Measurement of $\mathcal{B}(D_s^{+}\to l^{+}\nu )$ and the decay constant $f_{D_S^{+}}$

We examine $e^{+}{e}^{-}\to D_s^{-}{D}_s^{*+}$ and $D_s^{*-}{D}_s^{+}$ interactions at 4170 MeV using the CLEO-c detector in order to measure the decay constant $f_{D_s^{+}}$. We use the $D_s^{+}\to {\ell }^{+}\nu$ channel, where the ${\ell }^{+}$ designates either a ${\mu }^{+}$ or a ${\tau }^{+}$, when the ${\tau }^{+}\to {\pi }^{+}\overline{\nu }$. Analyzing both modes independently, we determine $\mathcal{B}(D_s^{+}\to {\mu }^{+}\nu )=(0.594\pm 0.066\pm 0.031)\%$, and $\mathcal{B}(D_s^{+}\to {\tau }^{+}\nu )=(8.0\pm 1.3\pm 0.4)\%$. We also analyze them simultaneously to find an effective value of ${\mathcal{B}}^{\mathrm{eff}}(D_s^{+}\to {\mu }^{+}\nu )=(0.638\pm 0.059\pm 0.033)\%$ and extract $f_{D_s^{+}}=274\pm 13\pm 7\mathrm{MeV}$. Combining with our previous determination of $\mathcal{B}(D^{+}\to {\mu }^{+}\nu )$, we also find the ratio $f_{D_s^{+}}/f_{D^{+}}=1.23\pm 0.11\pm 0.04$. We compare to current theoretical estimates. Finally, we find $\mathcal{B}(D_s^{+}\to e^{+}\nu )<1.3\times {10}^{-4}$ at 90% confidence level.

Figure 1

The decay diagram for $D_s^{+}\to {\ell }^{+}\nu$.

Figure 2

The beam constrained mass $m_{\mathrm{BC}}$ from Monte Carlo simulation of $e^{+}{e}^{-}\to D_s^{+}{D}_s^{*-}$, $D_s^{\pm }\to \varphi {\pi }^{\pm }$ at an $E_{\mathrm{CM}}$ of 4170 MeV. The narrow peak is from the $D_s^{+}$ and the wider one from $D_s^{*-}\to \gamma D_s^{-}$. (The distributions are not centered at the $D_s^{+}$ or $D_s^{*+}$ masses, because the reconstructed particles are assumed to have the energy of the beam.)

Figure 3

Invariant mass of $D_s^{-}$ candidates in the decay modes (a) $K^{+}{K}^{-}{\pi }^{-}$, (b) $K_S{K}^{-}$, (c) $\eta {\pi }^{-}$, (d) ${\eta }^{\prime }{\pi }^{-}$, (e) $\varphi {\rho }^{-}$, (f) ${\pi }^{+}{\pi }^{-}{\pi }^{-}$, (g) $K^{*-}{K}^{*0}$, and (h) $\eta {\rho }^{-}$, after requiring the total energy of the $D_s^{-}$ candidate to be consistent with the beam energy. The curves are fits to two-Gaussian signal functions plus a polynomial background.

Figure 4

The ${\mathrm{MM}}^{*2}$ distribution from events with a photon in addition to the $D_s^{-}$ tag for the modes: (a) $K^{+}{K}^{-}{\pi }^{-}$, (b) $K_S^0{K}^{-}$, (c) $\eta {\pi }^{-}$, (d) ${\eta }^{\prime }{\pi }^{-}$, (e) $\varphi {\rho }^{-}$, (f) ${\pi }^{+}{\pi }^{-}{\pi }^{-}$, (g) $K^{*-}{K}^{*0}$, and (h) $\eta {\rho }^{-}$. The curves are fits to the Crystal Ball function and a 5th order Chebyshev background function.

Figure 5

The ${\mathrm{MM}}^{*2}$ distribution from a sample of fully reconstructed $D_s^{-}{D}_s^{*+}$ events where one $D_s$ is ignored. The curve is a fit to the Crystal Ball function.

Figure 6

The ${\mathrm{MM}}^2$ resolution from Monte Carlo simulation for $D_s^{+}\to {\mu }^{+}\nu$ utilizing a $\varphi {\pi }^{-}$ tag and a $\gamma$ from either $D_s^{*}$ decay, both before the kinematic fit (a) and after (b).

Figure 7

The ${\mathrm{MM}}^2$ distribution for events with an identified $K^{+}$ track. The kinematic fit has been applied. The curve is a fit to the sum of two Gaussians centered at the square of the $K^0$ mass and a linear background.

Figure 8

The ${\mathrm{MM}}^2$ distribution from Monte Carlo simulation for $D_s^{+}\to {\tau }^{+}\nu$, ${\tau }^{+}\to {\pi }^{+}\overline{\nu }$ at an $E_{\mathrm{CM}}$ of 4170 MeV.

Figure 9

The ${\mathrm{MM}}^2$ distributions from data for events with a $D_s^{-}$ reconstructed in a tag mode, an additional positively charged track and no neutral energy clusters above 300 MeV. Case (i): the single track deposits $<300\mathrm{MeV}$ of energy in the calorimeter. The peak near zero is from $D_s^{+}\to {\mu }^{+}\nu$ events. Case (ii): track deposits $>300\mathrm{MeV}$ in the crystal calorimeter but is not consistent with being an electron. Case (iii): the track is identified as an electron.

Figure 10

The invariant ${\pi }^{+}{\pi }^0$ mass. The upper curve shows a fit using a background polynomial plus Gaussian signal functions, where the width of the Gaussian is fixed to a value determined by Monte Carlo simulation. The lower curve shows just the background polynomial.

Figure 11

The sum of case (i) and case (ii) ${\mathrm{MM}}^2$ distributions (histogram) compared to the predicted shape (curve) for the sum of $D_s^{+}\to {\mu }^{+}\nu$ and ${\tau }^{+}\nu$, ${\tau }^{+}\to {\pi }^{+}\overline{\nu }$. The curve is normalized to the total number of events below ${\mathrm{MM}}^2<0.2{\mathrm{GeV}}^2$.