Study of the semileptonic decay $D_s^{+}\to f_0(980)e^{+}\nu$ and implications for $B_s^0\to J/\psi f_0$

Using $e^{+}{e}^{-}\to D_s^{-}{D}_s^{*+}$ and $D_s^{*-}{D}_s^{+}$ interactions at 4170 MeV collected with the CLEO-c detector, we investigate the semileptonic decays $D_s^{+}\to f_0(980)e^{+}\nu$ and $D_s^{+}\to \varphi e^{+}\nu$. By examining the decay rates as functions of the four-momentum transfer squared $q^2$, we measure the ratio $[\frac{d\mathcal{B}}{d{q}^2}(D_s^{+}\to f_0(980)e^{+}\nu )\mathcal{B}(f_0\to {\pi }^{+}{\pi }^{-})]/[\frac{d\mathcal{B}}{d{q}^2}(D_s^{+}\to \varphi e^{+}\nu )\mathcal{B}(\varphi \to K^{+}{K}^{-})]$ at $q^2$ of zero to be $(42\pm 11)\%$. This ratio has been predicted to equal the rate ratio $[\mathcal{B}(B_s\to J/\psi f_0)\mathcal{B}(f_0\to {\pi }^{+}{\pi }^{-})]/[\mathcal{B}(B_s\to J/\psi \varphi )\mathcal{B}(\varphi \to K^{+}{K}^{-})]$, thus indicating that the $CP$ eigenstate $J/\psi f_0$ could be useful for measuring $CP$ violation via $B_s$ mixing. Assuming a simple pole model for the form factor $|f_{+}(q^2)|$ in the $f_0{e}^{+}\nu$ decay, we find a pole mass of $({1.7}_{-0.7}^{+4.5}\pm 0.2)\mathrm{GeV}$. We also determine the $f_0$ mass and width as $({977}_{-9}^{+11}\pm 1)$ and $({91}_{-22}^{+30}\pm 3)\mathrm{MeV}$, respectively. In addition, we present updated results for $\mathcal{B}(D_s^{+}\to f_0(980)e^{+}\nu )\mathcal{B}(f_0\to {\pi }^{+}{\pi }^{-})=(0.20\pm 0.03\pm 0.01)\%$ and $\mathcal{B}(D_s^{+}\to \varphi e^{+}\nu )=(2.36\pm 0.23\pm 0.13)\%$. Assuming that the $f_0$ wave function is a combination of strange and nonstrange quark-antiquark components, we use our measurement for $\mathcal{B}(D_s^{+}\to f_0(980)e^{+}\nu )$ to extract a value of the mixing angle that we find consistent with $\mid \overline{s}s⟩$ dominance, adding to the mystery as to why the $f_0$ decays predominantly to two pions rather than two kaons.

Figure 1

The Feynman diagram for semileptonic $D_s^{+}$ decay into a $\varphi$ or $f_0(980)$ meson.

Figure 2

(a) Invariant mass of $D_s^{-}$ candidates summed over all decay modes and fit to a two-Gaussian signal shape plus a straight line for the background. The vertical dotted-dashed lines indicate the $\pm 17.5\mathrm{MeV}$ definition of the signal region. (b) The ${\mathrm{MM}}^{*2}$ distribution summed over all modes. The curves are fits to the number of signal events using the Crystal Ball function and two 5th order Chebyshev background functions; the dashed curve shows the background from fake $D_s^{-}$ tags, while the dotted curve in (b) shows the sum of the backgrounds from multiple photon combinations and fake $D_s^{-}$ tags. The vertical dashed lines show the region of events selected for further analysis.

Figure 3

${\mathrm{MM}}^2$ distributions from Monte Carlo simulation for (a) $D_s^{+}\to f_0{e}^{+}\nu$ and (b) $D_s^{+}\to \varphi e^{+}\nu$. The curves are the sum of two CB functions with means fixed at zero, $n$ fixed at 1.8 and $\alpha$ values constrained to be equal, and a single Gaussian shape. (The parameters are listed in Table 2.)

Figure 4

(a) The ${\mathrm{MM}}^2$ distribution for ${\pi }^{+}{\pi }^{-}{e}^{+}\nu$. The dotted curve shows the signal, the long-dashed distributions are from sidebands of the $D_s^{-}$ candidate invariant mass distributions while the shorter-dashed curve in (a) shows the ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\gamma$ background level, while the dashed-dotted curve shows the (small) ${\pi }^{+}{\pi }^{-}{\pi }^0$ background from $\varphi e^{+}\nu$. The solid curve shows the total. (b) The ${\mathrm{MM}}^2$ distribution for $K^{+}{K}^{-}{e}^{+}\nu$, where the total and the small sideband background is shown.

Figure 5

Invariant mass distributions for (a) ${\pi }^{+}{\pi }^{-}$ and (b) $K^{+}{K}^{-}$ for semileptonic decays. The dotted curve shows the signal, the long-dashed distributions are from sidebands of the $D_s^{-}$ candidate invariant mass distributions while the shorter-dashed curve in (a) shows the ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\gamma$ background level, and the dashed-dotted curve in (a) shows the (small) ${\pi }^{+}{\pi }^{-}{\pi }^0$ background from $\varphi e^{+}\nu$. The solid curves show the totals.

Figure 6

The invariant ${\pi }^{+}{\pi }^{-}$ mass in the semileptonic mode for the different $q^2$ intervals in units of ${\mathrm{GeV}}^2$: (a) 0–0.2, (b) 0.2–0.4, (c) 0.4–0.6, (d) 0.6–0.8 and (e) 0.8–2.0. The fits to the data are described in the text. The dotted curves show the signal, the long-dashed distributions are from sidebands of the $D_s^{-}$ candidate invariant mass distributions, while the shorter-dashed curves show the ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\gamma$ background level, and the dashed-dotted curves show the (small) ${\pi }^{+}{\pi }^{-}{\pi }^0$ background from $\varphi e^{+}\nu$. The solid curves show the total.

Figure 7

The invariant $K^{+}{K}^{-}$ mass in the semileptonic mode for the different $q^2$ intervals in units of ${\mathrm{GeV}}^2$: (a) 0–0.2, (b) 0.2–0.4, (c) 0.4–0.6, (d) 0.6–0.8 and (e) 0.8–1.0. The solid curve show the total and the long-dashed distributions are from sidebands of the $D_s^{-}$ candidate invariant mass distributions.

Figure 8

The $q^2$ distributions for (a) $D_s^{+}\to f_0{e}^{+}\nu$ fit to a function allowing a varying pole mass and (b) $D_s^{+}\to \varphi e^{+}\nu$ fit to a function with form factors fixed to those measured by BABAR [3].

Figure 9

Invariant mass distribution for ${\pi }^{+}{\pi }^{-}$ in the semileptonic mode. The dotted curve shows the signal shape, a relativistic Breit-Wigner modified by phase space and form factor effects, the long-dashed distributions are from sidebands of the $D_s^{-}$ candidate invariant mass distributions while the shorter-dashed curve shows the ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\gamma$ background level, and the dashed-dotted curve shows the (small) ${\pi }^{+}{\pi }^{-}{\pi }^0$ background from $\varphi e^{+}\nu$. The solid curve shows the total.