Radiative and Dalitz decays of $\Upsilon(1S)$ in the light of the ATOMKI X17 anomaly

Technical Summary: Radiative and Dalitz Decays of $\Upsilon(1S)$ in the Light of the ATOMKI X17 Anomaly

Problem Statement
The ATOMKI Collaboration has reported a persistent anomaly in nuclear transitions ($^8$Be, $^4$He, $^{12}$C) characterized by an excess of electron-positron pairs at an invariant mass of approximately 17 MeV. This "X17" signal suggests the existence of a new light vector boson beyond the Standard Model (SM). While previous theoretical efforts have utilized Vector Meson Dominance (VMD) and Heavy Quark Effective Theory (HQET) to model Dalitz decays ($V \to P e^+ e^-$) in the charm sector, conflicting results and model limitations have hindered definitive conclusions. Specifically, recent BESIII observations of a $3.5\sigma$ excess in $D^{*0} \to D^0 e^+ e^-$ remain unexplained by current X17 hypotheses using standard models. This study addresses the need for independent theoretical explorations in the bottomonium sector to test the consistency of the X17 hypothesis at a higher mass scale and different kinematic regime.

Methodology
The authors employ the Covariant Confined Quark Model (CCQM) to calculate the radiative decay $\Upsilon(1S) \to \eta_b \gamma$ and the Dalitz decay $\Upsilon(1S) \to \eta_b e^+ e^-$. Unlike approaches relying on HQET or VMD, the CCQM is a phenomenological quark model based on an interacting Lagrangian between hadrons and their constituent quarks.

Key methodological components include:

• Formalism: The calculation utilizes the triangle Feynman diagram for the hadronic transition. The invariant matrix elements are parametrized by transition form factors $G_\gamma(q^2)$ and $G_X(q^2)$ for the photon and X17 boson, respectively.
• Model Parameters: The model employs Gaussian vertex functions to describe meson size, with parameters ($m_b$, $\Lambda_\Upsilon$, $\Lambda_{\eta_b}$, $\lambda$) fixed by fitting to experimental data for leptonic and electromagnetic decay constants. The bottom quark mass is set to $m_b = 5.05$ GeV.
• X17 Coupling: The study investigates the contribution of a hypothetical vector boson $X17$. In the absence of direct data for the third generation, the authors assume flavor universality within the down-type sector, setting the coupling $\epsilon_b = \epsilon_d = -2.5 \times 10^{-3}$, consistent with constraints from ATOMKI and NA48/2 experiments.
• Interference: The interference between the SM photon amplitude and the X17 amplitude is neglected due to the narrow width of the X17 boson, allowing the total width to be approximated as the sum of individual contributions ($\Gamma \approx \Gamma_\gamma + \Gamma_X$).

Key Contributions and Results
The paper provides the first CCQM-based predictions for the $\Upsilon(1S) \to \eta_b$ transition form factors and decay rates, offering a rigorous benchmark for bottomonium physics.

1. Standard Model Predictions:

   • Radiative Decay: The predicted decay width is $\Gamma(\Upsilon(1S) \to \eta_b \gamma) = 9.3(9)$ eV, corresponding to a branching fraction of $1.7(2) \times 10^{-4}$. These results show good agreement with other theoretical frameworks, including NRQCD ($\Gamma \approx 8.95$ eV) and Light-Front Quark Models ($\Gamma \approx 9.29$ eV).
   • Dalitz Decay: The predicted width is $\Gamma(\Upsilon(1S) \to \eta_b e^+ e^-) = 4.8(5) \times 10^{-2}$ eV, with a branching fraction of $8.9(9) \times 10^{-7}$.
   • Form Factors: The form factors are parametrized by a single-pole function, with $G_\gamma(0) = -0.128$ and $G_X(0) = -0.962 \times 10^{-3}$.

2. X17 Contribution Analysis:

   • The study calculates the ratio of the Dalitz decay rate to the radiative decay rate ($R_{ee}$).
   • The SM contribution is predicted to be $R^\gamma_{ee} = 5.1(5) \times 10^{-3}$.
   • The X17 contribution is predicted to be $R^X_{ee} = 5.0(5) \times 10^{-5}$.
   • The X17 contribution is found to be two orders of magnitude smaller than the SM background.

3. Comparative Sensitivity:

   • The authors compare the sensitivity of $\Upsilon(1S) \to \eta_b e^+ e^-$ to X17 effects against other channels. The sensitivity is found to be:
     • Similar to $B^{*0}_{(s)} \to B^{0}_{(s)} e^+ e^-$.
     • Larger than $D^{*0} \to D^0 e^+ e^-$, $B^{*+} \to B^+ e^+ e^-$, and $J/\psi \to \eta_c e^+ e^-$.
     • Smaller than $D^{*+}_{(s)} \to D^{+}_{(s)} e^+ e^-$.

Significance and Claims
The paper positions itself as a necessary extension of previous CCQM studies into the bottomonium sector. The authors claim that their work provides:

• Rigorous Hadronic Calculations: By avoiding VMD and HQET approximations, the CCQM offers a more fundamental calculation of the hadronic form factors required to definitively probe for X17 signals.
• Benchmark for Experiment: The calculated branching fractions and decay constants serve as a reference for upcoming experimental investigations at B-factories (specifically mentioning Belle II and LHCb).
• Sensitivity Assessment: The study concludes that while the $\Upsilon(1S) \to \eta_b e^+ e^-$ channel is a "promising probe" for the X17 vector boson, the X17 contribution is significantly suppressed relative to the SM background. This suggests that while the channel is viable for study, the signal-to-background ratio is challenging, requiring precise measurements to distinguish potential New Physics effects.

The authors maintain a modest tone, noting that the specific determination of the $b$-quark coupling ($\epsilon_b$) remains impossible without direct experimental data from B-meson or bottomonium Dalitz decays, and their assumption of universality serves as a standard baseline for sensitivity assessment.