Scalar and vector form factors of $D\to \pi (K)\ell \nu$ decays with $N_f=2+1+1$ twisted fermions

We present a lattice determination of the vector and scalar form factors of the $D\to \pi \ell \nu$ and $D\to K\ell \nu$ semileptonic decays, which are relevant for the extraction of the CKM matrix elements $|V_{cd}|$ and $|V_{cs}|$ from experimental data. Our analysis is based on the gauge configurations produced by the European Twisted Mass Collaboration with $N_f=2+1+1$ flavors of dynamical quarks, at three different values of the lattice spacing ($a\simeq 0.062,0.082,0.089\mathrm{fm}$) and with pion masses as small as 210 MeV. Quark momenta are injected on the lattice using nonperiodic boundary conditions. The matrix elements of both vector and scalar currents are determined for plenty of kinematical conditions in which parent and child mesons are either moving or at rest. Lorentz symmetry breaking due to hypercubic effects is clearly observed in the data and included in the decomposition of the current matrix elements in terms of additional form factors. After the extrapolations to the physical pion mass and to the continuum limit, we determine the vector and scalar form factors in the whole kinematical region from $q^2=0$ up to $q_{\max }^2=(M_D-M_{\pi (K)}{)}^2$ accessible in the experiments, obtaining a good overall agreement with experiments, except in the region at high values of $q^2$ where some deviations are visible. A set of synthetic data points, representing our results for $f_{+}^{D\pi (K)}(q^2)$ and $f_0^{D\pi (K)}(q^2)$ for several selected values of $q^2$, is provided and also the corresponding covariance matrix is available. At zero four-momentum transfer, we get $f_{+}^{D\to \pi }(0)=0.612(35)$ and $f_{+}^{D\to K}(0)=0.765(31)$. Using the experimental averages for $|V_{cd}|f_{+}^{D\to \pi }(0)$ and $|V_{cs}|f_{+}^{D\to K}(0)$, we extract $|V_{cd}|=0.2330(137)$ and $|V_{cs}|=0.945(38)$, respectively. The second row of the CKM matrix is found to be in agreement with unitarity within the current uncertainties: $|V_{cd}{|}^2+|V_{cs}{|}^2+|V_{cb}{|}^2=0.949(78)$.

Figure 1

Matrix elements $⟨{\widehat{V}}_{\mathrm{sp}}{⟩}_{\mathrm{imp}}$, $⟨{\widehat{V}}_0{⟩}_{\mathrm{imp}}$ and $⟨S{⟩}_{\mathrm{imp}}$ for the $D\to \pi$ case extracted from the ratios $R_{\mu }$ and $R_S$ [see Eqs. (19) and (20)] for the ensemble D20.48 with $\beta =2.10$, $L/a=48$, $a{\mu }_{\ell }=0.0020$, $a{\mu }_c=0.17$, ${\overrightarrow{p}}_D=-{\overrightarrow{p}}_{\pi }$ and $|{\overrightarrow{p}}_D|\simeq 150\mathrm{MeV}$. The meson masses are $M_{\pi }\simeq 254$ and $M_D\simeq 1933\mathrm{MeV}$. The horizontal red lines correspond to the plateau regions used to extract the matrix elements and to their central values and statistical errors.

Figure 2

Momentum dependence of the vector $f_{+}^{D\pi }$ (upper left panel), $f_{+}^{DK}$ (lower left panel) and scalar $f_0^{D\pi }$ (upper right panel), $f_0^{DK}$ (lower right panel) form factors in the case of the gauge ensemble A60.24 [18]. Different markers and colors distinguish different values of the child meson momentum. The simulated meson masses are $M_{\pi }\simeq 386$, $M_K\simeq 599$ and $M_D\simeq 2022\mathrm{MeV}$, while the lattice spacing and spatial size are $a\simeq 0.0885\mathrm{fm}$ and $L\simeq 2.13\mathrm{fm}$, respectively (see Table 1).

Figure 3

Results for the vector (left panels) and scalar (right panels) form factors in the case of the $D\to \pi$ (upper panels) and $D\to K$ (lower panels) semileptonic decays versus $q^2$ for the gauge ensembles A40.24 and A40.32, which correspond to $a\simeq 0.0885\mathrm{fm}$, $M_{\pi }\simeq 320$, $M_K\simeq 580$, $M_D\simeq 2020\mathrm{MeV}$ and two different lattice volumes $L/a=24$ (empty markers) and $L/a=32$ (filled markers), respectively. The different shape and color of the markers distinguish between different values of the child meson momentum.

Figure 4

Momentum dependence of the vector (left panel) and scalar (right panel) form factors regulating the semileptonic decay in which the parent and child mesons are two charmed PS mesons, $D_1$ and $D_2$, with masses close to the $D$-meson one. In this plot we have used the gauge ensemble A30.32, in which $D_1$ and $D_2$ have masses equal to 1718 and 1887 MeV, respectively. Different markers and colors distinguish different values of the child meson momentum.

Figure 5

Results for ${\mathrm{\Delta }}_{WTI}^{\mathrm{hyp}}$ [see Eq. (42)] versus $q^2$ (left panels) and $q^{[4]}$ (right panels) for the $D\to \pi$ (upper panels) and $D\to K$ (lower panels) transitions in the case of the gauge ensemble A30.32.

Figure 6

The same as in Fig. 2, but after correcting for the hypercubic effects determined in the global fitting procedure using Eqs. (33) and (39). Different markers and colors distinguish different values of the child meson momentum.

Figure 7

Left panel: the scalar form factor $f_0^{D\pi }(q^2)$ corresponding to the kinematical conditions with the $D$-meson at rest for the gauge ensemble D30.48. Hollow and filled points represent, respectively, the data before and after the removal of the hypercubic effects determined in the global fitting procedure. Right panel: the same as in the left panel, but for $f_0^{DK}(q^2)$ in the case of ensemble A40.32

Figure 8

Momentum dependencies of the Lorentz-invariant form factors $f_{+}(q^2)$ (orange bands) and $f_0(q^2)$ (cyan bands), extrapolated to the physical pion mass and to the continuum and infinite volume limits, for the $D\to \pi$ (left panel) and $D\to K$ (right panel) transitions, including their total uncertainties. For comparison, the values of $f_{+}^{D\pi (K)}(q^2)$ determined by BELLE, BABAR, CLEO and BESIII collaborations in Refs. [11, 12, 13, 14, 15] are shown. The bands correspond to the total ($\text{statistical}+\text{systematic}$) uncertainty at one standard-deviation level.

Figure 9

Comparison of the vector and scalar form factors, extrapolated to the physical pion mass and to the continuum and infinite volume limits, obtained either by choosing all the kinematical configurations and including the hypercubic terms (35) and (41) in the analysis (solid lines) or by limiting to the kinematical configurations corresponding to the $D$-meson rest frame without considering the subtraction of hypercubic effects (dashed lines). The bands correspond to the total uncertainty at one standard-deviation level.

Figure 10

Results for $|V_{cd}|$ and $|V_{cs}|$ obtained from leptonic and semileptonic $D$- and $D_s$-meson decays, represented, respectively, by green and red ellipses corresponding to a 68% probability contour. The solid ellipses are the results of Ref. [6] and of this work, obtained with $N_f=2+1+1$ dynamical quarks. The striped ellipses correspond to the latest FLAG results [3], which for the semileptonic decays are based on the LQCD results obtained in Refs. [38, 39] with $N_f=2+1$ dynamical quarks. The dashed line indicates the correlation between $|V_{cd}|$ and $|V_{cs}|$ that follows if the CKM-matrix is unitary.