A resolution of the inclusive flavor-breaking $\tau$ $|V_{us}|$ puzzle

TL;DR

This paper proposes a new method for extracting |V_{us}| from tau decay data, addressing previous inconsistencies and achieving results consistent with unitarity expectations.

Contribution

It introduces a novel sum rule approach that fits both |V_{us}| and higher-dimension condensates to data, reducing unphysical dependencies and resolving the longstanding puzzle.

Findings

New sum rule implementation yields higher |V_{us}| values.

Results are in agreement with K_{l3} determinations.

Addresses and corrects previous unphysical dependencies.

Abstract

We revisit the puzzle of $|V_{us}|$ values obtained from the conventional implementation of hadronic-$\tau$-decay-based flavor-breaking finite-energy sum rules lying $>3 \sigma$ below the expectations of three-family unitarity. Significant unphysical dependences of $|V_{us}|$ on the choice of weight, w, and upper limit, $s_0$, of the experimental spectral integrals entering the analysis are confirmed, and a breakdown of assumptions made in estimating higher dimension, $D>4$, OPE contributions is 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 $|V_{us}|$, 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 $s_0$-dependence of $|V_{us}|$ and to produce results $\sim 0.0020$ higher than those of the conventional implementation. With B-factory input, and using, in addition, dispersively constrained results for the $K\pi$ branching fractions, we find $\vert V_{us}\vert =0.2231(27)_{exp}(4)_{th}$, in excellent agreement with the result from $K_{\ell 3}$, and compatible within errors with the expectations of three-family unitarity, thus resolving the long-standing inclusive $\tau$ $|V_{us}|$ puzzle.

Tables

Table 1. Table 1: Single-weight fit | V u ​ s | subscript 𝑉 𝑢 𝑠 |V_{us}| error contributions for the w 2 subscript 𝑤 2 w_{2} , w 3 subscript 𝑤 3 w_{3} and w 4 subscript 𝑤 4 w_{4} FESRs, using 3-loop-truncated FOPT for the D = 2 𝐷 2 D=2 OPE series. Notation as described in the text.

Source	$\delta \left|V_{us}\right|$	$\delta \left|V_{us}\right|$	$\delta \left|V_{us}\right|$
	$w_2$ FESR	$w_3$ FESR	$w_4$ FESR
$\delta {\alpha }_s$	0.00002	0.00006	0.00006
$\delta m_s\left(2GeV\right)$	0.00008	0.00009	0.00008
$\delta ⟨m_s\overline{s}s⟩$	0.00035	0.00035	0.00035
$\delta \left(J=0sub\right)$	0.00009	0.00009	0.00009
${\delta }_{ud}^{exp}$	0.00027	0.00028	0.00028
${\delta }_{us}^{exp}$	0.00272	0.00273	0.00273

Equations

$$\frac{d{R}_{V/A;ij}}{ds}$$

$${\int }_0^{s_0}w(s)\Delta \rho (s)ds=-\frac{1}{2\pi i}{\oint }_{|s|=s_0}w(s)\Delta \Pi (-s)ds.$$

$$R_{V+A;ij}^w(s_0)\equiv {\int }_0^{s_0}ds\frac{w(s)}{w_{\tau }(s)}\frac{d{R}_{V+A;ij}^{(0+1)}(s)}{ds},$$

$$\delta R_{V+A}^w(s_0)\equiv \frac{R_{V+A;ud}^w(s_0)}{|V_{ud}{|}^2}-\frac{R_{V+A;us}^w(s_0)}{|V_{us}{|}^2},$$

$$|V_{us}|=\sqrt{R_{V+A;us}^w(s_0)/\left[\frac{R_{V+A;ud}^w(s_0)}{|V_{ud}{|}^2}-\delta R_{V+A}^{w,OPE}(s_0)\right]}.$$

$$-\frac{1}{2\pi i}{\oint }_{|s|=s_0}dsw(y){\left[\Delta \Pi (Q^2)\right]}_{D=2k+2}^{OPE}=(-1{)}^k{w}_k\frac{C_{2k+2}}{s_0^k}.$$

$${\left[\Delta \Pi (Q^2)\right]}_{D=2}^{OPE}$$

$${\left[\Delta \Pi (Q^2)\right]}_{D=4}^{OPE}=\frac{2\left[⟨m_u\bar{u}u⟩-⟨m_s\bar{s}s⟩\right]}{Q^4}\left(1-\bar{a}-\frac{13}{3}{\bar{a}}^2\right).$$

$$w_N(y)=1-\frac{N}{N-1}y+\frac{1}{N-1}{y}^N.$$

$$|V_{us}|=0.2231(27{)}_{exp}(4{)}_{th},$$

Full text

††preprint: ADP-17-07/T1013, YUPP-I/E-KM-17-02-2

I Introduction

With $|V_{ud}|=0.97417(21)$ ht14 as input and $|V_{ub}|$ negligible, 3-family unitary implies $|V_{us}|=0.2258(9)$. Direct determinations of $|V_{us}|$ from $K_{\ell 3}$ and $\Gamma[K_{\mu 2}]/\Gamma[\pi_{\mu 2}]$, using recent 2014 FlaviaNet experimental results kell3andKratiosvus and 2016 lattice input flag2016 for $f_{+}(0)$ and $f_{K}/f_{\pi}$, respectively, yield results, $|V_{us}|=0.2231(9)$ and $0.2253(7)$, in agreement with this expectation. In contrast, the most recent update hflav17 of the conventional implementation of the finite-energy sum rule (FESR) determination employing flavor-breaking (FB) combinations of inclusive strange and non-strange hadronic $\tau$ decay data gamizetal , yields $|V_{us}|=0.2186(21)$, $3.1\sigma$ below 3-family-unitarity expectations. A less discrepant, but still low, result, $0.2207(27)$, was obtained in Ref. aclp13 using the same conventional implementation but somewhat higher input $K\pi$ branching fractions (resulting from an analysis of $K\pi$ data imposing additional dispersive constraints on the timelike $K\pi$ form factors aclp13 ). The general FB FESR framework whose conventional implementation produces these low $|V_{us}|$ results is outlined below.

In the Standard Model, the differential distributions, $dR_{V/A;ij}/ds$, for flavor $ij=ud,\,us$, vector (V) or axial-vector (A) current-mediated decays, with $R_{V/A;ij}$ defined by $R_{V/A;ij}\,\equiv\,\Gamma[\tau^{-}\rightarrow\nu_{\tau}\,{\rm hadrons}_{V/A;ij}\,(\gamma)]/\Gamma[\tau^{-}\rightarrow\nu_{\tau}e^{-}{\bar{\nu}}_{e}(\gamma)]$, are related to the spectral functions, $\rho_{V/A;ij}^{(J)}$, of the $J=0,1$ scalar polarizations, $\Pi^{(J)}_{V/A;ij}$, of the corresponding current-current two-point functions, by tsai71

[TABLE]

where $y_{\tau}=s/m_{\tau}^{2}$, $w_{\tau}(y)=(1-y)^{2}(1+2y)$, $w_{L}(y)=2y(1-y)^{2}$, $S_{EW}$ is a known short-distance electroweak correction erler , and $V_{ij}$ is the flavor $ij$ CKM matrix element. The $J=0$ spectral functions, $\rho^{(0)}_{A;ud,us}(s)$, are dominated by the accurately known, chirally unsuppressed $\pi$ and $K$ pole contributions. The remaining, continuum contributions to $\rho^{(0)}_{V/A;ud,us}(s)$ are $\propto(m_{i}\mp m_{j})^{2}$, and hence negligible for $ij=ud$. For $ij=us$, they are small (though not totally negligible) and highly constrained, through the associated $ij=us$ scalar and pseudoscalar sum rules, by the known value of $m_{s}$, making possible mildly model-dependent determinations in the range $s\leq m_{\tau}^{2}$ relevant to hadronic $\tau$ decays jop ; mksps . Subtracting the resulting $J=0$ contributions from the RHS of Eq. (1) yields the $J=0+1$ analogue, $dR^{(0+1)}_{V/A;ij}/ds$, of $dR_{V/A;ij}/ds$, from which the $J=0+1$ spectral function combinations $\rho^{(0+1)}_{V/A;ud,us}(s)$ can be determined.

The inclusive $\tau$ determination of $|V_{us}|$ employs FB FESRs for the spectral function combination, $\Delta\rho(s)\,\equiv\,\rho^{(0+1)}_{V+A;ud}(s)\,-\,\rho^{(0+1)}_{V+A;us}(s)$ and associated polarization difference, $\Delta\Pi(Q^{2})\,\equiv\,\Pi_{V+A;ud}^{(0+1)}(Q^{2})\,-\,\Pi_{V+A;us}^{(0+1)(Q^{2})}$ gamizetal , with $Q^{2}\,=\,-s$. Generically, for any $s_{0}>0$ and any choice of analytic weight $w(s)$,

[TABLE]

For large enough $s_{0}$, the OPE is used on the RHS.

Defining the re-weighted integrals

[TABLE]

and using Eq. (2) to replace the FB difference

[TABLE]

with its OPE representation, one finds, solving for $|V_{us}|$ gamizetal ,

[TABLE]

The result is necessarily independent of $s_{0}$ and $w$ so long as all input is reliable. Assumptions employed in evaluating $\delta R^{w,OPE}_{V+A}(s_{0})$ can thus be tested for self-consistency by varying $w$ and $s_{0}$. OPE assumptions entering the conventional implementation of the FB FESR approach in fact produce $|V_{us}|$ displaying significant $w$- and $s_{0}$-dependence kmcwvus .

The low $|V_{us}|$ results noted above are produced by a conventional implementation of the general FB FESR framework, Eq. (5), in which a single $s_{0}$ ($s_{0}=m_{\tau}^{2}$) and single weight ($w=w_{\tau}$), are employed gamizetal . 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 $s_{0}$- and $w$-independence tests. Since $w_{\tau}$ has degree $3$, $\delta R^{w_{\tau},OPE}_{V+A}(s_{0})$ receives contributions up to dimension $D=8$. While $D=2$ and $4$ contributions, determined by $\alpha_{s}$ and the quark masses and condensates flag2016 ; bckd2ope ; ckd4ope ; PDG ; hpqcdcondratio , are known, $D>4$ contributions are not. In the conventional implementation, $D=6$ contributions are estimated using the vacuum saturation approximation (VSA) (see Ref. pp99 for the explicit expression) and $D=8$ contributions neglected gamizetal ; kmcwvus . These assumptions are potentially dangerous since the FB V+A VSA estimate involves a very strong double cancellation111A factor of $\sim 3$ reduction occurs when the individual $ud$ and $us$ V+A sums are formed, and a further factor of $\sim 6$ reduction in forming the FB $ud-us$ V+A difference. , and the VSA is known to be badly violated, in a channel-dependent manner, from studies in the non-strange sector dv7 .

Such assumptions can, in principle, be tested by varying $s_{0}$. Writing $D>4$ contributions to $\Delta\Pi(Q^{2})$ as $\sum_{D>4}C_{D}/Q^{D}$, with $C_{D}$ the effective dimension $D$ condensate, the integrated $D=2k+2$ OPE contribution to the RHS of Eq. (2), for a polynomial weight $w(y)=\sum_{n=0}w_{n}y^{n}$ with $y=s/s_{0}$, is, up to $\alpha_{s}$-suppressed logarithmic corrections,

[TABLE]

Problems with the assumptions employed for $C_{6}$ and $C_{8}$ in the conventional implementation will thus manifest themselves as an unphysical $s_{0}$-dependence in the $|V_{us}|$ results obtained using weights $w(y)$ with non-zero coefficients, $w_{2}$ and/or $w_{3}$, of $y^{2}$ and $y^{3}$.

Another potential issue for the FB FESR approach is the slow convergence of the $D=2$ OPE series. To four loops, neglecting $O(m^{2}_{u,d}/m^{2}_{s})$ corrections bckd2ope

[TABLE]

where $\bar{a}=\alpha_{s}(Q^{2})/\pi$, and $\bar{m}_{s}=m_{s}(Q^{2})$, $\alpha_{s}(Q^{2})$ are the running strange mass and coupling in the $\overline{MS}$ scheme. With $\bar{a}(m_{\tau}^{2})\simeq 0.1$, the ratio of $O(\bar{a}^{3})$ to $O(\bar{a}^{2})$ terms is $>1$ at the spacelike point on $|s|=s_{0}$ for all kinematically accessible $s_{0}$. Such slow “convergence” complicates choosing an appropriate truncation order and estimating the associated truncation uncertainty.

No apparent convergence problem exists for the $D=4$ series, which, to three loops, dropping numerically tiny $O(m_{q}^{4})$ terms, is given by ckd4ope

[TABLE]

The slow convergence of the $D=2$ OPE series and the reliability of conventional implementation assumptions for $C_{6}$ and $C_{8}$ will be investigated in the next section.

In the rest of the paper, the non-strange and strange spectral distributions entering the various FESRs considered are fixed using $\pi_{\mu 2}$, $K_{\mu 2}$ and SM expectations for the $\pi$ and $K$ pole contributions, recent ALEPH data for the continuum $ud$ V+A distribution aleph13 , Belle bellekspi and BaBar babarkmpi0 results for the $K^{-}\pi^{0}$ and $\bar{K}^{0}\pi^{-}$ distributions, BaBar results babarkpipiallchg for the $K^{-}\pi^{+}\pi^{-}$ distribution, Belle results bellekspipi for the $\bar{K}^{0}\pi^{-}\pi^{0}$ distribution, a combination of BaBar babar3K and Belle belle3K results for the very small $\bar{K}\bar{K}K$ distribution, and 1999 ALEPH results alephus99 for the combined “residual mode” distribution (the sum over contributions from those strange modes not remeasured by the B-factory experiments). BaBar and Belle exclusive strange mode distributions are given in unit-normalized form, with measured branching fractions required to set the overall scales. We work, in general, with 2016 HFAG HFAG2016 branching fractions. For the two $K\pi$ modes, however, we consider also the alternate results, $B[\tau\rightarrow K^{-}\pi^{0}\nu_{\tau}]=0.004707(181)$ and $B[\tau\rightarrow\bar{K}^{0}\pi^{-}\nu_{\tau}]=0.008566(299)$, obtained in Ref. aclp13 (ACLP), from an analysis imposing additional dispersive constraints on the timelike $K\pi$ form factors. The corresponding 2016 HFAG $K\pi$ results, obtained without the dispersive constraints, are $B[\tau\rightarrow K^{-}\pi^{0}\nu_{\tau}]=0.004327(149)$ and $B[\tau\rightarrow\bar{K}^{0}\pi^{-}\nu_{\tau}]=0.008386(141)$.

A plot of the latest version of the ALEPH $ud$ V+A spectral distribution may be found in Ref. aleph13 . The exclusive- and residual-mode contributions to the continuum $us$ V+A distribution, in the form, $|V_{us}|^{2}\,\tilde{\rho}_{V+A;us}(s)$, directly determinable from the experiment, are shown in Figure 1. For definiteness, the $K\pi$ points are shown with the 2016 HFAG $K\pi$ normalization. A global rescaling of $1.044$ is required to convert these to the alternate 2013 ACLP $K\pi$ normalization.

We base our central results on the additionally-constrained ACLP input choice, but quote results obtained using both $K\pi$ normalizations. Note that the publicly available ALEPH continuum $ud$ V+A distribution is normalized to a slightly older version of the inclusive $ud$ continuum branching fraction. A small rescaling ($0.5\%$ or less) is required to convert this to the normalization implied by the branching fractions we employ. The normalizations of the different components of the 1999 ALEPH residual mode distribution are also updated using HFAG 2016 branching fractions HFAG2016 .

II Testing conventional implementation assumptions

The conventional implementation assumptions, $C_{6}\simeq C_{6}^{VSA}$ and $C_{8}=0$, can be efficiently investigated using appropriately chosen $s_{0}$- and $w$-independence tests. A comparison of the results of the $w_{\tau}(y)=1-3y^{2}+2y^{3}$ and $\hat{w}(y)=1-3y+3y^{2}-y^{3}$ FESRs is particularly illuminating since the coefficients of $y^{2}$ in the two weights differ only by a sign. The corresponding integrated $D=6$ OPE contributions are thus identical in magnitude but opposite in sign. If, as the VSA estimate suggests, $D=6$ contributions are small for $w_{\tau}$, they must also be small for $\hat{w}$. Similarly, if integrated $D=8$ contributions are negligible for $w_{\tau}$, those for $\hat{w}$, which are $-1/2$ times as large, will also be negligible. If conventional implementation assumptions for $C_{6}$ and $C_{8}$ are reliable, the $|V_{us}|$ obtained from the $w_{\tau}$ and $\hat{w}$ FESRs should thus be in good agreement, and show good individual $s_{0}$ stability. In contrast, if these assumptions are not reliable, and $D=6$ and $D=8$ contributions are not both small, one should see $s_{0}$-instabilities of opposite signs in the two cases, and $s_{0}$-dependent differences in the results from the two FESRs which decrease with increasing $s_{0}$.

The central values of the results of this comparison, obtained using the ACLP and HFAG $K\pi$ normalizations, and, to be specific, the 3-loop-truncated contour-improved (CIPT) prescription CIPT for handling the integrated $D=2$ OPE series, are shown in the top left and bottom left panels of Figure 2, respectively, and clearly, in both cases, correspond to the second scenario. One should bear in mind that the results for a given weight but different $s_{0}$ are strongly correlated, as are the $w_{\tau}$ and $\hat{w}$ results at the same $s_{0}$. Neither changing the $D=2$ truncation order nor switching from CIPT to the alternate fixed-order (FOPT) $D=2$ prescription for the $D=2$ series serves to remove the strong, unphysical $s_{0}$ and weight dependences.

To understand the extent to which the $s_{0}$- and $w$-instabilities shown in Figure 2 are a problem for the conventional implementation $D>4$ condensate assumptions, it is useful to consider the differences between the $|V_{us}|$ obtained from the $\hat{w}$ and $w_{\tau}$ FESRs at the same $s_{0}$. If the conventional implementation assumptions are reliable these differences should be zero within errors. Fully propagating the $ud$ and $us$ experimental covariances, and adding independent sources of theory error in quadrature, we find, however, $\hat{w}$-$w_{\tau}$ differences of $0.0234\,(5)_{exp}\,(38)_{th}$ at $s_{0}=1.95\ GeV^{2}$, $0.0111\,(8)_{exp}\,(21)_{th}$ at $s_{0}=2.55\ GeV^{2}$, and $0.0064\,(16)_{exp}\,(13)_{th}$ at $s_{0}=3.15\ GeV^{2}$, clearly signalling problems with the conventional implementation assumptions. Similar conclusions follow from the observed $s_{0}$-instabilities. For example, the difference between the $\hat{w}$ FESR results at $s_{0}=2.55\ GeV^{2}$ and $3.15\ GeV^{2}$, which should once more be zero within errors, is instead $0.0039\,(5)_{exp}\,(8)_{th}$. A similarly discrepant result, $0.0096\,(8)_{exp}\,(19)_{th}$, is found for the difference between the $s_{0}=2.15\ GeV^{2}$ and $3.15\ GeV^{2}$ $\hat{w}$ results.

The top right and bottom right panels of Fig. 2 show the results of corresponding additional $w$- and $s_{0}$-independence tests involving the weights $w_{N}(y)$, $N=2,3,4$, with222The $w_{N}(y)$, like $w_{\tau}(y)$, have a double zero at $s=s_{0}$ ($y=1$). This serves to keep duality violating contributions safely small above $s\simeq 2\ GeV^{2}$ pinchingdvs .

[TABLE]

The upper solid lines in each case show the $w_{2}$, $w_{3}$ and $w_{4}$ results obtained using the conventional implementation treatment of $D>4$ OPE contributions and given $K\pi$ normalization, while the dashed-dotted show lines the corresponding results produced by the alternate implementation discussed below, in which the $D>4$ effective condensates are fit to experimental data. The corresponding conventional and alternate implementation $w_{\tau}$ results (represented by the lowest solid and dotted lines, respectively) are also included for comparison. The latter are obtained using the $D=6$ and $8$ effective condensates obtained from the alternate implementation $w_{2}$ and $w_{3}$ fits. The $s_{0}$-dependent, conventional implementation results for all of $w_{\tau}$, $\hat{w}$, $w_{2}$, $w_{3}$ and $w_{4}$ show evidence of converging toward a common value at $s_{0}>m_{\tau}^{2}$, as expected if the observed $s_{0}$-instabilities result from $D>4$ OPE contributions larger than those taken as input in the conventional implementation.

The impact of the slow convergence of the $D=2$ OPE series can be investigated by comparing OPE expectations to lattice results for $\Delta\Pi(Q^{2})$ over a range of Euclidean $Q^{2}=\,-q^{2}\,=\,-s$, using variously truncated versions of the $D=2$ OPE series. Lattice results were obtained using the RBC/UKQCD $n_{f}=2+1$, $32^{3}\times 64$, $1/a=2.38$ GeV, domain wall fermion ensemble with $m_{\pi}\sim 300$ MeV rbcukqcdfine11 . A tight cylinder cut, with a radius determined in a recent study of the extraction of $\alpha_{s}$ from lattice current-current two-point function data hlms15 , was imposed to suppress lattice artifacts at higher $Q^{2}$. The values of the light quark masses, $m_{u}=m_{d}\equiv m_{\ell}$ and $m_{s}$, for this ensemble, determined in Ref. rbcukqcdfine11 , were used for determining the corresponding OPE expectations.

We consider the comparison first for larger $Q^{2}$, where $D=2$ and $4$ contributions should dominate. The $D=2$ OPE contribution is determined using ensemble values of $m_{u}$ and $m_{s}$ rbcukqcdfine11 , the central PDG value for $\alpha_{s}$ PDG , and considering 2-, 3- and 4-loop truncation of the $D=2$ series. Both fixed scale, $\mu^{2}=4\ GeV^{2}$, and local scale, $\mu^{2}=Q^{2}$, choices for handling the logarithms in the truncated series are considered. The former choice is the analogue of the fixed order (FOPT) prescription for the $D=2$ FESR contour integrals, the latter the analogue of the alternate CIPT prescription.333The FOPT prescription evaluates weighted $D=2$ OPE integrals using a fixed-scale choice (usually $\mu^{2}=s_{0}$), the CIPT prescription CIPT using the local-scale choice, $\mu^{2}=Q^{2}$. For $D=4$ contributions, Eq. (8), we employ the Gell-Mann-Oakes-Renner (GMOR) relation for $\langle m_{u}\bar{u}u\rangle$ and fix $\langle m_{s}\bar{s}s\rangle$ using the ensemble value of $m_{s}/m_{\ell}$, translating the HPQCD result for $\langle\bar{s}s\rangle/\langle\bar{\ell}\ell\rangle$ at physical quark masses hpqcdcondratio , to that for the ensemble masses using NLO ChPT gl85 .

The comparisons obtained using the fixed- and local-scale versions of the $D=2$ series are shown in the left and right panels of Fig. 3, respectively. The best representation of the lattice results is provided by the 3-loop-truncated, fixed-scale version, which produces an excellent match over a wide range of $Q^{2}$, extending from near $\sim 10$ GeV2 down to just above $\sim 4$ GeV2, with the $Q^{2}$ dependence of the lattice results also favoring the fixed-scale over the alternate local-scale treatment444Note that both the lattice data and OPE results at different $Q^{2}$ are highly correlated. These correlations (and not just the errors on the individual OPE and lattice points) must be taken into account to assess the significance (or lack thereof) of the difference in the $Q^{2}$ dependences of the local-scale OPE and lattice results. The uncertainty on the $Q^{2}$ dependence is, in fact, strongly dominated by that on the input strange-to-light condensate ratio. Taking all correlations into account, one finds, for the fixed- and local-scale versions of the ratio of OPE to lattice values of the average slope between, for example, $Q^{2}\simeq 5\ GeV^{2}$ and $Q^{2}\simeq 9\ GeV^{2}$, the results $1.02(14)$ and $1.20(17)$, respectively, with $(13)$ and $(16)$ of the quoted errors coming from the uncertainty on the input strange-to-light condensate ratio. The $Q^{2}$ dependence of the lattice data thus favors the fixed-scale treatment of the $D=2$ series. .

Comparison to the lattice results also provides two further useful pieces of information. The left panel of Fig. 4 shows the comparison of the lattice results, the three-loop-truncated, fixed-scale $D=2$ series version of the $D=2+4$ OPE sum, and this same $D=2+4$ OPE sum now supplemented by the VSA estimate for $D=6$ contributions, in the lower $Q^{2}$ region. Below $\sim 4\ GeV^{2}$, the lattice results clearly require $D>4$ OPE contributions much larger than those assumed in the conventional implementation, confirming the conclusions reached already from the $w_{\tau}$-$\hat{w}$ FESR comparison above. The right panel shows the comparison of the lattice results and three-loop-truncated, fixed-scale $D=2$ series $D=2+4$ OPE sum, now with the conventionally estimated errors for the latter also displayed. These are obtained by combining in quadrature standard estimates for the $D=4$ truncation errors with uncertainties produced by those on the input $D=2$ and $4$ OPE parameters. Despite the apparently problematic convergence behavior of the $D=2$ series, conventional OPE error estimates are seen to provide an extremely conservative assessment of the uncertainty for the $D=2+4$ sum.

III An alternate implementation of the FB FESR approach

The results of the previous section suggest an obvious alternative to the conventional implementation of the FB FESR approach. First, the 3-loop-truncated FOPT treatment favored by the comparison to the high-$Q^{2}$ lattice results is employed for the $D=2$ OPE integrals555It is worth noting that the prescription of truncating at 3-loop order is also what one would arrive at were one to interpret the series as asymptotic and truncate it at its smallest term.. Second, since both lattice and continuum results suggest that conventional implementation assumptions for the effective $D>4$ condensates, $C_{D}$, are unreliable, we avoid such assumptions and instead fit the $C_{D}$ to data. FESRs based on the weights $w_{N}(y)$ are particularly convenient for use in fitting the $C_{D>4}$ since the $w_{N}$-weighted OPE integral involves only a single $D>4$ contribution, $(-1)^{N}\,C_{2N+2}/\left[(N-1)\,s_{0}^{N}\right]$. The $s_{0}$ dependence of the $w_{N}$-weighted spectral integrals in the region above $s_{0}\sim 2\ GeV^{2}$, where residual duality violations remain small, then provides sufficient information to allow both unknowns, $|V_{us}|$ and $C_{2N+2}$, entering the $w_{N}$ FESR to be determined.

Spectral distribution inputs employed in our analysis were outlined above666Note also that, when using the ACLP $K\pi$ normalization, we have, for consistency, implemented the long-distance electromagnetic corrections employed in arriving at the $K\pi$ branching fraction results obtained from the ACLP analysisthanksvcep .. On the OPE side, for the $D=2$ and $4$ contributions, we use PDG input for $\alpha_{s}$ PDG , FLAG input for the light and strange quark masses flag2016 , GMOR for the light-quark condensate gmor , and the HPQCD lattice result hpqcdcondratio for the ratio of strange to light quark condensates. The single-weight $w_{2}$, $w_{3}$ and $w_{4}$ FESR $|V_{us}|$ fit results obtained using our central (ACLP) choice of $K\pi$ normalization, $0.2228\,(27)_{exp}\,(4)_{th}$, $0.2230\,(27)_{exp}\,(4)_{th}$ and $0.2232\,(27)_{exp}\,(4)_{th}$, respectively, show a dramatically reduced weight dependence relative to those of the obtained using conventional implementation assumptions for the $D>4$ condensates. This is also true of the analogous results, $0.2205\,(23)_{exp}\,(4)_{th}$, $0.2208\,(23)_{exp}\,(4)_{th}$ and $0.2209\,(23)_{exp}\,(4)_{th}$, obtained using the alternate HFAG $K\pi$ normalization. It is worth commenting that, although the lattice results for Euclidean $Q^{2}$ favor the fixed-scale treatment of the $D=2$ series, and hence, by extension, the FOPT prescription for the weighted $D=2$ FESR integrals, the final results for $|V_{us}|$ are rather insensitive to choosing FOPT over CIPT. Explicitly, the alternate CIPT choice yields $0.2229\,(27)_{exp}\,(4)_{th}$ for all of the $w_{2}$, $w_{3}$ and $w_{4}$ FESRs when the ACLP $K\pi$ normalization is used and $0.2206\,(23)_{exp}\,(4)_{th}$ when the HFAG $K\pi$ normalization is used. The CIPT treatment, of course, generates slightly different fit results for the $C_{D>4}$, as expected, given that FOPT and CIPT represent different partial resummations of the presumably asymptotic $D=2$ series.

Given the excellent consistency of the individual $w_{2}$, $w_{3}$ and $w_{4}$ FESR determinations, we take our final result from a combined 3-weight fit. The central ACLP $K\pi$ normalization choice yields

[TABLE]

$0.0022$ higher than the result obtained using conventional implementation $D>4$ assumptions with the same experimental input. This result is in excellent agreement with the result from $K_{\ell 3}$ and compatible within errors with the expectations of 3-family unitarity. The combined 3-weight fit result, $|V_{us}|=0.2208\,(23)_{exp}\,(4)_{th}$, generated by the alternate (HFAG) choice of $K\pi$ normalization, similarly, lies $0.0020$ above the result obtained employing the same experimental input and conventional implementation assumptions for the $D>4$ condensates.

Table 1 shows the error budgets for the $w_{2}$, $w_{3}$ and $w_{4}$ fits employing the ACLP $K\pi$ normalization. Theory errors, resulting from uncertainties in the input parameters $\alpha_{s}$, $m_{s}$ and $\langle m_{s}\bar{s}s\rangle$, and the small $J=0$ continuum subtraction, are labelled by $\delta\alpha_{s}$, $\delta m_{s}$, $\delta\langle m_{s}\bar{s}s\rangle$ and $\delta(J=0\ sub)$, respectively, and shown above the horizontal line. Those induced by the covariances of the non-strange and strange experimental distributions $dR_{V+A;ud}/ds$ and $dR_{V+A;us}/ds$ are denoted $\delta^{exp}_{ud}$ and $\delta^{exp}_{us}$ and shown below the horizontal line. The $\delta_{us}^{exp}$ uncertainties strongly dominate the total errors.

From the lattice-OPE comparison discussed above, the estimates in the upper half of the table should provide a very conservative assessment of theoretical uncertainties. Combining the different components in quadrature yields a total theory error of $0.0004$ on $|V_{us}|$ for all three determinations. The new implementation of the FB FESR approach is thus competitive with the alternate $K_{\ell 3}$ and $\Gamma[K_{\mu 2}]/\Gamma[\pi_{\mu 2}]$ determinations from a theory error point of view, though improvements to the errors on the strange experimental distributions are required to make it fully competitive over all.

To test whether fitting the $D>4$ condensates has solved the problem of the $s_{0}$-instabilities found in the conventional implementation, we have rerun the $s_{0}$-dependent $w_{N}$ analyses, using the central fitted $C_{2N+2}$ values as input777It is worth noting that the central fitted $C_{2N+2>4}$ produce contributions to the $w_{N}$ FESRs which appear natural in size relative to the known $D=2$ and $4$ contributions. At $s_{0}=m_{\tau}^{2}$, for example, relative to the corresponding $D=2$ contributions, $D=4$ and $6$ contributions are $\sim 83\%$ and $-26\%$ for $w_{2}$, $D=4$ and $8$ contributions $\sim 67\%$ and $-11\%$ for $w_{3}$, and $D=4$ and $10$ contributions $\sim 61\%$ and $-5\%$ for $w_{4}$.. The dashed-dotted lines in the right panel of Fig. 2 show the results of this exercise. Using the fitted $C_{D>4}$ values dramatically reduces the $s_{0}$-instabilities of the conventional implementation versions of the same analyses, providing a strong self-consistency check on the new FB FESR implementation. The dotted line in this same panel shows the analogous $|V_{us}|$ results obtained from the $s_{0}$-dependent $w_{\tau}$ analysis using the fitted values of $C_{6}$ and $C_{8}$ as input. One again finds a dramatic reduction in the $s_{0}$ dependence, as well as excellent agreement with the results obtained using the other weights.

Errors on the $us$ distribution data limit the precision with which the $C_{2N+2}$ (which represent nuisance parameters for the determination of $|V_{us}|$) can be currently determined. It is, nonetheless, worth checking that the results for the FB condensates are compatible with an expected FB suppression relative to the corresponding flavor $ud$ V+A condensates. Comparing the results for $C_{6}$ and $C_{8}$ from our favored (FOPT) fits to those for the corresponding $ud$ V+A condensates, $C_{6,8}^{ud;V+A}$, obtained from the favored, $s_{min}=1.55\ GeV^{2}$, 3-weight, combined V&A FOPT fit of Ref. dv7 , we find, for the ratios of FB to non-FB $D=6$ and $8$ condensates, the results $0.50(16)(20)$ and $0.40(25)(19)$, respectively, where the first error, in each case, results from the uncertainty on the FB condensate $C_{2N+2}$ and the second from that on $C_{2N+2}^{ud;V+A}$. The results for the FB condensates are thus natural, and compatible with the expectation of FB suppression; the sizeable uncertainties, however, preclude going beyond these qualitative observations.

IV Conclusions

We have revisited the determination of $|V_{us}|$ from flavor-breaking finite-energy sum rule analyses of experimental inclusive non-strange and strange hadronic $\tau$ 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 also used lattice results to bring under better theoretical control the treatment of the potentially problematic $D=2$ OPE series entering these analyses. The new implementation, which employs the FOPT prescription for the integrated $D=2$ OPE series and requires fitting effective $D>4$ condensates to data, dramatically reduces the $w$- and $s_{0}$-instabilities found when conventional implementation assumptions are employed for the $D=6,\,8$ condensates. The $w$- and $s_{0}$-instabilities of the conventional implementation establish that the assumptions employed in that implementation are not self-consistent, and hence that the conventional implementation needs to be abandoned going forward.

It is worth commenting on the relation to earlier attempts to bring the unphysical $w$- and $s_{0}$-dependence of the results for $|V_{us}|$ under improved control. Refs. kmcw06 ; kmcw07 employed degree $8$, $10$ and $20$ weights constructed to simultaneously (i) emphasize $D=2$ contributions from the part of the contour with lower $|\alpha_{s}(Q^{2})|$, with the goal of improving the convergence of the $D=2$ series integrated using the CIPT prescription, and (ii) keep the coefficients $w_{N}$, $N\geq 2$ in $w(y)=\sum_{N}w_{N}y^{N}$, which govern $D>4$ OPE contributions, relatively small kmjkaltwts . Relative to the weights $w_{N}(y)$ employed above, the earlier weights have the disadvantage of producing large numbers of experimentally unconstrained $D>4$ contributions, several governed by coefficients larger than those appearing in the $w_{N}(y)$. Focusing on the “ACO” section of Table II of Ref. kmcw06 , which employs strange exclusive branching fractions closest to (if slightly higher than) those used here, we find, not surprisingly, $s_{0}$-dependences significantly larger than those found from the new implementation employing the lower degree $w_{N}$, which choices allow the relevant $D>4$ effective condensates to be fit to data, rather than neglected as in cases of the weights used in Refs. kmcw06 ; kmcw07 . Similarly, the weight-dependence of the $s_{0}=m_{\tau}^{2}$ $|V_{us}|$ results quoted in Ref. kmcw07 is significantly larger than that found from the new implementation, quoted above. The new implementation thus also supercedes those earlier attempts to address the same $w$- and $s_{0}$-dependence problems.

The new implementation produces results for $|V_{us}|$ $\sim 0.0020$ higher than those obtained analyzing the same data using conventional implementation assumptions for the $D=6$ and $8$ condensates. Taking into account the additional dispersive constraints incorporated by the ACLP $K\pi$ normalization, we find a result, Eq. (10), in excellent agreement with that obtained from $K_{\ell 3}$, and compatible within errors with the expectations of three-family unitarity, thus resolving the long-standing puzzle of the $>3\sigma$ low values of $|V_{us}|$ obtained from the conventional implementation of the FB FESR $\tau$ approach.

Roughly half of the increase from the $0.2186(21)$ conventional implementation result for $|V_{us}|$ quoted in Ref. hflav17 comes from the shift to the ACLP $K\pi$ normalization and half from the use of the new implementation strategy. The use of results for the $D>4$ condensates obtained from fits to data in place of the non-self-consistent conventional implementation assumptions for these condensates is a particularly important feature of the new implementation.

The FB FESR approach to the determination of $|V_{us}|$ has been shown to have very favorable theory errors. The limitations, at present, are entirely experimental in nature, with errors strongly dominated by those on the weighted inclusive strange spectral integrals. In this regard, it is worth noting that the errors on the lower-multiplicity exclusive-mode $K^{-}\pi^{0}$, $\bar{K}^{0}\pi^{-}$, $K^{-}\pi^{+}\pi^{-}$ and $\bar{K}^{0}\pi^{-}\pi^{0}$ contributions, all of which are based on the much higher statistics BaBar and Belle distribution data are, at present, dominated by the uncertainties on the corresponding branching fractions (which normalize the unit-normalized experimental distributions). Significant improvements to the overall error can thus be achieved through improvements to the strange exclusive-mode branching fractions without requiring simultaneous, experimentally more difficult, improvements to the associated differential distributions.

Acknowledgements.

Thanks to RBC/UKQCD for providing access to the data of Ref. rbcukqcdfine11 , used in the OPE-lattice study of the conventional FB FESR implementation. Lattice propagator inversions were performed on the STFC-funded “DiRAC” BG/Q system in the Advanced Computing Facility at the University of Edinburgh. The work of R.J.H., R.L. and K.M. is supported by the Natural Sciences and Engineering Research Council of Canada, that of J.M.Z. by Australian Research Council grants FT100100005 and DP140103067.