Chiral behavior of $K\to \pi l\nu$ decay form factors in lattice QCD with exact chiral symmetry

We calculate the form factors of the $K\to \pi l\nu$ semileptonic decays in three-flavor lattice QCD and study their chiral behavior as a function of the momentum transfer and the Nambu-Goldstone boson masses. Chiral symmetry is exactly preserved by using the overlap quark action, which enables us to directly compare the lattice data with chiral perturbation theory (ChPT). We generate gauge ensembles at a lattice spacing of 0.11 fm with four pion masses covering 290–540 MeV and a strange quark mass $m_s$ close to its physical value. By using the all-to-all quark propagator, we calculate the vector and scalar form factors with high precision. Their dependence on $m_s$ and the momentum transfer is studied by using the reweighting technique and the twisted boundary conditions for the quark fields. We compare the results for the semileptonic form factors with ChPT at next-to-next-to-leading order in detail. While many low-energy constants appear at this order, we make use of our data of the light meson electromagnetic form factors in order to control the chiral extrapolation. We determine the normalization of the form factors as $f_{+}(0)=0.9636(36){(}_{-35}^{+57})$ and observe reasonable agreement of their shape with experiment.

Figure 1

Jackknife error of three-point function $C_{4,{\varphi }_s,{\varphi }_s}^{K\pi }(\mathrm{\Delta }{x}_4,\mathrm{\Delta }{x}_4^{\prime },\mathbf{p},{\mathbf{p}}^{\prime })$ as a function of bin size. We plot data at $(m_{ud},m_s)=(0.015,0.080)$ with $\mathrm{\Delta }{x}_4=\mathrm{\Delta }{x}_4^{\prime }=9$, $\theta =0.00$, and ${\theta }^{\prime }=1.68$. See Sec. 2b for the definition and parameters of the three-point function.

Figure 2

Statistical fluctuation of three-point function $C_{\mu ,{\varphi }_s{\varphi }_s}^{K\pi }(\mathrm{\Delta }{x}_4,\mathrm{\Delta }{x}_4^{\prime };\mathbf{p},{\mathbf{p}}^{\prime })$ for $\mu =4$ (left panel) and 1 (right panel). We plot the value for each jackknife sample normalized by the statistical average. The horizontal axis represents the HMC trajectory count of the excluded configuration in the jackknife analysis. The data are obtained at $(m_l,m_s)=(0.015,0.080)$ with $\mathrm{\Delta }{x}_4=\mathrm{\Delta }{x}_4^{\prime }=10$, $\theta =0.00$, ${\theta }^{\prime }=1.68$. Triangles and circles are data before and after averaging over the temporal location of the source operator $x_4$, respectively.

Figure 3

Effective value of $f_0(t_{\max })$ estimated from double ratio $R_{\varphi {\varphi }^{\prime }}(\mathrm{\Delta }{x}_4,\mathrm{\Delta }{x}_4^{\prime })$ at $(m_l,m_s)=(0.050,0.080)$ (left panel) and (0.015,0.080) (right panel). Blue circles, squares and diamond show data with the smeared source and sink for different values of $\mathrm{\Delta }{x}_4+\mathrm{\Delta }{x}_4^{\prime }$. On the other hand, black triangles show data with local source and/or sink with $\mathrm{\Delta }{x}_4+\mathrm{\Delta }{x}_4^{\prime }$ kept fixed. All data are shifted along the horizontal axis so that the meson source and sink operators are located at $T/4-(\mathrm{\Delta }{x}_4+\mathrm{\Delta }{x}_4^{\prime })/2$ and $T/4+(\mathrm{\Delta }{x}_4+\mathrm{\Delta }{x}_4^{\prime })/2$, respectively. Solid and dashed lines show a constant fit to data with different values of $\mathrm{\Delta }{x}_4$ and $\mathrm{\Delta }{x}_4+\mathrm{\Delta }{x}_4^{\prime }$.

Figure 4

Vector (left panel) and scalar form factors (right panel) as a function of $t$ at $(m_l,m_s)=(0.050,0.080)$. Solid circles show the results at simulated $t$’s. Solid, dashed and dot-dashed lines are linear, quadratic (26) and free-pole fits (27), respectively. We also plot the fit with the $K^{*}$ pole (25) for $f_{+}$. The dotted lines show their statistical error. The value interpolated to $t=0$ is shown by the open diamond. The blue thicker lines show the fit to estimate the central value and statistical error of the interpolated value.

Figure 5

Same as Fig. 4, but for $(m_l,m_s)=(0.015,0.080)$.

Figure 6

Same as Fig. 4, but for $(m_l,m_s)=(0.050,0.060)$.

Figure 7

Chiral extrapolation of $f_{+}(0)$ as a function of $M_{\pi }^2$. Solid circles and squares show our data at $m_s=0.080$ and 0.060. The blue diamond is the value extrapolated to the physical point $(m_{l,\mathrm{phys}},m_{s,\mathrm{phys}})$. We note that the physical strange quark mass $m_{s,\mathrm{phys}}$ (diamond) is slightly off the simulated values (solid lines).

Figure 8

LEC-(in)dependent NLO and NNLO contributions to $f_{+}(0)$. Thick dashed and dot-dashed lines are the NLO and NNLO contributions, whereas the NNLO terms, $f_{+,4,L}(0)$, $f_{+,4,C}(0)$ and $f_{+,4,B}(0)$, are plotted by the thin dot-dot-dashed, dot-dashed and dot-dashed-dashed lines, respectively.

Figure 9

Effective value $c_{+,\pi \mathrm{K},\mathrm{eff}}^r$ as a function of $M_{\pi }^2$. Squares are slightly shifted along the horizontal axis for clarity. The solid and dotted lines show $c_{+,\pi K}^r$ obtained from the NNLO chiral fit and its uncertainty.

Figure 10

Chiral extrapolation of $f_{+}(t)$ as a function of $t$. Different symbols show data at different values of $m_l$, whereas the left and right panels are for $m_s=0.080$ and 0.060, respectively. Thin solid and dotted lines show the fit curve and its statistical uncertainty. We also plot those at the physical point $(m_{l,\mathrm{phys}},m_{s,\mathrm{phys}})$ by thick lines in the left panel at $m_s\sim m_{s,\mathrm{phys}}$.

Figure 11

Simultaneous chiral fit to $f_{+}(t)$ (left panel) and ${\tilde{f}}_0(t)$ (right panel) at $m_s=0.080$. Data at different values of $m_l$ are plotted by different symbols as a function of $t$. Thin solid and dotted line shows the fit curve and the statistical uncertainty, whereas thick lines show those at $(m_{l,\mathrm{phys}},m_{s,\mathrm{phys}})$.

Figure 12

Comparison of fit curves for $f_{+}(0)$ at $m_s=0.080\sim m_{s,\mathrm{phys}}$. The dashed, dotted and solid lines lines are reproduced from the fits to $f_{+}(0)$, $f_{+}(t)$ and the simultaneous fit to $f_{+}(t)$ and ${\tilde{f}}_0(t)$, respectively. For each fit, the thick line shows the central value, whereas its statistical error is shown by the two thin lines.

Figure 13

LEC-(in)dependent NLO and NNLO contributions to $f_{+}(t)$ (left panels) and ${\tilde{f}}_0(t)$ (right panels) as a function of $t$. The top, middle and bottom panels show data at $(m_l,m_s)=(0.050,0.080)$, (0.015,0.080) and the physical point $(m_{l,\mathrm{phys}},m_{s,\mathrm{phys}})$, respectively. The net NLO and NNLO contributions are plotted by thick blue dashed and red dot-dashed lines, respectively. Thin lines show their breakdown into LEC-(in)dependent terms.

Figure 14

Slope parameters ${\lambda }_{+}^{\prime }$ (left panel) and ${\lambda }_0^{\prime }$ (right panel) as a function of $M_{\pi }^2$. The blue solid and dotted lines are reproduced from our chiral fit based on NNLO ChPT, and represent the slopes at a simulated strange quark mass $m_s=0.080$. The black dashed (dot-dashed) line shows the contribution from the NLO (NNLO) correction to $f_{+}$ and ${\tilde{f}}_0$. The value extrapolated to the physical point $(m_{ud,\mathrm{phys}},m_{s,\mathrm{phys}})$ is plotted by the blue diamonds, whereas the red stars represent the experimental values (67). We also plot the values in Table 2 by blue circles ($m_s=0.080$) and squares (0.060).

Figure 15

Comparison of recent lattice estimates of $f_{+}(0)$. The filled circle shows our result (74). Recent results in $N_f=2+1$ [8, 9]. and $2+1+1$ [10, 11] QCD are plotted by open circles and squares, respectively.