Investigating the nature of light scalar mesons with semileptonic decays of $D$ mesons

We study the semileptonic decays of $D_s^{+}$, $D^{+}$, and $D^0$ mesons into the light scalar mesons [$f_0(500)$, $K_0^{*}(800)$, $f_0(980)$, and $a_0(980)$] and the light vector mesons [$\rho (770)$, $\omega (782)$, $K^{*}(892)$, and $\varphi (1020)$]. With the help of a chiral unitarity approach in coupled channels, we compute the branching fractions for scalar meson processes of the semileptonic $D$ decays in a simple way. Using current known values of the branching fractions, we make predictions for the branching fractions of the semileptonic decay modes with other scalar and vector mesons. Furthermore, we calculate the ${\pi }^{+}{\pi }^{-}$, $\pi \eta$, $\pi K$, and $K^{+}{K}^{-}$ invariant mass distributions in the semileptonic decays of $D$ mesons, which will help us clarify the nature of the light scalar mesons.

Figure 1

(a) Semileptonic decay of $D_s^{+}$ into $l^{+}{\nu }_l$ and a primary $s\overline{s}$ pair. (b) Semileptonic decay of $D_s^{+}$ into $l^{+}{\nu }_l$ and two pseudoscalar mesons $P$ with a hadronization.

Figure 2

Diagrammatic representation of the direct plus rescattering processes for two pseudoscalar mesons. The solid and dashed lines denote quarks and pseudoscalar mesons, respectively. The shaded ellipses indicate the hadronization of a quark-antiquark pair into two pseudoscalar mesons, while the open circle indicates the rescattering of two pseudoscalar mesons.

Figure 3

Meson–meson invariant mass distributions for the semileptonic decay $D_s^{+}\to PP{e}^{+}{\nu }_e$ with $PP={\pi }^{+}{\pi }^{-}$, $K^{+}{K}^{-}$, and ${\pi }^{-}{K}^{+}$ in $s$ wave. We multiply the ${\pi }^{-}{K}^{+}$ mass distribution, which is a Cabibbo-suppressed process, by 10.

Figure 4

Meson–meson invariant mass distributions for the semileptonic decay $D^{+}\to PP{e}^{+}{\nu }_e$ with $PP={\pi }^{+}{\pi }^{-}$, ${\pi }^0\eta$, $K^{+}{K}^{-}$, and ${\pi }^{+}{K}^{-}$ in $s$ wave. We multiply the ${\pi }^{+}{\pi }^{-}$, ${\pi }^0\eta$, and $K^{+}{K}^{-}$ mass distributions, which are Cabibbo-suppressed processes, by 10.

Figure 5

Meson–meson invariant mass distributions for the semileptonic decay $D^0\to PP{e}^{+}{\nu }_e$ with $PP={\pi }^{-}\eta$, $K^0{K}^{-}$, and ${\pi }^{-}{\overline{K}}^0$ in $s$ wave. We multiply the ${\pi }^{-}\eta$ and $K^0{K}^{-}$ mass distributions, which are Cabibbo-suppressed processes, by 10.

Figure 6

${\pi }^{+}{\pi }^{-}$ invariant mass distribution for the semileptonic decay $D_s^{+}\to {\pi }^{+}{\pi }^{-}{e}^{+}{\nu }_e$. The theoretical calculation is folded with the size of experimental bins, 25 MeV. The experimental data are taken from Ref. [27] and are scaled so that the fitted Breit-Wigner distribution (dashed line) reproduces the branching fraction of $\mathcal{B}[D_s^{+}\to f_0(980)e^{+}{\nu }_e,f_0(980)\to {\pi }^{+}{\pi }^{-}]=0.2\%$ by the Particle Data Group (see Table 1).

Figure 7

Differential decay width of the $D_s^{+}\to f_0(980)e^{+}{\nu }_e$ decay mode followed by $f_0(980)\to {\pi }^{+}{\pi }^{-}$, with $q^2=[M_{\text{inv}}^{(l\nu )}{]}^2$. The experimental data are taken from Ref. [27]. The experimental points should be rescaled dividing by about a factor of two if the absolute rate for the $f_0(980)$ production of the reanalysis of Ref. [47] were used.

Figure 8

Ratio of the experimental to theoretical branching fractions for the semileptonic $D$ decays into vector mesons.

Figure 9

Differential decay width of the $D_s^{+}\to \varphi (1020)e^{+}{\nu }_e$ decay mode followed by $\varphi (1020)\to K^{+}{K}^{-}$, with $q^2=[M_{\text{inv}}^{(l\nu )}{]}^2$. The experimental data are taken from Ref. [27]. The theoretical value is multiplied by the branching fraction of the $\varphi (1020)$ meson to $K^{+}{K}^{-}$, $\mathcal{B}[\varphi (1020)\to K^{+}{K}^{-}]=48.9\%$ [1].

Figure 10

$K^{+}{K}^{-}$ invariant mass distribution for the semileptonic decay $D_s^{+}\to K^{+}{K}^{-}{e}^{+}{\nu }_e$ both in $s$ and $p$ waves.

Figure 11

${\pi }^{+}{K}^{-}$ invariant mass distribution for the semileptonic decay $D^{+}\to {\pi }^{+}{K}^{-}{e}^{+}{\nu }_e$ both in $s$ and $p$ waves.