Study of the $D_s\to \phi \mathrm{\ell }{\nu }_{\mathrm{\ell }}$ semileptonic decay with (2+1)-flavor lattice QCD

We present a systematic lattice calculation of the $D_s\to \phi \mathrm{\ell }{\nu }_{\mathrm{\ell }}$ semileptonic decay using (2+1)-flavor Wilson-clover fermion configurations generated by the CLQCD collaboration. Seven gauge ensembles with different lattice spacings, from $0.052\text{ fm}$ to $0.105\text{ fm}$, and different pion masses, from about $210\text{ MeV}$ to $320\text{ MeV}$ are utilized, enabling us to take both the continuum limit and physical pion mass extrapolation. The ratios of form factors are obtained to be $r_V=1.614(19)$ and $r_2=0.741(31)$, with the precision improved by up to an order of magnitude compared to previous lattice studies. The branching fractions are given as $\mathcal{B}(D_s\to \phi e{\nu }_e)=2.493(66{)}_{\text{stat}}(31{)}_{\mathrm{\mid }{V}_{cs}\mathrm{\mid }}\times {10}^{-2}$ and $\mathcal{B}(D_s\to \phi \mu {\nu }_{\mu })=2.351(60{)}_{\text{stat}}(29{)}_{\mathrm{\mid }{V}_{cs}\mathrm{\mid }}\times {10}^{-2}$. The corresponding ratio of the branching fractions between the lepton $\mu$ and $e$ is given by ${\mathcal{R}}_{\mu \mathrm{/}e}=0.9432(13)$, which provides essential theoretical support for future high-precision experimental tests of the lepton flavor universality. The CKM matrix element $\mathrm{\mid }{V}_{cs}\mathrm{\mid }$ is also extracted to be $0.952(12{)}_{\text{stat}}(23{)}_{\text{PDG}}$ and $0.945(12{)}_{\text{stat}}(24{)}_{\text{PDG}}$ for the $\mu$ and $e$ channels, respectively.