$K\to \pi$ semileptonic form factors with $N_f=2+1+1$ twisted mass fermions

We present a lattice QCD determination of the vector and scalar form factors of the semileptonic $K\to \pi \ell \nu$ decay which are relevant for the extraction of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{us}|$ from experimental data. Our results are based on the gauge configurations produced by the European Twisted Mass Collaboration with $N_f=2+1+1$ dynamical fermions, which include in the sea, besides two light mass degenerate quarks, also the strange and the charm quarks. We use data simulated at three different values of the lattice spacing and with pion masses as small as 210 MeV. Our final result for the vector form factor at zero momentum transfer is $f_{+}(0)=0.9709(46)$, where the uncertainty is both statistical and systematic combined in quadrature. Using the latest experimental value of $f_{+}(0)|V_{us}|$ from $K_{\ell 3}$ decays, we obtain $|V_{us}|=0.2230(11)$, which allows us to test the unitarity constraint of the Standard Model below the permille level once the determination of $|V_{ud}|$ from superallowed nuclear $\beta$ decays is adopted. A slight tension with unitarity at the level of $\sim 2$ standard deviations is observed. Moreover, we present our results for the semileptonic scalar $f_0(q^2)$ and vector $f_{+}(q^2)$ form factors in the whole range of values of the squared four-momentum transfer $q^2$ measured in $K_{\ell 3}$ decays, obtaining a very good agreement with the momentum dependence of the experimental data. We provide a set of synthetic data points representing our results for the vector and scalar form factors at the physical point for several selected values of $q^2$.

Figure 1

Double ratios $R^{(S)}$ (black squares) and ${\overline{R}}^{(S)}$ (red dots), given by Eqs. (18) and (20), respectively, vs the Euclidean time $t/a$ in the cases of the ensembles A30.32 with $a{\mu }_s=0.0185$ (upper panel) and B55.32 with $a{\mu }_s=0.0180$ (lower panel). The meson masses are $(M_{\pi },M_K)\simeq (275,510)\mathrm{MeV}$ (upper panel) and $(M_{\pi },M_K)\simeq (375,545)\mathrm{MeV}$ (lower panel). The values of the pion and kaon momentum correspond to $|{\overrightarrow{p}}_K|=0$ and $|{\overrightarrow{p}}_{\pi }|\simeq 150\mathrm{MeV}$ for both panels. The black and red areas correspond to the values of the extracted matrix element $|⟨\pi (p_{\pi })|S|K(p_K)⟩{|}^2$ obtained using the plateau regions described later in the text and shown in the figure. The corresponding values of the squared 4-momentum transfer $q^2$ are close to $q^2=0$, namely $q^2\simeq 0.016{\mathrm{GeV}}^2$ (upper panel) and $q^2\simeq -0.003{\mathrm{GeV}}^2$ (lower panel).

Figure 2

Matrix elements $⟨{\widehat{V}}_{sp}{⟩}_{imp}$, $⟨{\widehat{V}}_0{⟩}_{imp}$ and $⟨S{⟩}_{imp}$ extracted from the ratios $R_{\mu }$ and ${\overline{R}}_S$ [see Eqs. (15) and (20)] for the ensemble A40.32 with $\beta =1.90$, $L/a=32$, $a{\mu }_l=0.0040$, $a{\mu }_s=0.0185$, ${\overrightarrow{p}}_K=-{\overrightarrow{p}}_{\pi }$ and $|{\overrightarrow{p}}_K|\simeq 150\mathrm{MeV}$. The meson masses are $M_{\pi }\simeq 315\mathrm{MeV}$ and $M_K\simeq 525\mathrm{MeV}$. The horizontal red lines correspond to the plateau regions used to extract the matrix elements and to their central values and statistical errors (see the text).

Figure 3

Left panel: results of the fit of our lattice data for the form factors $f_{+}(q^2)$ and $f_0(q^2)$ using the $z$-expansion (continuum lines), given by Eq. (36), compared to the ones obtained adopting a simple quadratic formula in $q^2$ (dashed lines). Right panel: $z$-expansion fit of the form factors using only the data corresponding to $a^2{q}^2>-0.01$ (continuum lines) or using all the data (dashed lines). Both panels correspond to the ensemble A60.24 with $\beta =1.90$, $L/a=24$, $a{\mu }_{\ell }=0.0060$ and $a{\mu }_s=0.0185$.

Figure 4

Chiral and continuum extrapolation of $f_{+}(0)$ based on the SU(2) ChPT fit given by Eq. (37) (left) and on the SU(3) ChPT fit given by Eqs. (38)–(41) (right).

Figure 5

Results for the vector (a) and scalar (b) form factors as functions of $q^2/M_K^2$ for the ensembles A40.24 and A40.32, which correspond to $\beta =1.90$ and $M_{\pi }\simeq 300\mathrm{MeV}$ at two different lattice volumes $L/a=24$ (red dots) and $L/a=32$ (blue squares), respectively. The dashed lines represent the results of a simple pole fit to the data for each lattice volume separately. The solid and dotted lines correspond to the infinite volume predictions obtained using Eq. (46) for the slopes in the two cases ${\alpha }_{\mathrm{eff}}=0$ and ${\alpha }_{\mathrm{eff}}=3/2$.

Figure 6

Vector (black squares) and scalar (red circles) form factors vs $q^2/M_K^2$ for the ensembles A50.32, B35.32 and D20.48. The solid lines represent the results of the SU(2) ChPT (right panel) and the modified $z$-expansion (left panel) fits (see the text). The dashed lines identify the uncertainties due to the statistical errors of the data and to the fitting procedure.

Figure 7

Scaling behavior of the vector (squares) and scalar (circles) form factors, interpolated at the references values $m_{\ell }=20\mathrm{MeV}$, $m_s=99.6\mathrm{MeV}$ and $q^2/M_K^2=-0.33,-0.11$, 0.14 corresponding to black, blue and red markers (or bottom to top), respectively. Solid and dashed lines represent the results of the SU(2) ChPT fit with its statistical uncertainty.

Figure 8

Comparison of the information for the dispersive parameters ${\mathrm{\Lambda }}_{+}$ and $\log (C)$ obtained in this work (solid ellipse) with the results of the $K_{\ell 3}$ experiments KTeV, KLOE, $\mathrm{NA}48/2$ and $\mathrm{ISTRA}+$ (dashed ellipses), taken from Refs. [2, 10], and with the updated FlaviaNet average of Ref. [10] (full ellipse). All the ellipses represent contours corresponding to a 68% likelihood.

Figure 9

Results for the vector (blue area) and scalar (red area) form factors, obtained at the physical point including both statistical and systematic uncertainties (see the text), multiplied by $|V_{us}|=0.2230$ [see Eq. (64)] vs $q^2$ in the range between $q^2=0$ and the physical kinematical end point $q^2=q_{\max }^2\simeq 0.129{\mathrm{GeV}}^2$. The black solid lines represent the results of the dispersive fit of the experimental data performed in Ref. [10].