Abstract
We revisit the puzzle of |Vus| values obtained from the conventional implementation of hadronic-τ- decay-based flavor-breaking finite-energy sum rules lying >3σ below the expectations of three-family unitarity. Significant unphysical dependences of |Vus| on the choice of weight, w, and upper limit, s0, of the experimental spectral integrals entering the analysis are confirmed, and a breakdown of assumptions made in estimating higher dimension, D>4, OPE contributions identified as the main source of these problems. A combination of continuum and lattice results is shown to suggest a new implementation of the flavor-breaking sum rule approach in which not only |Vus|, but also D>4 effective condensates, are fit to data. Lattice results are also used to clarify how to reliably treat the slowly converging D=2 OPE series. The new sum rule implementation is shown to cure the problems of the unphysical w- and s0-dependence of |Vus| and to produce results ∼0.0020 higher than those of the conventional implementation employing the same data. With B-factory input, and using, in addition, dispersively constrained results for the Kπ branching fractions, we find |Vus|=0.2231(27)exp(4)th, in excellent agreement with the result from Kℓ3, and compatible within errors with the expectations of three-family unitarity, thus resolving the long-standing inclusive τ|Vus| puzzle.
Highlights
With |Vud| = 0.97417(21) [1] as input and |Vub| negligible, 3-family unitary implies |Vus| = 0.2258(9)
We have revisited the determination of |Vus| from flavor-breaking finite-energy sum rule analyses of experimental inclusive non-strange and strange hadronic τ decay distributions, identifying an important systematic problem in the conventional implementation of this approach, and developing an alternate implementation which cures this problem
We have used lattice results to bring under better theoretical control the treatment of the potentially problematic D = 2 OPE series entering these analyses
Summary
The low |Vus| results noted above are produced by a conventional implementation of the general FB FESR framework, Eq (5), in which a single s0 (s0 = m2τ ) and single weight (w = wτ ), are employed [5] This restriction allows the ij = ud and us spectral integrals to be determined from the inclusive ud and us branching fractions alone, but precludes carrying out s0- and w-independence tests. Problems with the assumptions employed for C6 and C8 in the conventional implementation will manifest themselves as an unphysical s0-dependence in the |Vus| results obtained using weights w(y) with non-zero coefficients, w2 and/or w3, of y2 and y3 Another potential issue for the FB FESR approach is the slow convergence of the D = 2 OPE series. The normalizations of the different components of the 1999 ALEPH residual mode distribution are updated using HFAG 2016 branching fractions [26]
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