Lattice QCD calculation of the $D_s^{*}$ radiative decay with ($2+1$)-flavor Wilson-clover ensembles

We perform a lattice calculation on the radiative decay of $D_s^{*}$ using the ($2+1$)-flavor Wilson-clover gauge ensembles generated by CLQCD collaboration. A method allowing us to calculate the form factor with zero transfer momentum is proposed and applied to the radiative transition $D_s^{*}\to D_s\gamma$ and the Dalitz decay $D_s^{*}\to D_s{e}^{+}{e}^{-}$. After a continuum extrapolation using three lattice spacings, we obtain $\mathrm{\Gamma }(D_s^{*}\to D_s\gamma )=0.0549(54)\mathrm{keV}$, where the error is purely statistical. The result is consistent with previous lattice calculations but with a error reduced to only a fifth of the before. The Dalitz decay rate is also calculated for the first time and the ratio with the radiative transition is found to be $R_{ee}=0.624(3)\%$. A total decay width of $D_s^{*}$ can then be determined as 0.0587(54) keV taking into account the experimental branching fraction. Combining with the most recent experimental measurement on the branching fraction of the purely leptonic decay $D_s^{+,*}\to e^{+}{\nu }_e$, we obtain the quantity $f_{D_s^{*}}|V_{cs}|=({190.5}_{-41.7_{\mathrm{stat}}}^{+55.1}\pm {12.6}_{\mathrm{syst}})\mathrm{MeV}$, where the stat. is only the statistical error from the experiment, and syst. results from the experimental systematic uncertainty and the lattice statistical error. Our result leads to an improved systematic uncertainty compared to ${42.7}_{\mathrm{syst}}$ obtained using previous lattice prediction of total decay width 0.070(28) keV as the input.

Figure 1

For the ensembles of C24P29, C32P30, and C48P32 from left to right, the mass spectra of $D_s^{*}$ and $D_s$ particles extracted from the two-point functions calculated using wall source propagators (top) and the energy levels of $D_s$ particle with different momentum $\overrightarrow{p}=2\pi \overrightarrow{n}/L$, $|\overrightarrow{n}{|}^2=0$, 1, 2, 3, 4 extracted from two-point functions calculated using point source propagators (bottom). The horizontal gray bands denote the fitting ranges.

Figure 2

The dispersion relation of $D_s$ meson for the ensembles C24P29, C32P30, and C48P32, respectively.

Figure 3

The transition factors $V_{c/s}(q^2)$ with different momentum $\overrightarrow{p}=2\pi \overrightarrow{n}/L$, $|\overrightarrow{n}{|}^2=0$, 1, 2, 3, 4. The horizontal gray bands denote the fitting results.

Figure 4

The momentum extrapolation for transition factors $V_{\mathrm{eff}}(q^2)$ with different momentum $\overrightarrow{p}=2\pi \overrightarrow{n}/L$, $|\overrightarrow{n}{|}^2=0$, 1, 2, 3, 4. The black squares denote the on-shell transition factor $V_{\mathrm{eff}}(0)$.

Figure 5

The lattice results of $V_{\mathrm{eff}}(0)$ as a function of the lattice spacing. The errors of lattice spacing have been included in the continuous limit, which are presented by the horizontal error bars. The symbols of the blue cross and black triangle indicate the results given by HPQCD and this work, respectively.

Figure 6

The lattice results of $R_{ee}$ as a function of the lattice spacing. The black triangle denotes the result in the continuous limit $a^2\to 0$.