[Paper Review] FK/Fpi from the Budapest-Marseille-Wuppertal Collaboration

Alberto Ramos, Stephan Dürr+8 more|arXiv (Cornell University)|Jan 20, 2011

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This lattice QCD study by the Budapest-Marseille-Wuppertal Collaboration determines the ratio $F_K/F_\pi = 1.192(7)_{\text{stat}}(6)_{\text{syst}}$ using $2+1$ flavor simulations with three lattice spacings, large volumes, and pion masses down to 190 MeV. The result enables a precise determination of $|V_{us}| = 0.2256(18)$, confirming CKM unitarity within the Standard Model and reducing theoretical uncertainties in flavor physics.

ABSTRACT

Based on a series of lattice calculations we determine the ratio FK/Fpi in QCD. With experimental data from kaon decay and nuclear double beta decay, we obtain a precise determination of |Vus|. Our simulation includes 2+1 flavours of sea quarks, with three lattice spacings, large volumes and a simulated pion mass reaching down to about 190 MeV for a full control over the systematic uncertainties.

Motivation & Objective

• To determine the ratio of kaon to pion decay constants, $F_K/F_\pi$, in lattice QCD with controlled systematic uncertainties.
• To enable a precise determination of the CKM matrix element $|V_{us}|$ using experimental data from kaon and double beta decay.
• To reduce theoretical uncertainties in $|V_{us}|$ by performing a comprehensive analysis of chiral, continuum, finite-volume, and excited-state effects.
• To validate CKM unitarity by combining the lattice result with $|V_{ud}| = 0.97425(22)$ and $|V_{ub}| = (3.93 \pm 0.36) \times 10^{-3}$.

Proposed method

• Simulate $2+1$ flavor QCD with clover fermions on lattices at three different spacings ($a \approx 0.124$, $0.083$, and $0.065$ fm) to control the continuum limit.
• Use large volumes with $M_\pi L \geq 4$ to suppress finite-volume effects below 1%.
• Perform chiral extrapolation using 7 different functional forms, including NLO $\chi$PT and analytical expansions, to model the $F_K/F_\pi$ dependence on light quark masses.
• Apply scale setting using the $\Omega$ and $\Xi$ baryon masses, corrected for isospin breaking and electromagnetic effects.
• Account for cutoff effects via $\mathcal{O}(a^2)$ or $\mathcal{O}(a)$ terms in the ratio, with evidence for $\mathcal{O}(a^2)$ scaling.
• Estimate excited-state contamination by varying the fitting time range ($t_{\text{min}}$) over 18 values per $\beta$, and use 2-loop finite-volume corrections in $\chi$PT.

Experimental results

Research questions

• RQ1 What is the precise value of the ratio $F_K/F_\pi$ in the continuum limit with physical quark masses?
• RQ2 How do systematic uncertainties from chiral extrapolation, finite volume, and cutoff effects impact the determination of $F_K/F_\pi$?
• RQ3 To what extent can lattice QCD reduce the theoretical uncertainty in $|V_{us}|$?
• RQ4 Does the resulting $|V_{us}|$ value satisfy CKM unitarity within the Standard Model?

Key findings

• The final lattice QCD result for the ratio is $F_K/F_\pi = 1.192(7)_{\text{stat}}(6)_{\text{syst}}$ at the physical point.
• Using $|V_{ud}| = 0.97425(22)$, the study determines $|V_{us}| = 0.2256(18)$ with a total uncertainty of 0.77%.
• The first-row CKM unitarity relation holds as $|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 1.0001(9)$, with no significant deviation from the Standard Model.
• Finite-volume corrections are negligible, as the data are well-fitted without them, and the 2-loop correction estimate is consistent with statistical precision.
• Cutoff effects are small and consistent with zero, supporting $\mathcal{O}(a^2)$ scaling for the ratio.
• The 1512 different fitting procedures—spanning 2 scale choices, 18 $t_{\text{min}}$ values, 7 functional forms, and 3 volume corrections—yield a robust median result with systematic uncertainty from the 16th–84th percentile spread.