[Paper Review] A resolution of the puzzle of low V_us values from inclusive flavor-breaking sum rule analyses of hadronic tau decay

Renwick J. Hudspith, Randy Lewis+3 more|arXiv (Cornell University)|Nov 26, 2015

Particle physics theoretical and experimental studies3 references3 citations

TL;DR

This paper resolves the long-standing puzzle of low $|V_{us}|$ values from inclusive $ au$ decay sum rules by introducing a new implementation that fits $D>4$ operator product expansion (OPE) condensates to data, eliminating unphysical $s_0$- and weight-dependence. The new method yields $|V_{us}| = 0.2228(23)_{\text{exp}}(5)_{\text{th}}$, in excellent agreement with $K_{\ell 3}$ decays and 3-family unitarity, resolving a $3.6\sigma$ discrepancy in conventional analyses.

ABSTRACT

Continuum and lattice methods are used to investigate systematic issues in the sum rule determination of $V_{us}$ using inclusive hadronic $τ$ decay data. Results for $V_{us}$ employing assumptions for $D>4$ OPE contributions used in previous conventional implementations of this approach are shown to display unphysical dependence on the sum rule weight, $w$, and choice of upper limit, $s_0$, of the relevant experimental spectral integrals. Continuum and lattice results suggest a new implementation of the sum rule approach with not just $\vert V_{us}\vert$, but also $D>4$ effective condensates, fit to data. Lattice results are also shown to provide a quantitative assessment of truncation uncertainties for the slowly converging $D=2$ OPE series. The new sum rule implementation yields $\vert V_{us}\vert$ results free of unphysical $s_0$- and $w$-dependences and $\sim 0.0020$ higher than that obtained using the conventional implementation. With preliminary new experimental results for the $Kπ$ branching fraction, the resulting $\vert V_{us}\vert$ is in excellent agreement with that based on $K_{\ell 3}$, and compatible within errors with expectations from three-family unitarity.

Motivation & Objective

• To resolve the persistent $3.6\sigma$ discrepancy between inclusive $ au$ decay determinations of $|V_{us}|$ and the 3-family unitarity expectation.
• To identify and correct systematic errors in conventional sum rule implementations arising from unphysical $s_0$- and weight-dependence in $|V_{us}|$ results.
• To develop a new sum rule approach that fits $D>4$ OPE condensates directly to experimental data, reducing reliance on untested theoretical assumptions.
• To provide a quantitative assessment of truncation uncertainties in the $D=2$ OPE series using lattice QCD.
• To achieve a $|V_{us}|$ determination consistent with both $K_{\ell 3}$ decays and unitarity, resolving long-standing tensions in the CKM matrix.

Proposed method

• Proposes a new implementation of finite-energy sum rules (FESRs) where $|V_{us}|$ and $D>4$ OPE condensates are simultaneously fitted to inclusive $ au$ decay spectral data.
• Uses multiple weight functions $w_N(s)$ and varying $s_0$ to test stability and reduce systematic biases in $|V_{us}|$ determination.
• Employs continuum and lattice QCD methods to assess the convergence and truncation uncertainties of the $D=2$ OPE series, favoring the fixed-order perturbation theory (FOPT) approach.
• Applies dispersive and sum rule techniques to constrain small, doubly-chirally-suppressed $J=0$ contributions in the $us$ channel.
• Combines results from multiple FESRs ($w_2$, $w_3$, $w_4$) to extract a robust, stable $|V_{us}|$ value with combined experimental and theoretical error estimates.
• Uses updated experimental $K\pi$ branching fractions from B-factory data to improve normalization and reduce systematic errors.

Experimental results

Research questions

• RQ1 Why do conventional inclusive $ au$ decay sum rule analyses yield $|V_{us}|$ values significantly below the 3-family unitarity expectation?
• RQ2 What causes the unphysical $s_0$- and weight-dependence in conventional $|V_{us}|$ determinations?
• RQ3 Can fitting $D>4$ OPE condensates to data instead of assuming their values eliminate systematic biases in $|V_{us}|$ extraction?
• RQ4 How do lattice QCD results inform the treatment of the slowly converging $D=2$ OPE series?
• RQ5 Is the new sum rule implementation capable of producing a $|V_{us}|$ value consistent with both $K_{\ell 3}$ decays and CKM unitarity?

Key findings

• The new sum rule implementation yields $|V_{us}| = 0.2228(23)_{\text{exp}}(5)_{\text{th}}$, which is $\sim 0.0020$ higher than the conventional $w_{\tau}$ implementation and resolves the $3.6\sigma$ discrepancy with unitarity.
• The new method eliminates unphysical $s_0$- and weight-dependence in $|V_{us}|$, with excellent stability across different $w_N$ weights and $s_0$ values.
• Lattice QCD results confirm the FOPT treatment of the $D=2$ OPE series as more reliable than the conventional CIPT approach, reducing truncation uncertainty.
• The dominant experimental uncertainty in the new determination arises from the $us$ V+A spectral integral, particularly the branching fraction normalization of exclusive strange modes.
• The final $|V_{us}|$ result is in excellent agreement with $K_{\ell 3}$-based determinations ($0.2235(4)$) and compatible with 3-family unitarity ($0.2258(9)$).
• The theoretical error of $0.0005$ is the smallest among current $|V_{us}|$ determination methods, indicating high robustness of the new approach.