Study of the $D_s \to φ\ell ν_\ell$ semileptonic decay with (2+1)-flavor lattice QCD

TL;DR

This work reports a comprehensive lattice QCD study of the $D_s\to\phi\,\ell\nu_\ell$ semileptonic decay using $(2+1)$-flavor Wilson-clover configurations across seven ensembles to control the continuum and physical-pion limits. The authors introduce a scalar-function method to extract the vector and axial form factors $V,A_0,A_1,A_2$, achieving high-precision determinations and obtaining the form-factor ratios $r_V=1.614(19)$ and $r_2=0.741(31)$. They determine the branching fractions for $D_s\to\phi e\nu_e$ and $D_s\to\phi\mu\nu_\mu$, with ${\mathcal B}(D_s\to\phi e\nu_e)=2.493(66)_{stat}(31)_{|V_{cs}|}\times 10^{-2}$ and ${\mathcal B}(D_s\to\phi\mu\nu_\mu)=2.351(60)_{stat}(29)_{|V_{cs}|}\times 10^{-2}$, and a lepton-flavor ratio ${\mathcal R}_{\mu/e}=0.9432(13)$. By comparing to PDG inputs, they extract $|V_{cs}|=0.952(12)_{stat}(23)_{PDG}$ (\mu) and $0.945(12)_{stat}(24)_{PDG}$ (e), consistent with CKM unitarity. The scalar-function approach enables robust control of rotational symmetry breaking and excited-state contamination, and the results offer a precise theoretical benchmark for future experimental tests of lepton flavor universality and CKM physics.

Abstract

We present a systematic lattice calculation of the $D_s \to φ\ell ν_\ell$ 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 $\mathcal{B}(D_s\toφeν_e)=2.493(66)_{\text{stat}}(31)_{|V_{cs}|}\times 10^{-2}$ and $\mathcal{B}(D_s\toφμν_μ)=2.351(60)_{\text{stat}}(29)_{|V_{cs}|}\times 10^{-2}$. The corresponding ratio of the branching fractions between the lepton $μ$ and $e$ is given by $\mathcal{R}_{μ/e}=0.9432(13)$, which provides essential theoretical support for future high-precision experimental tests of the lepton flavor universality. The CKM matrix element $|V_{cs}|$ is also extracted to be $0.952(12)_{\text{stat}}(23)_{\text{PDG}}$ and $0.945(12)_{\text{stat}}(24)_{\text{PDG}}$ for the $μ$ and $e$ channels, respectively.

Figures (23)

• Figure 1: Definition of the angular variables in the $D_s^+\to K^+K^-\ell^+\nu_\ell$ decay.
• Figure 2: The lattice results of $\phi$ mass $m$ and decay constant $f_\phi$ as a function of the lattice spacing $a$ and pion mass $m_\pi$.
• Figure 3: The lattice results of the $D_s\to\phi$ form factors and the extrapolation fitting. The shaded regions correspond to the final results at the continuum limit and physical pion mass.
• Figure 4: The lattice results of form factors at $q^2=0$ as a function of the lattice spacing $a$ and pion mass $m_\pi$.
• Figure 5: The comparison of the $r_V$ (top) and $r_2$ (bottom) calculated in this work, measured by experiments, and those given by different theoretical predictions.