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

Abstract: The leptonic decays of the $D_s<sup>+$-meson have received considerable attention in recent years. In this work, we perform a precise calculation of the decay constant $f_{D_s<sup>+}$ using the QCD sum rules method within the background field theory framework. In our calculation, we fully include the quark propagator contributions up to dimension-six condensates. By adopting two different constraint schemes, we obtain $f_{D_s^{+}^{\text{(I)}}} = 253.0_{-3.1}^{+3.3}\ \text{MeV}$ and $f_{D_s^{+}^{{\text{(II)}}}} = 251.8_{-1.3}^{+1.4}\ \text{MeV}$, respectively, both of which are in good agreement with existing theoretical and experimental results. The conventional scheme follows the standard Borel window criteria, while the derivative scheme reduces the dependence of the decay constant on the Borel parameter through an auxiliary function. Based on these decay constants and incorporating the NLO electroweak radiative corrections, we further calculate the branching fractions for the three leptonic decay channels for both schemes. Combined with the latest branching fraction $$\mathcal{B}(D_s⁺ \to μ⁺ ν<em>μ)$$ from the PDG, we extract the CKM matrix elements $$|V</em>{cs}|^{\text{(I)}} = 0.967 \pm 0.012$$ and $|V_{cs}|^{{\text{(II)}}} = 0.970 \pm 0.005$ from the two schemes, respectively.

• The paper Uses QCD sum rules with background field theory (BFT) to determine the decay constant of $D_s^+$ mesons, yielding f_{D_s^+}= 253.0+3.3−3.1 for conventional and fD_s^+= 251.8+1.4−1.3 for scheme II, improving upon previous inaccuracies.
• The analysis includes a full quark propagator expansion and handles infrared divergences through dimensional regularization, focusing on the non-perturbative nature of massive $s$-quarks.
• The research predicts leptonic branching fractions and extracts $|V_{cs}| = 0.967–0.970$, laying groundwork for future experimental measurements and theoretical consistency in the BFT context.

Motivation and context

The purely leptonic decays $D_s^+ \to \ell^+\nu_\ell$ proceed through $c\bar{s}$ annihilation into a virtual $W^+$ boson and depend on a single non-perturbative quantity, the decay constant $f_{D_s^+}$, making them a clean channel for extracting the CKM element $|V_{cs}|$. The paper addresses two persistent problems: (i) the spread among experimental determinations of $f_{D_s^+}$ — from Belle's early $275 \pm 16 \pm 12$ MeV through CLEO-c to BESIII's $251.1 \pm 2.4 \pm 3.0$ MeV — remains unresolved; and (ii) within QCD sum rules, previous results range from roughly $240$ to $259$ MeV depending on perturbative order and treatment of the strange-quark mass. The authors perform a QCD sum rule calculation in the background field theory (BFT) framework, retaining the full quark propagator expansion up to dimension-six condensates and keeping the finite $c\bar{s}$0-quark mass throughout.

Theoretical framework

The analysis uses the pseudoscalar interpolating current $c\bar{s}$1 coupled to the correlator $c\bar{s}$2, chosen to suppress higher-twist and excited-state contamination in the BFT setup. The quark propagators are expanded in background gluon fields, generating contributions from $c\bar{s}$3, $c\bar{s}$4, $c\bar{s}$5, $c\bar{s}$6, and $c\bar{s}$7. Infrared divergences from the light-quark momentum region are regularized dimensionally ($c\bar{s}$8), evaluated via hypergeometric functions, and renormalized into the decay constant in the $c\bar{s}$9 scheme. The authors emphasize that the massless-light-quark approximation used for the $W^+$0 is inadequate here because of the sizable $W^+$1, so the full $W^+$2-quark mass dependence is retained in all spectral densities.

Two Borel-window constraint schemes

Scheme I follows conventional criteria: continuum contribution below 35%, dimension-six condensate contribution below 1%, and plateau stability of $W^+$3 versus $W^+$4. With $W^+$5 (near the squared mass of the first excited state $W^+$6), the authors find that the finite $W^+$7-quark mass suppresses the dimension-six term to $W^+$8, so the lower bound of the window cannot be fixed by the convergence criterion; they instead take a value $W^+$9 below the upper bound as central. This yields $f_{D_s^+}$0 and

$f_{D_s^+}$1

Scheme II introduces an auxiliary function $f_{D_s^+}$2 and imposes the condition $f_{D_s^+}$3, which holds when $f_{D_s^+}$4. This determines the Borel window without prior assumptions about its boundaries and further suppresses higher-dimensional condensates (to $f_{D_s^+}$5). With $f_{D_s^+}$6 and $f_{D_s^+}$7,

$f_{D_s^+}$8

Over equal window widths of $f_{D_s^+}$9, the variation of $|V_{cs}|$0 is about $|V_{cs}|$1 in scheme I but only $|V_{cs}|$2 in scheme II, demonstrating the improved stability of the derivative constraint. Both results are consistent with HFLAV'24 and PDG'26 averages; scheme I agrees with CLEO'09 and BESIII'24, while lattice QCD predictions (roughly 246.8–249.9 MeV) sit systematically below both.

Branching fractions and extraction of $|V_{cs}|$3

Using the standard SM expression for the helicity-suppressed leptonic rate, including the Sirlin short-distance electroweak correction of $|V_{cs}|$4, the authors predict:

Channel	Scheme I	Scheme II
$|V_{cs}|$5	$|V_{cs}|$6	$|V_{cs}|$7
$|V_{cs}|$8	$|V_{cs}|$9	$f_{D_s^+}$0
$f_{D_s^+}$1	$f_{D_s^+}$2	$f_{D_s^+}$3

The $f_{D_s^+}$4 and $f_{D_s^+}$5 predictions agree with the latest PDG and HFLAV averages and with BESIII measurements; the $f_{D_s^+}$6 channel prediction matches the LQCD'19 central value and serves as guidance for future searches, since only upper limits exist experimentally. Combining $f_{D_s^+}$7 with the PDG'26 value of $f_{D_s^+}$8 gives

$f_{D_s^+}$9

both consistent with FLAG'24, HFLAV'24, and PDG'26. Notably, the scheme II uncertainty ($275 \pm 16 \pm 12$0) is competitive with the world averages themselves, indicating that the theoretical error from the sum-rule side no longer dominates this extraction. The comparison also highlights a mild tension: recent BESIII extractions from the $275 \pm 16 \pm 12$1 channel trend upward (up to $275 \pm 16 \pm 12$2 combined in 2026), while older lattice-based extractions cluster near $275 \pm 16 \pm 12$3.

Limitations and open questions

Several caveats bear directly on the precision claims. First, in scheme I the lower edge of the Borel window cannot be set by OPE convergence because the dimension-six contribution is already negligible; the choice of taking $275 \pm 16 \pm 12$4 slightly below the upper bound is a pragmatic prescription rather than a derived criterion. Second, the derivative scheme II requires choosing the continuum threshold by an external criterion ($275 \pm 16 \pm 12$5), so threshold dependence is not eliminated but transferred. Third, the calculation includes condensates only up to dimension six and perturbative corrections at leading order in $275 \pm 16 \pm 12$6; earlier work showed that $275 \pm 16 \pm 12$7 terms can shift the central value by nearly 20 MeV, so higher-order perturbative corrections remain an unquantified systematic. Finally, the residual tension between lattice QCD values (~247–250 MeV) and the present sum-rule results (~252–253 MeV), and between different BESIII $275 \pm 16 \pm 12$8 extractions, is left unresolved.

Conclusion

This work provides a complete dimension-six QCD sum rule determination of $275 \pm 16 \pm 12$9 in the background field framework with explicit $251.1 \pm 2.4 \pm 3.0$0-quark mass effects, obtaining $251.1 \pm 2.4 \pm 3.0$1 MeV under conventional constraints and $251.1 \pm 2.4 \pm 3.0$2 MeV under a new derivative-based stability criterion that halves the Borel-parameter sensitivity. The resulting branching fractions for the $251.1 \pm 2.4 \pm 3.0$3 and $251.1 \pm 2.4 \pm 3.0$4 channels agree with current experimental averages, and the extracted $251.1 \pm 2.4 \pm 3.0$5–$251.1 \pm 2.4 \pm 3.0$6 is consistent with global fits at percent-level precision. The main open issues are the unquantified impact of higher-order perturbative corrections and the persistent few-MeV offset relative to lattice QCD determinations.