A resolution of the inclusive flavor-breaking τ |V_{us}| puzzle
Renwick J. Hudspith, Randy Lewis, Kim Maltman, James Zanotti
aa r X i v : . [ h e p - ph ] M a r ADP-17-07/T1013, YUPP-I/E-KM-17-02-2
A resolution of the inclusive flavor-breaking τ | V us | puzzle Renwick J. Hudspith ∗ and Randy Lewis † Department of Physics and Astronomy, York University,4700 Keele St., Toronto, ON CANADA M3J 1P3
Kim Maltman ‡ Department of Mathematics and Statistics, York University,4700 Keele St., Toronto, ON CANADA M3J 1P3 § James Zanotti ¶ CSSM, Department of Physics, University of Adelaide, Adelaide, SA 5005 AUSTRALIA
Abstract
We revisit the puzzle of | V us | values obtained from the conventional implementation ofhadronic- τ -decay-based flavor-breaking finite-energy sum rules lying > σ below the expec-tations of three-family unitarity. Significant unphysical dependences of | V us | on the choice ofweight, w , and upper limit, s , of the experimental spectral integrals entering the analysis areconfirmed, and a breakdown of assumptions made in estimating higher dimension, D >
4, OPEcontributions identified as the main source of these problems. A combination of continuumand lattice results is shown to suggest a new implementation of the flavor-breaking sum ruleapproach in which not only | V us | , but also D > D = 2 OPE series.The new sum rule implementation is shown to cure the problems of the unphysical w - and s -dependence of | V us | and to produce results ∼ . Kπ branching fractions, we find | V us | = 0 . exp (4) th ,in excellent agreement with the result from K ℓ , and compatible within errors with the expec-tations of three-family unitarity, thus resolving the long-standing inclusive τ | V us | puzzle. PACS numbers: 12.15.Hh,11.55.Hx,12.38.Gc ∗ [email protected] † [email protected] ‡ [email protected] § Alternate address: CSSM, Department of Physics, University of Adelaide, Adelaide, SA 5005 AUS-TRALIA ¶ [email protected] . INTRODUCTION With | V ud | = 0 . | V ub | negligible, 3-family unitary implies | V us | = 0 . | V us | from K ℓ and Γ[ K µ ] / Γ[ π µ ], usingrecent 2014 FlaviaNet experimental results [2] and 2016 lattice input [3] for f + (0) and f K /f π , respectively, yield results, | V us | = 0 . . τ decay data [5], yields | V us | = 0 . . σ below 3-family-unitarity expectations. A less discrepant, but stilllow, result, 0 . Kπ branching fractions (resulting from an analysis of Kπ data imposing additional dispersive constraints on the timelike Kπ form factors [6]).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 ≡ Γ[ τ − → ν τ hadrons V/A ; ij ( γ )] / Γ[ τ − → ν τ e − ¯ ν e ( γ )], are related to the spectralfunctions, ρ ( J ) V/A ; ij , of the J = 0 , ( J ) V/A ; ij , of the correspondingcurrent-current two-point functions, by [7] dR V/A ; ij ds = 12 π | V ij | S EW m τ h w τ ( y τ ) ρ (0+1) V/A ; ij ( s ) − w L ( y τ ) ρ (0) V/A ; ij ( s ) i ≡ π | V ij | S EW m τ (1 − y τ ) ˜ ρ V/A ; ij ( s ) , (1)where y τ = s/m τ , w τ ( y ) = (1 − y ) (1 + 2 y ), w L ( y ) = 2 y (1 − y ) , S EW is a known short-distance electroweak correction [8], and V ij is the flavor ij CKM matrix element. The J = 0 spectral functions, ρ (0) A ; ud,us ( s ), are dominated by the accurately known, chirallyunsuppressed π and K pole contributions. The remaining, continuum contributions to ρ (0) V/A ; ud,us ( s ) are ∝ ( m i ∓ m j ) , and hence negligible for ij = ud . For ij = us , theyare 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 possiblemildly model-dependent determinations in the range s ≤ m τ relevant to hadronic τ decays [9, 10]. 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 + 1spectral function combinations ρ (0+1) V/A ; ud,us ( s ) can be determined.The inclusive τ determination of | V us | employs FB FESRs for the spectral functioncombination, ∆ ρ ( s ) ≡ ρ (0+1) V + A ; ud ( s ) − ρ (0+1) V + A ; us ( s ) and associated polarization difference,∆Π( Q ) ≡ Π (0+1) V + A ; ud ( Q ) − Π (0+1)( Q ) V + A ; us [5], with Q = − s . Generically, for any s > w ( s ), Z s w ( s )∆ ρ ( s ) ds = − πi I | s | = s w ( s )∆Π( − s ) ds . (2)For large enough s , the OPE is used on the RHS.Defining the re-weighted integrals R wV + A ; ij ( s ) ≡ Z s ds w ( s ) w τ ( s ) dR (0+1) V + A ; ij ( s ) ds , (3)and using Eq. (2) to replace the FB difference δR wV + A ( s ) ≡ R wV + A ; ud ( s ) | V ud | − R wV + A ; us ( s ) | V us | , (4)with its OPE representation, one finds, solving for | V us | [5], | V us | = s R wV + A ; us ( s ) / (cid:20) R wV + A ; ud ( s ) | V ud | − δR w,OP EV + A ( s ) (cid:21) . (5)The result is necessarily independent of s and w so long as all input is reliable. As-sumptions employed in evaluating δR w,OP EV + A ( s ) can thus be tested for self-consistency byvarying w and s . OPE assumptions entering the conventional implementation of the FBFESR approach in fact produce | V us | displaying significant w - and s -dependence [11].The low | V us | results noted above are produced by a conventional implementation ofthe general FB FESR framework, Eq. (5), in which a single s ( s = m τ ) and single weight( w = w τ ), are employed [5]. This restriction allows the ij = ud and us spectral integralsto be determined from the inclusive ud and us branching fractions alone, but precludescarrying out s - and w -independence tests. Since w τ has degree 3, δR w τ ,OP EV + A ( s ) receivescontributions up to dimension D = 8. While D = 2 and 4 contributions, determined by α s and the quark masses and condensates [3, 12–15], are known, D > D = 6 contributions are estimated usingthe vacuum saturation approximation (VSA) (see Ref. [16] for the explicit expression)and D = 8 contributions neglected [5, 11]. These assumptions are potentially dangeroussince the FB V+A VSA estimate involves a very strong double cancellation , and theVSA is known to be badly violated, in a channel-dependent manner, from studies in thenon-strange sector [17].Such assumptions can, in principle, be tested by varying s . Writing D > Q ) as P D> C D /Q D , with C D the effective dimension D condensate, the A factor of ∼ ud and us V+A sums are formed, and a furtherfactor of ∼ ud − us V+A difference. D = 2 k + 2 OPE contribution to the RHS of Eq. (2), for a polynomial weight w ( y ) = P n =0 w n y n with y = s/s , is, up to α s -suppressed logarithmic corrections, − πi I | s | = s ds w ( y ) (cid:2) ∆Π( Q ) (cid:3) OP ED =2 k +2 = ( − k w k C k +2 s k . (6)Problems with the assumptions employed for C and C in the conventional implemen-tation will thus manifest themselves as an unphysical s -dependence in the | V us | resultsobtained using weights w ( y ) with non-zero coefficients, w and/or w , of y and y .Another potential issue for the FB FESR approach is the slow convergence of the D = 2 OPE series. To four loops, neglecting O ( m u,d /m s ) corrections [12] (cid:2) ∆Π( Q ) (cid:3) OP ED =2 = 32 π ¯ m s Q (cid:20) a + 19 . a + 208 . a (cid:21) , (7)where ¯ a = α s ( Q ) /π , and ¯ m s = m s ( Q ), α s ( Q ) are the running strange mass andcoupling in the M S scheme. With ¯ a ( m τ ) ≃ .
1, the ratio of O (¯ a ) to O (¯ a ) terms is > | s | = s for all kinematically accessible s . Such slow“convergence” complicates choosing an appropriate truncation order and estimating theassociated truncation uncertainty.No apparent convergence problem exists for the D = 4 series, which, to three loops,dropping numerically tiny O ( m q ) terms, is given by [13] (cid:2) ∆Π( Q ) (cid:3) OP ED =4 = 2 [ h m u ¯ uu i − h m s ¯ ss i ] Q (cid:18) − ¯ a −
133 ¯ a (cid:19) . (8)The slow convergence of the D = 2 OPE series and the reliability of conventionalimplementation assumptions for C and C will be investigated in the next section.In the rest of the paper, the non-strange and strange spectral distributions enter-ing the various FESRs considered are fixed using π µ , K µ and SM expectations forthe π and K pole contributions, recent ALEPH data for the continuum ud V+A dis-tribution [18], Belle [19] and BaBar [20] results for the K − π and ¯ K π − distributions,BaBar results [21] for the K − π + π − distribution, Belle results [22] for the ¯ K π − π dis-tribution, a combination of BaBar [24] and Belle [25] results for the very small ¯ K ¯ KK distribution, and 1999 ALEPH results [23] for the combined “residual mode” distri-bution (the sum over contributions from those strange modes not remeasured by theB-factory experiments). BaBar and Belle exclusive strange mode distributions are givenin unit-normalized form, with measured branching fractions required to set the overallscales. We work, in general, with 2016 HFAG [26] branching fractions. For the two Kπ modes, however, we consider also the alternate results, B [ τ → K − π ν τ ] = 0 . B [ τ → ¯ K π − ν τ ] = 0 . Kπ form factors. Thecorresponding 2016 HFAG Kπ results, obtained without the dispersive constraints, are B [ τ → K − π ν τ ] = 0 . B [ τ → ¯ K π − ν τ ] = 0 . ]1e-061e-050.00010.0010.010.1 [ | V u s | ρ ~ u s ( s )] e x c l K π K - π + π − K π − π residual FIG. 1: Exclusive- and residual-mode contributions to the continuum | V us | ˜ ρ V + A ; us ( s ) distri-bution, with 2016 HFAG normalization for the Kπ points. A plot of the latest version of the ALEPH ud V+A spectral distribution may be foundin Ref. [18]. The exclusive- and residual-mode contributions to the continuum us V+Adistribution, in the form, | V us | ˜ ρ V + A ; us ( s ), directly determinable from the experiment,are shown in Figure 1. For definiteness, the Kπ points are shown with the 2016 HFAG Kπ normalization. A global rescaling of 1 .
044 is required to convert these to the alternate2013 ACLP Kπ normalization.We base our central results on the additionally-constrained ACLP input choice, butquote results obtained using both Kπ normalizations. Note that the publicly availableALEPH continuum ud V+A distribution is normalized to a slightly older version of theinclusive ud continuum branching fraction. A small rescaling (0 .
5% or less) is requiredto convert this to the normalization implied by the branching fractions we employ. Thenormalizations of the different components of the 1999 ALEPH residual mode distributionare also updated using HFAG 2016 branching fractions [26].
II. TESTING CONVENTIONAL IMPLEMENTATION ASSUMPTIONS
The conventional implementation assumptions, C ≃ C V SA and C = 0, can be ef-ficiently investigated using appropriately chosen s - and w -independence tests. A com-parison of the results of the w τ ( y ) = 1 − y + 2 y and ˆ w ( y ) = 1 − y + 3 y − y FESRsis particularly illuminating since the coefficients of y in the two weights differ only by5 sign. The corresponding integrated D = 6 OPE contributions are thus identical inmagnitude but opposite in sign. If, as the VSA estimate suggests, D = 6 contributionsare small for w τ , they must also be small for ˆ w . Similarly, if integrated D = 8 contri-butions are negligible for w τ , those for ˆ w , which are − / C and C are reliable, the | V us | obtained from the w τ and ˆ w FESRs should thus be in good agreement, and showgood individual s stability. In contrast, if these assumptions are not reliable, and D = 6and D = 8 contributions are not both small, one should see s -instabilities of oppositesigns in the two cases, and s -dependent differences in the results from the two FESRswhich decrease with increasing s .The central values of the results of this comparison, obtained using the ACLP andHFAG Kπ normalizations, and, to be specific, the 3-loop-truncated contour-improved(CIPT) prescription [27] for handling the integrated D = 2 OPE series, are shown inthe 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 givenweight but different s are strongly correlated, as are the w τ and ˆ w results at the same s .Neither changing the D = 2 truncation order nor switching from CIPT to the alternatefixed-order (FOPT) D = 2 prescription for the D = 2 series serves to remove the strong,unphysical s and weight dependences.To understand the extent to which the s - and w -instabilities shown in Figure 2 area problem for the conventional implementation D > | V us | obtained from the ˆ w and w τ FESRsat the same s . If the conventional implementation assumptions are reliable these dif-ferences should be zero within errors. Fully propagating the ud and us experimentalcovariances, and adding independent sources of theory error in quadrature, we find, how-ever, ˆ w - w τ differences of 0 . exp (38) th at s = 1 . GeV , 0 . exp (21) th at s = 2 . GeV , and 0 . exp (13) th at s = 3 . GeV , clearly signalling prob-lems with the conventional implementation assumptions. Similar conclusions follow fromthe observed s -instabilities. For example, the difference between the ˆ w FESR resultsat s = 2 . GeV and 3 . GeV , which should once more be zero within errors, isinstead 0 . exp (8) th . A similarly discrepant result, 0 . exp (19) th , is found forthe difference between the s = 2 . GeV and 3 . GeV ˆ w results.The top right and bottom right panels of Fig. 2 show the results of correspondingadditional w - and s -independence tests involving the weights w N ( y ), N = 2 , ,
4, with w N ( y ) = 1 − NN − y + 1 N − y N . (9)The upper solid lines in each case show the w , w and w results obtained using theconventional implementation treatment of D > Kπ nor-malization, while the dashed-dotted show lines the corresponding results produced by The w N ( y ), like w τ ( y ), have a double zero at s = s ( y = 1). This serves to keep duality violatingcontributions safely small above s ≃ GeV [28]. D > w τ results (represented by the lowest solid and dotted lines, respectively) are alsoincluded for comparison. The latter are obtained using the D = 6 and 8 effective con-densates obtained from the alternate implementation w and w fits. The s -dependent,conventional implementation results for all of w τ , ˆ w , w , w and w show evidence of con-verging toward a common value at s > m τ , as expected if the observed s -instabilitiesresult from D > [GeV ]0.220.2220.2240.2260.2280.23 | V u s | w τ (y)w^ (y) [GeV ]0.220.2220.2240.2260.2280.23 | V u s | w (y), VSA D=6w (y), VSA D=6w (y), VSA D=6w (y), fitted C w (y), fitted C w (y), fitted C w τ (y), VSA D=6w τ (y), fitted C , C [GeV ]0.2180.220.2220.2240.2260.228 | V u s | w τ (y)w^ (y) [GeV ]0.2180.220.2220.2240.2260.228 | V u s | w (y), VSA D=6w (y), VSA D=6w (y), VSA D=6w (y), fitted C w (y), fitted C w (y), fitted C w τ (y), VSA D=6w τ (y), fitted C , C FIG. 2: Left panels: conventional implementation w τ (bottom curve) and ˆ w (top curve) resultsfor | V us | . Right panels: w N and w τ FESR results obtained using the fixed-order (FOPT) D = 2prescription. Solid lines show, top to bottom, conventional implementation results for w , w , w and w τ . Dashed-dotted lines show, bottom to top, w , w and w results, and the dottedline w τ results, obtained using central C , fit values from the alternative FB FESR analysesdescribed in the text. Figures in the first row show results obtained using the ACLP Kπ normalization, those in the second row those obtained using the HFAG Kπ normalization D = 2 OPE series can be investigatedby comparing OPE expectations to lattice results for ∆Π( Q ) over a range of Euclidean Q = − q = − s , using variously truncated versions of the D = 2 OPE series. Latticeresults were obtained using the RBC/UKQCD n f = 2 + 1, 32 ×
64, 1 /a = 2 .
38 GeV,domain wall fermion ensemble with m π ∼
300 MeV [29]. A tight cylinder cut, with aradius determined in a recent study of the extraction of α s from lattice current-currenttwo-point function data [30], was imposed to suppress lattice artifacts at higher Q . Thevalues of the light quark masses, m u = m d ≡ m ℓ and m s , for this ensemble, determinedin Ref. [29], were used for determining the corresponding OPE expectations.We consider the comparison first for larger Q , where D = 2 and 4 contributions shoulddominate. The D = 2 OPE contribution is determined using ensemble values of m u and m s [29], the central PDG value for α s [14], and considering 2-, 3- and 4-loop truncationof the D = 2 series. Both fixed scale, µ = 4 GeV , and local scale, µ = Q , choices forhandling the logarithms in the truncated series are considered. The former choice is theanalogue of the fixed order (FOPT) prescription for the D = 2 FESR contour integrals,the latter the analogue of the alternate CIPT prescription. For D = 4 contributions,Eq. (8), we employ the Gell-Mann-Oakes-Renner (GMOR) relation for h m u ¯ uu i and fix h m s ¯ ss i using the ensemble value of m s /m ℓ , translating the HPQCD result for h ¯ ss i / h ¯ ℓℓ i at physical quark masses [15], to that for the ensemble masses using NLO ChPT [31]. [GeV ]0.0020.0030.004 Q ∆ Π τ ( Q ) [ G e V ] Lattice data2-loop D=2 truncation3-loop D=2 truncation4-loop D=2 truncation [GeV ]0.0020.0030.004 Q ∆ Π τ ( Q ) [ G e V ] Lattice data2-loop D=2 truncation3-loop D=2 truncation4-loop D=2 truncation
FIG. 3: Comparison of lattice results and D = 2 + 4 OPE expectations for Q ∆Π τ ( Q ), foreither fixed-scale (left panel) or local-scale (right panel) treatments of the D = 2 series. The comparisons obtained using the fixed- and local-scale versions of the D = 2 seriesare shown in the left and right panels of Fig. 3, respectively. The best representation ofthe lattice results is provided by the 3-loop-truncated, fixed-scale version, which produces The FOPT prescription evaluates weighted D = 2 OPE integrals using a fixed-scale choice (usually µ = s ), the CIPT prescription [27] using the local-scale choice, µ = Q .
8n excellent match over a wide range of Q , extending from near ∼
10 GeV down tojust above ∼ , with the Q dependence of the lattice results also favoring thefixed-scale over the alternate local-scale treatment .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 region. Below ∼ GeV , the lattice results clearly require D > w τ - ˆ w FESR comparison above. Theright 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 errorsfor the latter also displayed. These are obtained by combining in quadrature standardestimates for the D = 4 truncation errors with uncertainties produced by those on theinput D = 2 and 4 OPE parameters. Despite the apparently problematic convergencebehavior of the D = 2 series, conventional OPE error estimates are seen to provide anextremely conservative assessment of the uncertainty for the D = 2 + 4 sum. Note that both the lattice data and OPE results at different Q are highly correlated. These correla-tions (and not just the errors on the individual OPE and lattice points) must be taken into account toassess the significance (or lack thereof) of the difference in the Q dependences of the local-scale OPEand lattice results. The uncertainty on the Q dependence is, in fact, strongly dominated by thaton the input strange-to-light condensate ratio. Taking all correlations into account, one finds, for thefixed- and local-scale versions of the ratio of OPE to lattice values of the average slope between, forexample, Q ≃ GeV and Q ≃ GeV , the results 1 . . Q dependence of the lattice data thus favors the fixed-scale treatment of the D = 2 series. [GeV ]0.0030.0040.005 Q ∆ Π τ ( Q ) [ G e V ] Lattice data3-loop D=2+D=4 OPE3-loop D=2 + D=4 + VSA D=6 OPE [GeV ]0.0020.0030.004 Q ∆ Π τ ( Q ) [ G e V ] Lattice dataCentral 3-loop D=2 + D=4 OPE3-loop D=2 + D=4 OPE +_ 1 σ FIG. 4: Left panel: Comparison of lower- Q lattice results to D = 2 + 4 and D = 2 + 4 + 6OPE expectations (fixed-scale, 3-loop truncation for D = 2, VSA for D = 6). Right panel:Lattice results and the D = 2+4 OPE sum at larger Q , with conventional OPE error estimates(fixed-scale, 3-loop-truncated D = 2). III. AN ALTERNATE IMPLEMENTATION OF THE FB FESR APPROACH
The results of the previous section suggest an obvious alternative to the conventionalimplementation of the FB FESR approach. First, the 3-loop-truncated FOPT treatmentfavored by the comparison to the high- Q lattice results is employed for the D = 2 OPEintegrals . Second, since both lattice and continuum results suggest that conventionalimplementation assumptions for the effective D > C D , are unreliable, weavoid 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> since the w N -weighted OPEintegral involves only a single D > − N C N +2 / (cid:2) ( N − s N (cid:3) . The s dependence of the w N -weighted spectral integrals in the region above s ∼ GeV ,where residual duality violations remain small, then provides sufficient information toallow both unknowns, | V us | and C N +2 , entering the w N FESR to be determined.Spectral distribution inputs employed in our analysis were outlined above . On theOPE side, for the D = 2 and 4 contributions, we use PDG input for α s [14], FLAGinput for the light and strange quark masses [3], GMOR for the light-quark conden-sate [33], and the HPQCD lattice result [15] for the ratio of strange to light quarkcondensates. The single-weight w , w and w FESR | V us | fit results obtained using It is worth noting that the prescription of truncating at 3-loop order is also what one would arrive atwere one to interpret the series as asymptotic and truncate it at its smallest term. Note also that, when using the ACLP Kπ normalization, we have, for consistency, implemented thelong-distance electromagnetic corrections employed in arriving at the Kπ branching fraction resultsobtained from the ACLP analysis[32]. Kπ normalization, 0 . exp (4) th , 0 . exp (4) th and 0 . exp (4) th , respectively, show a dramatically reduced weight dependencerelative to those of the obtained using conventional implementation assumptions forthe D > . exp (4) th ,0 . exp (4) th and 0 . exp (4) th , obtained using the alternate HFAG Kπ nor-malization. It is worth commenting that, although the lattice results for Euclidean Q favor the fixed-scale treatment of the D = 2 series, and hence, by extension, the FOPTprescription for the weighted D = 2 FESR integrals, the final results for | V us | are ratherinsensitive to choosing FOPT over CIPT. Explicitly, the alternate CIPT choice yields0 . exp (4) th for all of the w , w and w FESRs when the ACLP Kπ normalizationis used and 0 . exp (4) th when the HFAG Kπ normalization is used. The CIPTtreatment, of course, generates slightly different fit results for the C D> , as expected,given that FOPT and CIPT represent different partial resummations of the presumablyasymptotic D = 2 series.Given the excellent consistency of the individual w , w and w FESR determina-tions, we take our final result from a combined 3-weight fit. The central ACLP Kπ normalization choice yields | V us | = 0 . exp (4) th , (10)0 . D > K ℓ and compatible within errors with the expectations of 3-family unitarity.The combined 3-weight fit result, | V us | = 0 . exp (4) th , generated by the alternate(HFAG) choice of Kπ normalization, similarly, lies 0 . D > w , w and w fits employing the ACLP Kπ normalization. Theory errors, resulting from uncertainties in the input parameters α s , m s and h m s ¯ ss i , and the small J = 0 continuum subtraction, are labelled by δα s , δm s , δ h m s ¯ ss i and δ ( J = 0 sub ), respectively, and shown above the horizontal line. Thoseinduced by the covariances of the non-strange and strange experimental distributions dR V + A ; ud /ds and dR V + A ; us /ds are denoted δ expud and δ expus and shown below the horizontalline. The δ expus uncertainties strongly dominate the total errors.From the lattice-OPE comparison discussed above, the estimates in the upper half of thetable should provide a very conservative assessment of theoretical uncertainties. Com-bining the different components in quadrature yields a total theory error of 0 . | V us | for all three determinations. The new implementation of the FB FESR approachis thus competitive with the alternate K ℓ and Γ[ K µ ] / Γ[ π µ ] determinations from a the-ory error point of view, though improvements to the errors on the strange experimentaldistributions are required to make it fully competitive over all.To test whether fitting the D > s -instabilities found in the conventional implementation, we have rerun the s -dependent11 ABLE I: Single-weight fit | V us | error contributions for the w , w and w FESRs, using 3-loop-truncated FOPT for the D = 2 OPE series. Notation as described in the text.Source δ | V us | δ | V us | δ | V us | w FESR w FESR w FESR δα s δm s (2 GeV ) 0.00008 0.00009 0.00008 δ h m s ¯ ss i δ ( J = 0 sub ) 0.00009 0.00009 0.00009 δ expud δ expus w N analyses, using the central fitted C N +2 values as input . The dashed-dotted lines inthe right panel of Fig. 2 show the results of this exercise. Using the fitted C D> valuesdramatically reduces the s -instabilities of the conventional implementation versions ofthe same analyses, providing a strong self-consistency check on the new FB FESR imple-mentation. The dotted line in this same panel shows the analogous | V us | results obtainedfrom the s -dependent w τ analysis using the fitted values of C and C as input. Oneagain finds a dramatic reduction in the s dependence, as well as excellent agreementwith the results obtained using the other weights.Errors on the us distribution data limit the precision with which the C N +2 (whichrepresent nuisance parameters for the determination of | V us | ) can be currently deter-mined. It is, nonetheless, worth checking that the results for the FB condensates arecompatible with an expected FB suppression relative to the corresponding flavor ud V+A condensates. Comparing the results for C and C from our favored (FOPT) fits tothose for the corresponding ud V+A condensates, C ud ; V + A , , obtained from the favored, s min = 1 . GeV , 3-weight, combined V&A FOPT fit of Ref. [17], we find, for the ratiosof FB to non-FB D = 6 and 8 condensates, the results 0 . . C N +2 and the second from that on C ud ; V + A N +2 . The results for the FB con-densates are thus natural, and compatible with the expectation of FB suppression; thesizeable uncertainties, however, preclude going beyond these qualitative observations. It is worth noting that the central fitted C N +2 > produce contributions to the w N FESRs whichappear natural in size relative to the known D = 2 and 4 contributions. At s = m τ , for example,relative to the corresponding D = 2 contributions, D = 4 and 6 contributions are ∼
83% and − w , D = 4 and 8 contributions ∼
67% and −
11% for w , and D = 4 and 10 contributions ∼ −
5% for w . V. CONCLUSIONS
We have revisited the determination of | V us | from flavor-breaking finite-energy sumrule analyses of experimental inclusive non-strange and strange hadronic τ decay distri-butions, identifying an important systematic problem in the conventional implementationof this approach, and developing an alternate implementation which cures this problem.We have also used lattice results to bring under better theoretical control the treatmentof the potentially problematic D = 2 OPE series entering these analyses. The new imple-mentation, which employs the FOPT prescription for the integrated D = 2 OPE seriesand requires fitting effective D > w - and s -instabilities found when conventional implementation assumptions are employed forthe D = 6 , w - and s -instabilities of the conventional implementationestablish 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 -dependence of the results for | V us | under improved control. Refs. [34, 35] em-ployed degree 8, 10 and 20 weights constructed to simultaneously (i) emphasize D = 2contributions from the part of the contour with lower | α s ( Q ) | , with the goal of im-proving the convergence of the D = 2 series integrated using the CIPT prescription,and (ii) keep the coefficients w N , N ≥ w ( y ) = P N w N y N , which govern D > w N ( y ) employed above,the earlier weights have the disadvantage of producing large numbers of experimentallyunconstrained D > w N ( y ). Focusing on the “ACO” section of Table II of Ref. [34], whichemploys strange exclusive branching fractions closest to (if slightly higher than) thoseused here, we find, not surprisingly, s -dependences significantly larger than those foundfrom the new implementation employing the lower degree w N , which choices allow therelevant D > s = m τ | V us | results quoted in Ref. [35] is significantly larger than that found from the new im-plementation, quoted above. The new implementation thus also supercedes those earlierattempts to address the same w - and s -dependence problems.The new implementation produces results for | V us | ∼ . D = 6 and 8 condensates. Taking into account the additional dispersive constraintsincorporated by the ACLP Kπ normalization, we find a result, Eq. (10), in excellentagreement with that obtained from K ℓ , and compatible within errors with the expec-tations of three-family unitarity, thus resolving the long-standing puzzle of the > σ low values of | V us | obtained from the conventional implementation of the FB FESR τ approach.Roughly half of the increase from the 0 . | V us | quoted in Ref. [4] comes from the shift to the ACLP Kπ normalization andhalf from the use of the new implementation strategy. The use of results for the D > | V us | has been shown to have veryfavorable theory errors. The limitations, at present, are entirely experimental in na-ture, with errors strongly dominated by those on the weighted inclusive strange spectralintegrals. In this regard, it is worth noting that the errors on the lower-multiplicity exclusive-mode K − π , ¯ K π − , K − π + π − and ¯ K π − π contributions, all of which are basedon the much higher statistics BaBar and Belle distribution data are, at present, domi-nated by the uncertainties on the corresponding branching fractions (which normalize theunit-normalized experimental distributions). Significant improvements to the overall er-ror can thus be achieved through improvements to the strange exclusive-mode branchingfractions without requiring simultaneous, experimentally more difficult, improvements tothe associated differential distributions. Acknowledgments
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