$V_{cs}$ from $D_s\to \varphi \ell \nu$ semileptonic decay and full lattice QCD

We determine the complete set of axial and vector form factors for the $D_s\to \varphi \ell \nu$ decay from full lattice QCD for the first time. The valence quarks are implemented using the highly improved staggered quark action and we normalize the appropriate axial and vector currents fully nonperturbatively. The $q^2$ and angular distributions we obtain for the differential rate agree well with those from the BABAR experiment and, from the total branching fraction, we obtain $V_{cs}=1.017(63)$, in good agreement with that from $D\to K\ell \nu$ semileptonic decay. We also find the mass and decay constant of the $\varphi$ meson in good agreement with experiment, showing that its decay to $K\overline{K}$ (which we do not include here) has at most a small effect. We include an Appendix on nonperturbative renormalization of the complete set of staggered vector and axial vector bilinears needed for this calculation.

Figure 1

Diagram to show the angles used for the differential rate for $D_s\to \varphi \ell \nu$. $\ell \nu$ are drawn in the virtual $W^{*}$ rest frame and $K^{+}{K}^{-}$ in the $\varphi$ rest frame. The angles are defined in the $D_s$ rest frame, however [3].

Figure 2

A schematic diagram of the 3-point function for $D_s\to \varphi \ell \nu$ decay. Different currents, $J$, are inserted at the vertex, as described in the text, to obtain specific form factors.

Figure 3

Upper plot: The difference in mass between the $s\overline{s}$ vector $\varphi$ and pseudoscalar ${\eta }_s$ as a function of sea light quark mass in units of the physical strange quark mass for coarse lattices (red open triangles) and fine (green open circle). The “experimental” result is plotted with a black burst. This is obtained from the experimental result for $M_{\varphi }$ and the lattice QCD result for $M_{{\eta }_s}$ in the continuum and chiral limits [11]. The gray shaded band gives the lattice result in the continuum and chiral limits from a simple fit described in the text. Lower plot: A similar plot for the $\varphi$ and ${\eta }_s$ decay constants. The black bursts denote the experimental result for the $\varphi$ obtained from its leptonic width [19] and the result from lattice QCD for the ${\eta }_s$ [11].

Figure 4

Lines and bands show the form factors from our lattice QCD calculation extrapolated to the physical point as a function of $q^2$ for $q^2=0$ to $q_{\max }^2$. The form factor $A_1(q^2)$ is shown in black, $A_2(q^2)$ in blue, $V(q^2)$ in red and $A_0(q^2)$ in green. We also plot the raw lattice results from Table 3 for each form factor. The bursts are the lattice data from Set 1, squares from Set 2 and circles Set 3. At $q_{\max }^2$, the $\varphi$ meson is at rest and only $A_1(q^2)$ contributes to the decay. We plot the lattice data for both the local and 1-link determinations of $A_1(q^2)$.

Figure 5

The $D_s\to \varphi \ell \nu$ helicity amplitudes constructed from our extrapolated form factors. They are shown as $p_{\varphi }{q}^2{|H_{\pm }(q^2)|}^2$ and $p_{\varphi }{q}^2{|H_0(q^2)|}^2$, including the kinematic factors that appear in the differential decay rate. At the $q^2=q_{\max }^2$ end of the distribution, $p_{\varphi }\to 0$ and each of the helicity amplitudes vanishes.

Figure 6

Helicity amplitude contributions to $D_s\to \varphi \ell \nu$ which are suppressed by the lepton mass. Here we plot the contributions from $H_0(q^2)$ and $H_t(q^2)$ which are enhanced at small values of $q^2$. They are plotted as they appear in the differential decay rate for muons, i.e. $m_{\mu }^2{p}_{\varphi }{|H_i(q^2)|}^2$. The scale is the same as in Fig. 5. For semileptonic decay with electrons in the final state, these contributions are further suppressed.

Figure 7

Decay distributions for $D_s\to \varphi \ell \nu$ with $\varphi \to K^{+}{K}^{-}$ for each of the kinematic variables in the decay. The decay angles are shown in Fig. 1 and described in the text. The lattice results are shown with the red points with error bars and the experimental results by blue blocks. The experimental errors are not plotted, but are of similar size to the lattice errors.

Figure 8

Comparison of lattice QCD with CLEO’s determination [6] of the helicity amplitudes for $D\to K^{*}$. We assume that the form factors are insensitive to the spectator quark mass, so we can construct the $D\to K^{*}$ helicity amplitudes using the same form factors as $D_s\to \varphi$. The data plotted is normalized such that $q^2{|H_0(q^2)|}^2\to 1$ as $q^2\to 0$.

Figure 9

Comparison of the values of $V_{cs}$ obtained here from $D_s\to \varphi$ decay and those from $D_s$ leptonic decay [8] and $D\to K$ semileptonic decay [1]. We also show the result from CKM unitarity [19].

Figure 10

The ratio of axial vector and pseudoscalar amplitudes plotted against meson momentum in lattice units for pseudoscalar mesons made from $c$ and $s$ valence quarks. The results come from coarse lattices, set 2. This ratio is proportional to the momentum and the renormalization factor for the axial vector operator can be extracted from the gradient of the line, given by $\frac{m_{01}+m_{02}}{M_{P_0}^2}\frac{1}{Z}$.

Figure 11

The ratio of the axial vector to pseudoscalar amplitudes in the case where the charm and strange propagators carry the same phase at the boundary, $\theta$. In this case, the total meson momentum is zero and we expect this ratio to be zero. The ratio is plotted against the momentum carried (in opposite directions) by each propagator. The blue squares are for the coarse Set 2 and the red circles for fine Set 3. These gauge configurations have approximately the same physical size, so the same $\theta$ corresponds to the same physical momentum on each.

Figure 12

The ratio of the axial vector to pseudoscalar amplitudes in the case where the charm and strange propagators carry the same twist. In this case, the total meson momentum is zero and we expect this ratio to be zero. The ratio is plotted against $a^2$. Sets 2 and 3 have different lattice spacing, but the same physical size so the same twist $\theta$ on each corresponds to the same physical momentum. The red and blue data points are for the cases in which both the strange and charm propagators carry $\theta =4,7$.

Figure 13

The ratio of axial vector to pseudoscalar fit amplitudes plotted against $D_s$ total momentum (given by the difference of phases carried by the $c$ and $s$ quarks), where the 1-link axial vector operator is given by Eq. (b4). The different symbols indicate the momentum (phase at the boundary) carried by the charm quark. Results are for the coarse lattices, set 2. Now the amplitude ratio agrees for points where the meson total momentum is same.