[Paper Review] D to K and D to pi semileptonic form factors from Lattice QCD

J. Koponen, C. T. H. Davies+1 more|arXiv (Cornell University)|Aug 30, 2012

Particle physics theoretical and experimental studies2 references3 citations

TL;DR

This lattice QCD study computes $D\to K\ell\nu$ and $D\to\pi\ell\nu$ semileptonic form factors with high statistical precision using $N_f=2+1$ HISQ lattices and twisted boundary conditions to access the full $q^2$ range. It finds the form factors are insensitive to the spectator quark, enabling accurate extraction of $|V_{cs}| = 0.965(14)$, in excellent agreement with experiment and unitarity-based PDG values.

ABSTRACT

We present a very high statistics study of D and D_s semileptonic decay form factors on the lattice. We work with MILC N_f=2+1 lattices and use the Highly Improved Staggered Quark action (HISQ) for both the charm and the strange and light valence quarks. We use both scalar and vector currents to determine the form factors f_0(q^2) and f_+(q^2) for a range of D and D_s semileptonic decays, including D to pi and D to K. By using a phased boundary condition we are able to tune accurately to q^2=0 and explore the whole q^2 range allowed by kinematics. We can thus compare the shape in q^2 to that from experiment and extract the CKM matrix element |V_cs|. We show that the form factors are insensitive to the spectator quark: D to K and D_s to eta_s form factors are essentially the same, which is also true for D to pi and D_s to K within 5%. This has important implications when considering the corresponding B/B_s processes.

Motivation & Objective

• To compute semileptonic form factors for $D\to K\ell\nu$ and $D\to\pi\ell\nu$ decays from first principles using lattice QCD.
• To determine the $q^2$-dependence of form factors $f_0(q^2)$ and $f_+(q^2)$ across the full kinematic range using twisted boundary conditions.
• To test the spectator quark independence of form factors by comparing $D\to K\ell\nu$ with $D_s\to\eta_s\ell\nu$ and $D\to\pi\ell\nu$ with $D_s\to K\ell\nu$.
• To extract $|V_{cs}|$ by combining lattice form factors with experimental differential decay rates, avoiding reliance on CKM unitarity.
• To validate the lattice results by comparing predicted decay rates in $q^2$ bins with experimental data.

Proposed method

• Uses $N_f=2+1$ MILC lattices with the Highly Improved Staggered Quark (HISQ) action for charm, strange, and light valence quarks.
• Computes 2-point and 3-point correlation functions with scalar and vector currents to extract $f_0(q^2)$ and $f_+(q^2)$ form factors.
• Applies twisted boundary conditions to tune precisely to $q^2 = 0$ and explore the entire $q^2$ range up to $(M_D - M_K)^2$.
• Employs multi-exponential fits with Bayesian priors to suppress excited-state contamination and reduce systematic errors.
• Uses the $z$-expansion to perform continuum and chiral extrapolations, transforming $q^2$-dependent form factors into a power series in $z$.
• Combines lattice form factors with experimental $d\Gamma/dq^2$ data to extract $|V_{cs}|^2$ in each $q^2$ bin and fits to a constant value.

Experimental results

Research questions

• RQ1 How do the $D\to K\ell\nu$ and $D_s\to\eta_s\ell\nu$ form factors compare, given they differ only in the spectator quark?
• RQ2 To what extent are the semileptonic form factors of $D$ and $D_s$ mesons dependent on the spectator quark mass?
• RQ3 Can lattice QCD accurately predict the $q^2$-dependence of $f_+(q^2)$ for $D\to K\ell\nu$ to allow a precise extraction of $|V_{cs}|$?
• RQ4 How well do the lattice-predicted decay rates in $q^2$ bins agree with experimental measurements?
• RQ5 Does the lattice result for $|V_{cs}|$ agree with the unitarity-based PDG value without assuming CKM unitarity?

Key findings

• The form factors $f_0(q^2)$ and $f_+(q^2)$ for $D\to K\ell\nu$ and $D_s\to\eta_s\ell\nu$ are indistinguishable within uncertainties, indicating spectator quark independence.
• The form factors for $D\to\pi\ell\nu$ and $D_s\to K\ell\nu$ differ by less than 5%, confirming mild spectator dependence.
• The lattice calculation achieves sub-3% accuracy for $f_+(q^2)$ in $D\to K\ell\nu$, with better than 2% precision in the $q^2$-region near zero.
• The $z$-expansion provides a reliable parametrization for continuum and chiral extrapolation, enabling precise form factor extrapolation.
• The extracted value of $|V_{cs}| = 0.965(14)$ (preliminary) is consistent with the PDG value of $0.97344(16)$, which assumes CKM unitarity.
• Lattice predictions for $D\to K\ell\nu$ decay rates in $q^2$ bins show excellent agreement with experimental data from CLEO, BaBar, Belle, and BESIII.