Novel j Vus j Determination Using Inclusive Strange τ Decay and Lattice Hadronic Vacuum Polarization Functions

SUPA, School of Physics, The University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom York University, 4700 Keele Street, Toronto, Ontario, Canada M3J IP3 Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom CSSM, University of Adelaide, Adelaide, SA 5005 Australia Physics Department, Nara Women’s University, Nara 630-8506, Japan School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom

The conventional FB FESR implementation employs assumptions for unknown dimension D ¼ 6 and 8 operator product expansion (OPE) condensates which turn out to fail self-consistency tests [9].An alternate implementation, fitting D > 4 condensates to data, yields results passing these tests and compatible with determinations from other sources [9].The resulting error is dominated by uncertainties on the relevant weighted inclusive flavor us spectral integrals and a factor > 2 larger than that of K-decay-based approaches.Improved branching fractions (BFs) used in normalizing low-multiplicity us exclusive-mode Belle and BABAR distributions would help, but ∼25% errors on higher-multiplicity us "residual mode" contributions [10], involving modes not remeasured at the B factories, preclude a factor of 2 improvement [9,11].
This Letter presents a novel dispersive approach to determining jV us j using inclusive strange hadronic τ decay data, hadronic vacuum polarization (HVP) functions computed on the lattice, and weight functions, We show examples of such ω N which strongly suppress spectral contributions from the high-multiplicity us "residual" region without blowing up errors on the related lattice HVP combinations.The approach yields jV us j in good agreement with K-decay analysis results and three-family CKM unitarity expectations.The lattice error is comparable to the experimental one, and the total error is less than that of the inclusive FB FESR τ decay determination.
New inclusive determination.-Theconventional inclusive FB τ decay determination is based on the FESR relation [12,13] Z connecting, for any s 0 and analytic ωðsÞ, the relevant FB combination, ΔΠð−sÞ ¼ Π us ð−sÞ − Π ud ð−sÞ, of spin J ¼ 0, 1 HVPs and associated spectral function ΔρðsÞ ¼ ð1=πÞImΔΠð−sÞ.Experimental data are used on the LHS and, for large enough s 0 , the OPE on the RHS.In the SM, the differential distribution, dR V=A;ij =ds, associated with the flavor ij ¼ ud, us vector (V) or axial vector (A) current-induced decay ratio where and S EW is a known short-distance electroweak correction [15,16].Experimental dR ij;V=A =ds distributions thus determine, up to factors of jV ij j 2 , combinations of the ρ ðJÞ ij;V=A .The low jV us j noted above results from a conventional implementation [17] of Eq. (1) which employs fixed s 0 ¼ m 2 τ and ω ¼ ω τ and assumptions for experimentally unknown D ¼ 6 and 8 condensates.With s 0 ¼ m 2 τ and ω ¼ ω τ , inclusive nonstrange and strange BFs determine the ud and us spectral integrals.Testing D ¼ 6 and 8 assumptions by varying s 0 and/or ω, however, yields jV us j with significant unphysical s 0 -and ω dependence, motivating an alternate implementation employing variable s 0 and ω which allows a simultaneous fit of jV us j and the D > 4 condensates.Significantly larger (now stable) jV us j are found, the conventional implementation results jV us j ¼ 0.2186ð21Þ [7] and 0.2207 (27) [8], shifting up to 0.2208 (23) and 0.2231 (27) [9], respectively, with the new implementation.us spectral integral uncertainties dominate the error, with current ∼25% residual mode contribution errors precluding a competitive determination [9].
Motivated by this limitation, we switch to generalized dispersion relations involving the experimental us V þ A inclusive distribution and weights, us;VþA ðsÞ: ð3Þ For N ≥ 3, the associated HVP combination With Πus ðQ 2 k Þ measured on the lattice, dR us;VþA =ds used to fix s < m 2 τ spectral integral contributions, and s > m 2 τ contributions approximated using perturbative quantum chromodynamics (pQCD), one has Choosing uniform pole spacing Δ, ω N can be characterized by Δ, N, and the pole-interval midpoint, With large enough N, and all Q 2 k below ∼1 GeV 2 , spectral integral contributions from s > m 2 τ and the higher-s, larger-error part of the experimental distribution can be strongly suppressed.Increasing N lowers the error of the LHS in Eq. ( 5) but increases the relative RHS error.With results insensitive to modest changes of Δ, we fix Δ ¼ 0.2=ðN − 1Þ GeV 2 , ensuring ω N with the same C but different N have poles spanning the same Q 2 range.C and N are varied to minimize the error on jV us j.
We employ the following us spectral input: K μ2 or τ → Kν τ [7] for K pole contributions, unit-normalized Belle or BABAR distributions for Kπ [18,19] 23], the most recent Heavy Flavor Averaging Group (HFLAV) BFs [7], and 1999 ALEPH results [10], modified for current BFs, for the residual mode distribution.Multiplication of a unit-normalized distribution by the ratio of corresponding exclusive mode to electron BFs converts that distribution to the corresponding contribution to dR us;VþA ðsÞ=ds.The dispersively constrained Kπ BFs of Ref. [8] (ACLP) provide an alternate Kπ normalization.In what follows, we illustrate the lattice approach using the HFLAV Kπ normalization.Alternate results using the ACLP normalization are given in Ref. [24].
Lattice calculation method.-Wecompute the two-point functions of the flavor us V and A currents, J and q the lattice momentum, qμ ¼ 2 sin q μ =2.FV corrections to this infinite volume result are discussed below.We use lattice HVPs measured on the near-physical quark mass, 2 þ 1 flavor 48 3 × 96 and 64 3 × 128 Möbius domain wall fermion ensembles of the RBC and UKQCD collaborations [27], employing all-mode averaging (AMA) [28,29] to reduce costs.Slight u, d, s mass mistunings are corrected by measuring the HVPs with partially quenched (PQ) physical valence quark masses [27], also using AMA.
Fω N in Eq. ( 5) can be decomposed into four contributions, FðJÞ V=A;ω N , labeled by the spin J ¼ 0 or 1, and current type, V or A.  9) and ( 4) With finite lattice temporal extent, finite-time effects may exist.Increasing N increases the level of cancellation and relative weight of large-t contributions on the RHS of Eq. ( 5).The restrictions 0.1 GeV 2 < C < 1 GeV 2 and N ≤ 5, chosen to strongly suppress large-t contributions, allow us to avoid modeling the large-t behavior.Figure 1 shows, as an example, the large-t plateaus of the partial sums L ðJÞ V=A;ω ðtÞ, obtained in all four channels, for N ¼ 4, C ¼ 0.5 GeV 2 on the 48 3 × 96 ensemble.
The upper panel of Fig. 2 shows the relative sizes of the four C-dependent lattice contributions, V ðJÞ , A ðJÞ , for N ¼ 4. The lower panel, similarly, shows the relative sizes of different contributions to the weighted us spectral integrals.Kπ denotes the sum of K − π 0 and K0 π − contributions, pQCD the contribution from s > m 2 τ , evaluated using the five-loop-truncated pQCD form [30,31]. Varying C (and N) varies the level of suppression of the pQCD and higher-multiplicity contributions, the relative size of K and Kπ contributions, and hence the level of "inclusiveness" of the analysis.The stability of jV us j under such variations provides additional systematic cross-checks.
Analysis and results.-TheA ð0Þ channel produces the largest RHS contribution to Eq. ( 5).On the LHS, the K pole dominates ρ ð0Þ us;A ðsÞ, with continuum contributions doubly chirally suppressed.Estimated LHS continuum A ð0Þ contributions, obtained using sum-rule Kð1460Þ and Kð1830Þ decay constant results [32], are numerically negligible for the ω N we employ.An "exclusive" A ð0Þ analysis relating FA ð0Þ w N to the K-pole contribution RK K obtained from either K μ2 or Γ½τ → Kν τ .Because the simulations underlying FA ð0Þ w N are isospin symmetric, we correct γ K for leading-order electromagnetic (EM) and strong isospinbreaking (IB) effects [4,8].With PDG τ lifetime [6] and HFLAV τ → Kν τ BF [7] input, γ K ½τ K ¼ 0.0012061ð167Þ exp ð13Þ IB GeV 2 .γ K ½τ K is employed in our main, fully τ-based analysis.The more precise result γ K ½K μ2 ¼ 0.0012347ð29Þ exp ð22Þ IB [6] from Γ½K μ2 can also be used if SM dominance is assumed.Exclusive analysis jV us j results are independent of C for C < 1 GeV 2 (confirming tiny continuum A ð0Þ contributions) and agree with the results, jV us j ¼ 0.2233ð15Þ exp ð12Þ th and 0.2260ð3Þ exp ð12Þ th , obtained using jV us j ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15551ð83Þ GeV [27] and γ K ¼ γ K ½τ K and γ K ½K μ2 , respectively.See Ref. [24] for further details.
For the fully inclusive analysis, statistical and systematic uncertainties are reduced by employing 2f 2 K ω N ðm 2 K Þ, with measured f K , for the K pole A ð0Þ channel contribution.The residual, continuum A ð0Þ contributions are compatible with zero within errors, as anticipated above.IB corrections, beyond those applied to γ K , are numerically relevant only for Kπ.We account for (i) π 0 -η mixing, (ii) EM effects, and (iii) IB in the phase space factor, with π 0 -η mixing numerically dominant, evaluating these corrections, and their uncertainties, from the results presented in Ref. [8].A 2% uncertainty, estimated using results from a study of duality violations in the SUð3Þ F -related flavor ud channels [33], is assigned to pQCD contributions.Because our analysis is optimized for ω N strongly suppressing higher-multiplicity and s > m 2 τ contributions, such an uncertainty plays a negligible role in our final error.
Several systematic uncertainties enter the lattice computation.With an assumed continuum extrapolation linear in a 2 but only two lattice spacings, Oða 4 Þ discretization uncertainties must be estimated.For the ω N we employ, the two ensembles yield Fω N differing by less than (typically significantly less than) 10%, compatible with ∼Ca 2 or smaller Oða 2 Þ errors.Anticipating a further ∼Ca 2 reduction of Oða 4 Þ relative to Oða 2 Þ corrections, we estimate residual Oða 4 Þ continuum extrapolation uncertainties to be ∼0.1Ca 2 f , with a −1 f ¼ 2.36 GeV [27] the smaller of the two lattice spacings.We also take into account the lattice scale setting uncertainty.The dominant FV effect is expected to come from Kπ loop contributions in the V ð1Þ channel, which we estimate using a lattice regularized version of finite-volume chiral perturbation theory (ChPT).It is known, from Ref. [34], that one-loop ChPT for HVPs involving the light u, d quarks yields a good semiquantitative representation of observed FV effects [35]; we thus expect it to also work well for the flavor us case considered here, where FV effects involving the heavier s quark should be suppressed relative to those in the purely light u, d quark sector.The result of our one-loop ChPT estimate is a 1% FV correction.We thus assign a 1% FV uncertainty to our V ð1Þ channel contributions [37].Regarding the impact of the slight u, d, s sea-quark mass mistunings on the PQ results, the shift from slightly mistuned unitary to PQ shifted-valence-mass results for Fðω N Þ corresponds to shifts in jV us j of < 0.4% for both ensembles.With masses and decay constants typically much less sensitive to sea-quark mass shifts than to the same valence-quark mass shifts, we expect sea-mass PQ effects to be at the sub-∼0.1% level, and hence negligible on the scale of the other errors in the analysis.Figure 3 shows the C dependence of relative, nondata, inclusive analysis error contributions.K labels the f Kinduced A ð0Þ uncertainty, other that induced by the statistical error on the sum of V ð1Þ , V ð0Þ , A ð1Þ , and tiny continuum A ð0Þ channel contributions.The statistical error dominates for low C, the discretization error for large C.
Figure 4 shows our jV us j results.These agree well for different N, and C < 1 GeV 2 .The slight trend toward lower central values for weights less strongly suppressing high-s spectral contributions (N ¼ 3 and higher C) suggests the residual mode distribution may be somewhat underestimated due to missing higher-multiplicity contributions.Such missing high-s strength would also lower the jV us j obtained from FB FESR analyses.Table I lists relative spectral integral contributions for selected ω N .Note the significantly larger (6.8% and 21%) residual mode and pQCD contributions for N ¼ 3 and C ¼ 1 GeV 2 .Restricting C to < 1 GeV 2 keeps these from growing further and helps control higher-order discretization errors.The error budget for various sample weight choices is summarized in Table II.
Our optimal inclusive determination is obtained for N ¼ 4, C ¼ 0.7 GeV 2 , where residual mode and pQCD contributions are highly suppressed, and yields results jV us j ¼ 0.2228ð15Þ exp ð13Þ th ; for γ K ½τ K 0.2245ð11Þ exp ð13Þ th ; for γ K ½K μ2 ; ð11Þ consistent with determinations from K physics and three-family unitarity.Theoretical (lattice) errors are comparable to experimental ones, and combined errors improve on those of the corresponding inclusive FB FESR determinations.A comparison to the results of other determinations is given in Fig. 5.

Conclusion and discussion
.-We have presented a novel method for determining jV us j using inclusive strange hadronic τ decay data.Key advantages over the related FB FESR approach employing the same us data are (i) the use of systematically improvable precision lattice data in place of the OPE, and (ii) the existence of weight functions that more effectively suppress spectral contributions from the larger-error, high-s region without blowing up theory errors.The results provide not only the most accurate inclusive τ decay sum rule determination of jV us j but also evidence that high-s region systematic errors may be underestimated in the alternate FB FESR approach.The combined experimental uncertainty can be further reduced through improvements to the experimental τ → K, Kπ BFs, whereas the largest of the current theoretical errors, that due to lattice statistics, is improvable by straightforward lattice computational effort.Such future improvements will help constrain the flavor dependence of any new physics contributions present in hadronic tau decays, contributions expected to be present at some level if the apparent violation of lepton flavor universality seen recently in semileptonic b → c decays involving τ persists.
We are grateful to RBC/UKQCD for fruitful discussion and support.We thank the Benasque centre for hospitality at the Benasque workshop "High-precision QCD at low energy," where this project started V=A;ω N ðtÞ, where L ðJÞ V=A;ω N ðtÞ ¼ P t l¼−t ω ðJÞ N ðlÞC ðJÞ V=A ðlÞ.From Eqs. (
. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (Grant No. FP7/2007-2013)/ERC Grants No. 279757 and No. STFC ST/P000711/1.The authors gratefully acknowledge computing time granted through the STFC-funded DiRAC facility (Grants No. ST/ K005790/1, No. ST/K005804/1, No. ST/K000411/1, and ), dR us;VþA =ds directly determines jV 2 us jρ us ðsÞ, with

TABLE I .
Lattice jV us j error contributions for N ¼ 4.FIG.4.jV us j vs C for N ¼ 3, 4, 5. N ¼ 3, 5 results are shifted horizontally for presentational clarity and statistical and systematic errors added in quadrature.The determination using γ½τ K and lattice f K is shown for comparison.Sample relative spectral integral contributions.