Characterize the properties of $D_s^+$-meson decay constant and leptonic decays by using QCD sum rules within background field theory framework

Technical Summary: Characterization of $D_s^+$-Meson Decay Constant and Leptonic Decays via QCD Sum Rules

Problem Statement
The leptonic decay of the $D_s^+$-meson serves as a critical platform for the precise extraction of the Cabibbo–Kobayashi–Maskawa (CKM) matrix element $|V_{cs}|$. While experimental measurements of branching fractions for the $\mu^+\nu_\mu$ and $\tau^+\nu_\tau$ channels have improved, the theoretical determination of the decay constant $f_{D_s^+}$ remains a limiting factor. Existing experimental determinations of $f_{D_s^+}$ show unresolved discrepancies, and theoretical predictions from Lattice QCD and QCD Sum Rules (QCDSR) vary, with Lattice results generally clustering lower (246–250 MeV) than some experimental averages. Furthermore, previous QCDSR calculations often relied on massless approximations for the strange quark or lacked higher-order condensate contributions, introducing potential deviations in the theoretical prediction.

Methodology
The authors employ the QCD sum rules approach within the background field theory framework to calculate the $D_s^+$ decay constant. The methodology involves the following key steps:

1. Correlation Function and OPE: A two-point correlation function is constructed using an interpolating current $J(x) = \bar{c}(x)\not{z}\gamma_5 s(x)$. The Operator Product Expansion (OPE) is performed in the deep Euclidean region, incorporating non-perturbative vacuum effects via the background field expansion of the quark propagator.
2. Condensate Inclusion: The calculation fully includes quark propagator contributions up to dimension-six condensates. Crucially, the authors retain the strange quark mass ($m_s$) effects throughout the calculation, avoiding the massless approximation often used for $D^+$ mesons, which is deemed insufficient for the heavier $D_s^+$ system.
3. Infrared Regularization: Dimensional regularization ($D=4-2\epsilon$) combined with Feynman parametrization is used to handle infrared divergences arising from the vanishing light-quark mass limit. These divergences are renormalized into the decay constant using the $\overline{MS}$ scheme.
4. Constraint Schemes: To determine the Borel window ($M^2$) and minimize dependence on the Borel parameter, two distinct constraint schemes are applied:
   • Scheme I (Conventional): Adheres to standard criteria: continuum contribution $<35\%$, dimension-six condensate contribution $<1\%$, and stability of $f_{D_s^+}$ within the window.
   • Scheme II (Derivative): Introduces an auxiliary function $\xi(1/M^2)$ and utilizes the derivative condition $X_{mD_s^+} = m_{D_s^+}^2$ (where the decay constant is ideally independent of $M^2$). This scheme reduces the weight of higher-dimensional condensates and determines the Borel window without prior assumptions on its boundaries.
5. Branching Fractions and $|V_{cs}|$: Using the calculated decay constants, the authors compute branching fractions for $D_s^+ \to \ell^+\nu_\ell$ ($\ell=e, \mu, \tau$), incorporating Next-to-Leading Order (NLO) electroweak radiative corrections. These are then used to extract $|V_{cs}|$ using the latest PDG branching fraction data.

Key Results
The study yields the following numerical results:

• Decay Constants:
  • Scheme I: $f_{D_s^+}^{(I)} = 253.0^{+3.3}_{-3.1}$ MeV.
  • Scheme II: $f_{D_s^+}^{(II)} = 251.8^{+1.4}_{-1.3}$ MeV.
  • The Scheme II result demonstrates significantly reduced sensitivity to the Borel parameter (variation of $\pm 0.6\%$ vs. $\pm 1.3\%$ in Scheme I) and a smaller uncertainty range.
• Branching Fractions:
  • Predictions for $D_s^+ \to \mu^+\nu_\mu$ and $D_s^+ \to \tau^+\nu_\tau$ are consistent with recent BESIII and HFLAV averages.
  • For the helicity-suppressed $D_s^+ \to e^+\nu_e$ channel, the predicted branching fractions are $(1.267^{+0.033}_{-0.031}) \times 10^{-7}$ (Scheme I) and $(1.255^{+0.014}_{-0.013}) \times 10^{-7}$ (Scheme II), aligning with Lattice QCD predictions.
• CKM Matrix Element $|V_{cs}|$:
  • Extracted from the $D_s^+ \to \mu^+\nu_\mu$ channel using the PDG'26 branching fraction:
    • $|V_{cs}|^{(I)} = 0.967 \pm 0.012$
    • $|V_{cs}|^{(II)} = 0.970 \pm 0.005$

Significance and Claims
The paper claims that the inclusion of dimension-six condensates and the explicit treatment of the strange quark mass within the background field framework lead to a more complete and reliable determination of $f_{D_s^+}$.

• Validation: Both calculated decay constants are in good agreement with existing experimental results (specifically CLEO'09 and BESIII'24) and theoretical averages (FLAG'24, HFLAV'24, PDG'26).
• Methodological Improvement: The derivative constraint scheme (Scheme II) is highlighted as a robust method for reducing the subjective uncertainty associated with Borel window selection, yielding a more stable and precise result.
• Standard Model Consistency: The extracted values of $|V_{cs}|$ are consistent with the latest world averages and other experimental determinations, supporting the validity of the Standard Model in this sector.
• Utility: The precise predictions for the $e^+\nu_e$ channel provide a theoretical reference for future experimental searches, which are currently limited by helicity suppression.

The authors conclude that their work provides a valuable reference for probing new physics beyond the Standard Model by offering a refined theoretical baseline for purely leptonic $D_s^+$ decays.