Dₛ⁺ Radiative Decay: γK*(892)⁺ Insights

• The paper presents the first dedicated search for Dₛ⁺→γK*(892)⁺ decay, setting an upper limit on the branching fraction at 2.3×10⁻⁴ using advanced double-tag techniques.
• It explains how the short-distance c→uγ transition is suppressed by the GIM mechanism while long-distance weak annihilation and VMD effects can enhance the decay rate.
• The study employs rigorous signal extraction and systematic uncertainty analysis, laying the groundwork for future high-luminosity experiments in charm physics.

The radiative decay $D_s^+\to\gamma K^*(892)^+$ is a flavor-changing electromagnetic transition in the charm sector, representing a key probe of Standard Model (SM) processes and their long-distance and short-distance dynamics. This decay arises from $c\to u\gamma$ transitions, with potential enhancements from weak annihilation and vector-meson dominance (VMD) mechanisms. The first dedicated experimental search for $D_s^+\to\gamma K^*(892)^+$ has been performed by the BESIII Collaboration, utilizing a substantial $e^+e^-$ collision dataset and advanced double-tagging techniques to set an upper limit on the branching fraction at the $10^{-4}$ level (Collaboration et al., 23 Jan 2026).

1. Theoretical Framework and Physics Motivation

In the SM, the short-distance $c\to u\gamma$ transition is highly suppressed by Glashow–Iliopoulos–Maiani (GIM) mechanism, yielding a radiative branching ratio from perturbative “penguin” diagrams of only $$\mathcal{B}_{\mathrm{peng}}(D_s^+\to\gamma K^*(892)^+)\sim\mathcal{O}(10^{-8})$$ [Fajfer et al., Eur. Phys. J. C 6 (1999) 471]. However, long-range effects—predominantly from weak annihilation topologies with VMD or final-state rescattering—can enhance the rate by up to four orders of magnitude, leading to SM predictions for $\mathcal{B}(D_s^+\to\gamma K^*(892)^+)$ in the range $\mathcal{O}(10^{-4})$ [Altmannshofer & Archilli (2022), de Boer & Hiller JHEP 08 (2017) 091, Lyon & Zwicky Phys. Rev. D 106 (2022) 053001, Burdman et al. Phys. Rev. D 52 (1995) 6383]. The dominant long-distance mechanism proceeds via a weak annihilation ($c\bar{s}\to u\bar{d}$) followed by emission of a virtual vector meson that converts to a real photon via VMD. Specific model predictions (in units of $c\to u\gamma$0) for $c\to u\gamma$1 include:

• HSI+WA: $c\to u\gamma$2–$c\to u\gamma$3 [de Boer–Hiller]
• LCSR: $c\to u\gamma$4 [Lyon–Zwicky]
• Hybrid long-distance: $c\to u\gamma$5–$c\to u\gamma$6 [Fajfer–Singer]
• VMD: $c\to u\gamma$7–$c\to u\gamma$8 [Burdman et al.]

No observation at or above these levels would constrain the non-local hadronic mechanisms in the SM.

2. Experimental Dataset and Detector Description

The search for $c\to u\gamma$9 exploits an integrated luminosity of $D_s^+\to\gamma K^*(892)^+$0, collected by BESIII at center-of-mass energies of 4.128–4.226 GeV, partitioned into four data groups (4.128/4.157, 4.178, 4.189–4.219, and 4.226 GeV). The BESIII detector features:

• A 1 T solenoidal magnet.
• A multilayer drift chamber (MDC), providing momentum resolution ($D_s^+\to\gamma K^*(892)^+$1 at 1 GeV/$D_s^+\to\gamma K^*(892)^+$2) and $D_s^+\to\gamma K^*(892)^+$3 for charged particle identification.
• Time-of-flight (TOF) counters, with time resolutions of $D_s^+\to\gamma K^*(892)^+$4 (barrel) and $D_s^+\to\gamma K^*(892)^+$5 (endcap).
• A CsI(Tl) electromagnetic calorimeter (EMC), with energy resolution $D_s^+\to\gamma K^*(892)^+$6 (barrel) and $D_s^+\to\gamma K^*(892)^+$7 (endcap) at 1 GeV.
• Muon detection in the instrumented flux return.

These subsystems provide the necessary kinematic and PID information for high-efficiency charm hadron reconstruction.

3. Event Selection and Decay Reconstruction

BESIII employs a double-tag (DT) technique. On the tag side, $D_s^+\to\gamma K^*(892)^+$8 candidates are fully reconstructed via standard hadronic decay modes, using tight vertexing and PID in the MDC and TOF, as well as $D_s^+\to\gamma K^*(892)^+$9 and $e^+e^-$0 reconstruction with EMC information. The recoil mass against the single-tag $e^+e^-$1 ensures selection of events consistent with $e^+e^-$2 production:

$e^+e^-$3

requiring $e^+e^-$4 to match $e^+e^-$5 within specific windows.

On the signal side, candidate events require:

• The highest-energy photon in EMC not matched to a track ($e^+e^-$6 GeV).
• $e^+e^-$7 reconstruction via $e^+e^-$8 ($e^+e^-$9 GeV/$10^{-4}$0) or $10^{-4}$1 ($10^{-4}$2 GeV/$10^{-4}$3).
• Veto of extra $10^{-4}$4 or $10^{-4}$5 candidates to suppress backgrounds from $10^{-4}$6 and $10^{-4}$7.

The analysis does not rely on the more common $10^{-4}$8 and $10^{-4}$9 variables, but uses $c\to u\gamma$0 and tag mass.

4. Signal Extraction and Statistical Procedure

A simultaneous unbinned maximum-likelihood fit is performed in the two-dimensional space of $c\to u\gamma$1 for both $c\to u\gamma$2 decay channels, with the isospin-constrained ratio $c\to u\gamma$3. The total probability density function (PDF) is:

$c\to u\gamma$4

where:

• $c\to u\gamma$5 is derived from MC and convolved with a double-Gaussian resolution function.
• $c\to u\gamma$6 models backgrounds from $c\to u\gamma$7 events.
• $c\to u\gamma$8 accounts for continuum and combinatorial backgrounds.

The helicity angle $c\to u\gamma$9 distribution for the signal ($$\mathcal{B}_{\mathrm{peng}}(D_s^+\to\gamma K^*(892)^+)\sim\mathcal{O}(10^{-8})$$0) provides additional discrimination power. Signal efficiencies are determined by large-scale exclusive MC simulations for each tag mode and energy group, yielding an average double-tag efficiency of $$\mathcal{B}_{\mathrm{peng}}(D_s^+\to\gamma K^*(892)^+)\sim\mathcal{O}(10^{-8})$$1 (inclusive of $$\mathcal{B}_{\mathrm{peng}}(D_s^+\to\gamma K^*(892)^+)\sim\mathcal{O}(10^{-8})$$2 and $$\mathcal{B}_{\mathrm{peng}}(D_s^+\to\gamma K^*(892)^+)\sim\mathcal{O}(10^{-8})$$3 branching fractions).

5. Systematic Uncertainties

Multiplicative systematic uncertainties originate from tracking ($$\mathcal{B}_{\mathrm{peng}}(D_s^+\to\gamma K^*(892)^+)\sim\mathcal{O}(10^{-8})$$4 for $$\mathcal{B}_{\mathrm{peng}}(D_s^+\to\gamma K^*(892)^+)\sim\mathcal{O}(10^{-8})$$5), PID ($$\mathcal{B}_{\mathrm{peng}}(D_s^+\to\gamma K^*(892)^+)\sim\mathcal{O}(10^{-8})$$6 for $$\mathcal{B}_{\mathrm{peng}}(D_s^+\to\gamma K^*(892)^+)\sim\mathcal{O}(10^{-8})$$7), photon reconstruction ($$\mathcal{B}_{\mathrm{peng}}(D_s^+\to\gamma K^*(892)^+)\sim\mathcal{O}(10^{-8})$$8), $$\mathcal{B}_{\mathrm{peng}}(D_s^+\to\gamma K^*(892)^+)\sim\mathcal{O}(10^{-8})$$9 and $\mathcal{B}(D_s^+\to\gamma K^*(892)^+)$0 reconstruction ($\mathcal{B}(D_s^+\to\gamma K^*(892)^+)$1, $\mathcal{B}(D_s^+\to\gamma K^*(892)^+)$2 respectively), MC statistics ($\mathcal{B}(D_s^+\to\gamma K^*(892)^+)$3), and selection mass windows for $\mathcal{B}(D_s^+\to\gamma K^*(892)^+)$4 and $\mathcal{B}(D_s^+\to\gamma K^*(892)^+)$5 candidates. The overall relative uncertainties sum to $\mathcal{B}(D_s^+\to\gamma K^*(892)^+)$6 ($\mathcal{B}(D_s^+\to\gamma K^*(892)^+)$7 mode) and $\mathcal{B}(D_s^+\to\gamma K^*(892)^+)$8 ($\mathcal{B}(D_s^+\to\gamma K^*(892)^+)$9 mode). Additive uncertainties are estimated by varying fitting procedures, background shapes, and yields; the most conservative limit is adopted.

6. Results and Implications

No statistically significant signal is observed, with fit yields of $\mathcal{O}(10^{-4})$0 ($\mathcal{O}(10^{-4})$1) and $\mathcal{O}(10^{-4})$2 ($\mathcal{O}(10^{-4})$3) events. The $\mathcal{O}(10^{-4})$4 confidence-level upper limit on the branching fraction is

$\mathcal{O}(10^{-4})$5

as determined by integration of the profile likelihood ($\mathcal{O}(10^{-4})$6) convolved with the systematic uncertainty [Stenson physics/0605236]. This bound is above, but approaches, the upper edge of the predicted SM range for long-distance dominated processes ($\mathcal{O}(10^{-4})$7–$\mathcal{O}(10^{-4})$8). No theoretical scenario is excluded.

A plausible implication is that future high-luminosity flavor factories (Belle II, Super τ-Charm Facility) will be required to decisively access the SM-calculable regime for this decay. Progress in both experimental precision and theoretical control of long-distance contributions will be necessary for conclusive SM tests (Collaboration et al., 23 Jan 2026).