancies hint at the existence of new, unseen forces or particles that interact differently with heavy matter than with light matter, potentially opening a door to a deeper understanding of the cosmos.

A team of researchers from Hebei University and the University of Lisbon has taken a closer look at these anomalies by testing them against a specific theoretical framework called the U(1)XSSM. This model is an extension of the widely accepted Minimal Supersymmetric Standard Model, which already proposes that every known particle has a heavier, invisible partner. The new model adds an extra layer of symmetry and introduces a new, heavy force-carrying particle, along with additional Higgs-like particles and right-handed neutrinos. The researchers used this framework to calculate how these new ingredients would influence the decay of bottom quarks into charm quarks and tau leptons. They did not perform a physical experiment in a laboratory; instead, they built a complex mathematical simulation to see if the new particles could explain the strange ratios observed in real-world data.

The study focused on four specific decay processes, comparing how often a tau lepton is produced versus a lighter electron or muon. In the standard view, these ratios should be nearly identical, but the new model suggested that the presence of the extra particles could shift these numbers. The researchers ran their calculations across a wide range of possible values for the masses and interaction strengths of these new particles, keeping them within the limits set by previous experiments. They found that the model could indeed alter the predicted outcomes. For one specific decay involving a particle called the J/psi, the new physics could boost the ratio to a value around 1.3, bringing it closer to the higher values seen in some experimental measurements. For another decay involving a lambda baryon, the model predicted a suppression, lowering the ratio to approximately 0.85.

The investigation revealed that not all the new parameters in the model were equally important. The researchers discovered that the mass of a specific heavy partner particle, known as a chargino, and a parameter controlling the mixing of new Higgs fields played the dominant roles in shaping the results. When these values were adjusted, the predicted ratios changed significantly. In contrast, other parameters, such as the strength of certain interactions involving neutrinos, had a much smaller effect, acting more like fine-tuning knobs than the main drivers of the change. The team also observed that the sign of a mass parameter could flip the nature of the effect, turning a suppression into an enhancement, which highlights the delicate balance required for these new forces to manifest.

While the results show that this extended model can produce numbers that differ from the standard predictions and move closer to experimental data, the researchers noted that it does not perfectly match every measurement yet. The model suggests that the observed deviations are not random errors but could be the result of new particles interacting with the heaviest leptons. The study concludes that the U(1)XSSM provides a viable and helpful framework for exploring these rare decay processes. It demonstrates that by introducing a new gauge symmetry and its associated particles, physicists can generate the necessary corrections to the decay rates, offering a plausible explanation for the tension between current observations and the long-held Standard Model. The work serves as a guide for future experiments, indicating which parameters are most critical to measure as scientists continue to search for the hidden layers of reality.

Technical Summary: Lepton Flavor Universality in $b \to c\ell\nu$ within the U(1)XSSM

Problem Statement
The Standard Model (SM) posits Lepton Flavor Universality (LFU), asserting that electroweak interactions are identical for all three generations of charged leptons ( $e, \mu, \tau$ ), differing only by mass effects. However, persistent anomalies in $b$ -hadron decays, specifically the charged-current transitions $b \to c\ell\nu$ , have suggested potential deviations from SM predictions. While recent measurements from LHCb and Belle have reduced tensions, significant experimental uncertainties remain. The paper addresses the need to interpret these potential deviations using a specific Beyond the Standard Model (BSM) framework: the U(1) extension of the Minimal Supersymmetric Standard Model (U(1)XSSM). The goal is to determine if this model can accommodate the observed anomalies in ratios such as $R_{J/\psi}$ , $R_{D_s}$ , $R_{D^*_s}$ , and $R_{\Lambda_c}$ better than the SM.

Methodology
The authors employ a systematic approach combining theoretical derivation and numerical analysis:

Theoretical Framework: The study utilizes the U(1)XSSM, which extends the MSSM by an additional $U(1)_X$ gauge symmetry. This extension introduces a heavy neutral gauge boson ( $Z'$ ), an extended Higgs sector (including singlet Higgs superfields $\hat{\eta}, \hat{\bar{\eta}}, \hat{S}$ ), and three generations of right-handed neutrino superfields. The model addresses the $\mu$ -problem and generates neutrino masses via the seesaw mechanism.
Effective Field Theory (EFT): At the low-energy scale, the heavy BSM particles are integrated out. The $b \to c\ell\nu$ transition is described by a dimension-6 effective Lagrangian containing five independent Lorentz-invariant four-fermion operators: vector left-handed ( $O_{VL}$ ), vector right-handed ( $O_{VR}$ ), scalar left-handed ( $O_{SL}$ ), scalar right-handed ( $O_{SR}$ ), and tensor ( $O_T$ ).
Calculation of Wilson Coefficients: The authors calculate the contributions to the Wilson coefficients ( $C_{VL}, C_{VR}, C_{SL}, C_{SR}, C_T$ $C_{V L}, C_{V R}, C_{S L}, C_{S R}, C_{T}$ ) arising from one-loop diagrams in the U(1)XSSM. These include:
- Penguin-type diagrams: Involving exchanges of neutralinos, charginos, and sleptons/sneutrinos.
- Box-type diagrams: Involving exchanges of charginos, neutralinos, squarks, and sleptons.
- The calculations utilize dimensional regularization and the modified minimal subtraction ( $\overline{MS}$ ) scheme to handle divergences.
Numerical Analysis: The derived Wilson coefficients are used to compute the ratios $R_{J/\psi}/R_{J/\psi}^{SM}$ $R_{J / ψ} / R_{J / ψ}^{S M}$ , $R_{D_s}/R_{D_s}^{SM}$ $R_{D_{s}} / R_{D_{s}}^{S M}$ , $R_{D^*_s}/R_{D^*_s}^{SM}$ $R_{D_{s}^{*}} / R_{D_{s}^{*}}^{S M}$ , and $R_{\Lambda_c}/R_{\Lambda_c}^{SM}$ $R_{Λ_{c}} / R_{Λ_{c}}^{S M}$ . The analysis involves:
- Constraints: Adhering to experimental limits on the lightest CP-even Higgs mass ( $m_h \approx 125.13$ GeV), the new angle $\tan\beta_\eta < 1.5$ , and lower bounds on sparticle masses (neutralinos $> 116$ GeV, others at the TeV scale).
- Parameter Scanning: Performing one-dimensional scans and two-dimensional scatter plots to analyze the sensitivity of the observables to key parameters, including trilinear couplings ( $l_W, T_\nu, T_e$ ), soft SUSY breaking masses ( $M_2, M_E, M_L$ ), and Yukawa couplings ( $Y_X$ ).

Key Contributions and Results

Derivation of Analytical Formulas: The paper provides the full analytical expressions for the Wilson coefficients in the U(1)XSSM, accounting for the specific particle content and mixing matrices (neutralinos, charginos, sneutrinos, and squarks) unique to this model.
Impact of Parameters:
- Dominant Parameters: The parameter $l_W$ $l_{W}$ (related to Higgs mass matrices) and the gaugino mass $M_2$ $M_{2}$ are identified as dominant control parameters.
  - For $R_{J/\psi}$ , small values of $l_W$ and $M_2$ lead to constructive interference, significantly enhancing the ratio. The numerical analysis indicates the value can reach "around 1.3" in the discussion, though specific plots show maxima ranging from approximately 1.17 to 1.25 depending on the parameter set. This brings the result closer to experimental data than the SM.
  - For $R_{\Lambda_c}$ , the ratio is suppressed. The analysis finds the value can be "lowered to around 0.85," representing a floor for the suppression, while specific numerical results in the figures range between 0.87 and 0.90.
- Subdominant Parameters: Trilinear couplings like $(T_\nu)_{ii}$ and $(T_e)_{ii}$ , as well as slepton mass parameters like $(M_E)_{ii}$ , are found to be less sensitive. They provide sub-leading corrections or induce step-like features rather than continuous tuning of the results.
- Interference Effects: The study highlights that the sign of parameters (e.g., $M_2$ ) can flip the interference phase between SM and new physics amplitudes, qualitatively altering the direction of the correction (enhancement vs. suppression).
Model Performance: While the U(1)XSSM cannot perfectly match all experimental data simultaneously, the numerical results demonstrate that with specific combinations of parameter values, the model yields better agreement with experimental trends for certain observables (particularly $R_{J/\psi}$ ) compared to the SM. The tensor operator $C_T$ is noted as a dominant contributor to quadratic terms, making $R_{D^*_s}$ a primary discriminator for tensor-type new physics.

Significance
The paper concludes that the U(1)XSSM is a viable and helpful framework for studying $b \to c\ell\nu$ processes. The presence of additional gauge bosons, extended Higgs sectors, and right-handed neutrinos provides rich flavor structures capable of modifying the Wilson coefficients of the effective Lagrangian. The study confirms that the model offers a feasible theoretical explanation for the discrepancies between experimental measurements and SM predictions, particularly by allowing for significant deviations in $R_{J/\psi}$ and $R_{\Lambda_c}$ through the interplay of specific SUSY parameters. The work underscores the importance of future precision measurements (e.g., at the Electron-Ion Collider) in constraining the Wilson coefficients and further testing the parameter space of such supersymmetric extensions.

Technical Summary: Lepton Flavor Universality in b→cℓνb \to c\ell\nub→cℓν within the U(1)XSSM

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Technical Summary: Lepton Flavor Universality in $b \to c\ell\nu$ within the U(1)XSSM