The inclusive flavor-breaking τ-based sum rule determination of Vus

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Nuclear and Particle Physics Proceedings 287–288 (2017) 25–28 www.elsevier.com/locate/nppp

The inclusive flavor-breaking τ-based sum rule determination of Vus K. Maltmana,b , R.J. Hudspithc , T. Izubuchid , R. Lewisc , H. Ohkie , J.M. Zanottib a Math

and Statistics, York University, Toronto, ON Canada M3J 1P3 University of Adelaide, Adelaide, SA Australia 5005 c Physics and Astronomy, York University, Toronto, ON Canada M3J 1P3 d Physics Department, Brookhaven National Laboratory, Upton, NY, 11973, USA e RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY, 11973, USA b CSSM,

Abstract We revisit the puzzle of the > 3σ low (c.f. three-family-unitarity expections) determination of Vus from the conventional implementation of flavor-breaking (FB) finite-energy sum rules (FESRs) employing inclusive hadronic τ decay data. Problems with this implementation are identified and an alternative implementation which cures them described. Using this new implementation and preliminary BaBar results for the exclusive τ → K − π0 ντ branching fraction, we find Vus = 0.2228(23)exp (6)th , in excellent agreement with results from other sources. Limitations on the near-term possibilities for reducing experimental errors in this approach are highlighted, and a new proposal for improving this situation via dispersive analyses of strange hadronic τ data using lattice data as input described. Keywords: Vus , sum rules 1. Introduction With |Vud | = 0.97417(21) [1] as input, three-familyunitary implies |Vus | = 0.2258(9). Direct determinations from K3 and Γ[Kμ2 ]/Γ[πμ2 ], using recent 2014 FlaviaNet experimental results [2] and 2016 lattice input [3], yield results, |Vus | = 0.2231(9) and 0.2253(7), in reasonable agreement with this expectation. In contrast, the most recent update of the conventional implementation of the FB FESR hadronic τ decay approach [4] yields a result, |Vus | = 0.2176(21) [5] 3.6σ below this expectation. In the Standard Model, the differential distributions, dRV/A;i j /ds, for flavor i j = ud, us, vector (V) or axialvector (A) current-mediated decays, where RV/A;i j ≡ Γ[τ− → ντ hadronsV/A;i j (γ)]/Γ[τ− → ντ e− ν¯ e (γ)], are related to the spectral functions, ρ(J) V/A;i j , of the J = 0, 1 scalar polarizations, Π(J) V/A;i j , of the corresponding current-current two-point functions, by [6] dRV/A;i j ds

=

12π2 |Vi j |2 S EW (1 − yτ )2 ρ(s) ˜ , (1) m2τ

http://dx.doi.org/10.1016/j.nuclphysbps.2017.03.037 2405-6014/© 2017 Elsevier B.V. All rights reserved.

(0) ˜ = (1+2yτ )ρ(1) with yτ = s/m2τ , ρ(s) V/A;i j (s)+ρV/A;i j (s), S EW a known short-distance electroweak correction, and Vi j the flavor i j CKM matrix element. When rewritten in terms of kinematic-singularity-free combinations, the dominant ρ(0+1) V/A;i j appears in combination with the “kinematic weight” wτ (y) = (1 − y)2 (1 + 2y). The π and K pole contributions are not chirally suppressed and dominate ρ(0) A;ud,us (s). The remaining, doubly-chirallysuppressed continuum J = 0 contributions are negligible for i j = ud, and small for i j = us. Estimating the latter using the related i j = us scalar and pseudoscalar sum rules [7, 8], ρ(0+1) V/A;ud,us (s) can be determined from the experimental dRV/A;i j /ds distributions. The inclusive τ determination of |Vus | employs FB FESRs for the polarization difference, ΔΠ ≡ Π(0+1) V+A;ud − (0+1) ΠV+A;us , and associated spectral function, Δρ ≡ (0+1) ρ(0+1) V+A;ud − ρV+A;us [4], generically,  s0  1 w(s)Δρ(s) ds = − w(s)ΔΠ(s) ds , (2) 2πi |s|=s0 0

valid for any s0 > 0 and analytic w(s). For large enough

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26

s0 , the OPE can be used on the RHS. On the LHS, for general w, subtracting J = 0 contributions yields the J = 0+1 analogue, dR(0+1) V/A;i j /ds, of dRV/A;i j /ds. Defining the re-weighted integrals RwV+A;i j (s0 ) ≡

s0

ds 0

w(s) wτ (s)

dR(0+1) V+A;i j (s) ds

,

(3)

and using Eq. (2) to replace the FB difference δRwV+A (s0 ) ≡

RwV+A;ud (s0 ) |Vud |2

−

RwV+A;us (s0 ) |Vus |2

wτ(y) ^ (y) w

0.23

|Vus|



Figure 1: |Vus | as a function of s0 from the wτ and wˆ FESRs, with conventional implementation OPE assumptions as input.

,

0.225

0.22

(4)

with its OPE representation, one finds [4],   w  RV+A;ud (s0 ) w,OPE |Vus | = RwV+A;us (s0 )/ − δR (s ) 0 . V+A |Vud |2 (5) Self-consistency tests are provided by the s0 - and windependence of this result. The > 3σ low |Vus | results discussed above result from a conventional implementation of Eq. (5) [4] with s0 = m2τ and w = wτ . With this choice, the spectral integrals can be obtained from the inclusive nonstrange and strange branching fractions, though s0 - and w-independence tests are impossible. wτ has degree τ ,OPE (s0 ), therefore, OPE contributions up 3, and δRwV+A to dimension D = 8. D = 2 and 4 contributions, which involve α s and the quark masses and condensates [3, 9, 10, 11], are known, while experimentally unknown D = 6 condensates are estimated with the vacuum saturation approximation (VSA), and D = 8 contributions neglected [4, 12]. The very strong cancellation in the FB D = 6 VSA estimate and known crudeness of the VSA in the ud sector [13] make the D = 6 and 8 assumptions of the conventional implementation potentially dangerous. There is also a potential issue with the slow convergence of the D = 2 OPE series which, to 4-loops, neglecting O(m2u,d /m2s ) corrections, has the form [9]   ¯s 7 3 m 2 3 a ¯ + 19.93¯ a + 208.75¯ a 1 + , (6) 3 2π2 Q2 with a¯ = α s (Q2 )/π, and m ¯ s = m s (Q2 ), α s (Q2 ) the running MS strange mass and coupling. With a¯ (m2τ )  0.1, the O(¯a3 ) term is larger than the O(¯a2 ) one at the spacelike point on the contour for all s0 accessible in τ decays. The D > 4 assumptions of the conventional implementation are testable by comparing the results of the wτ (y) = 1 − 3y2 + 2y2 and w(y) ˆ = 1 − 3y + 3y2 − y3 (y = s/s0 ) FESRs, which have integrated D = 6 OPE contributions equal in magnitude but opposite in sign

0.215 2

2.5 2 s0 [GeV ]

3

and integrated D = 8 contributions for wˆ −1/2 those of wτ . Small D = 6 and negligible D = 8 contributions for wτ , thus mean similarly small D > 4 contributions for w. ˆ The two FESRS should then produce compatible, s0 -stable |Vus | results. A breakdown of conventional implementation D > 4 assumptions, in contrast, will produce s0 -instabilities of opposite sign for the two FESRs and an output |Vus | difference decreasing with increasing s0 . The results of the comparison, shown in Fig. 1, obviously support the second scenario, not the first. The D = 2 convergence issue was investigated by comparing OPE expectations to n f = 2 + 1 RBC/UKQCD lattice data [14] for ΔΠ(Q2 ). An excellent match of lattice and D = 2 + 4 OPE results was observed over the broad high-Q2 interval 4 GeV 2 < Q2 < 10 GeV 2 when the D = 2 series was evaluated using 3loop truncation and fixed (rather than local) scale treatment of logarithmic contributions [15].1 Conventional D = 2 + 4 OPE error estimates were also seen to be extremely conservative [15]. Below Q2 ∼ 4 GeV 2 , much larger deviations of the D = 2 + 4 OPE sum from the lattice data are seen than expected from conventional implementation D > 4 assumptions [15]. 2. A new FB FESR implementation An alternate FB FESR implementation [15] is suggested by the observations above. One uses the 3-looptruncated FOPT version of D = 2 OPE contributions favored by lattice data and fits the effective D > 4 OPE 1 The fixed- and local-scale treatments are the analogues of the “fixed-order” (FOPT) and “contour-improved” (CIPT) FESR D = 2 series prescriptions.

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condensates, C D , rather than making assumptions about them. FESRs based on the weights 1 N N y+ y , (7) N−1 N−1 which involve only a single unknown D = 2N + 2 > 4 OPE contribution, are particularly convenient. The 1/s0N scaling of D = 2N + 2 OPE contributions allows both |Vus | and C2N+2 to be obtained from a wN FESR fit. Weighted non-strange and strange spectral integrals are determined as follows. Pion and kaon pole contributions are evaluated using πμ2 , Kμ2 and SM expectations and continuum ud spectral integrals using the ALEPH ud V+A distribution [16]. Continuum us V+A spectral integrals are obtained by summing over exclusive modes, with Belle [17] and BaBar [18, 19] results used for the K¯ 0 π− and K − π0 distributions, BaBar [20] and Belle [21] results for the K − π+ π− and K¯ 0 π− π0 distributions, and 1999 ALEPH results [22] for the combined distribution of exclusive strange modes not remeasured by at B-factories. Two different possibilities exist for the K − π0 ντ branching fraction which normalizes the corresponding exclusive distribution: 0.00433(15) from the 2014 HFAG summer fit [23], and 0.00500(14) from a preliminary BaBar thesis update [19]. The latter is favored by BaBar, whose earlier result dominates the 2014 HFAG average. Central results below are obtained using the latter choice. Figure 2 shows the w2,3,4 FESR results for |Vus | as a function of s0 . The solid lines were obtained using standard implementation OPE assumptions/input, the dashed lines using as D > 4 OPE input the central results for the D > 4 effective condensates C2N+2 obtained from the new-implementation wN FESR fit. Switching from conventional implementation assumptions to the fitted values of the effective condensates C D>4 is seen to completely cure the s0 - and w-instablility problems of the conventional implementation. With the |Vus | from the different wN FESRs agreeing well, we obtain our final result by performing a combined 3-weight fit. With the favored preliminary BaBar branching fraction as normalization for the exclusive K − π0 distribution, we find [15]

Figure 2: |Vus | from the conventional (solid lines) and new implementations (dashed lines) of the wN FESRs.

wN (y) = 1 −

|Vus | = 0.2228(5)th (23)exp ,

(8)

where the uncertainty in m s s¯ s dominates the theory error and the strange exclusive distribution errors the experimental error [15]. The result agrees well with that from K3 , and also, within errors, with 3-family unitarity expectations.2 Compared to the > 3σ low conven2 With the K − π0

distribution instead normalized by the HFAG 2014

27

w2(y), VSA D=6 w3(y), VSA D=6 w4(y), VSA D=6 w2(y), fitted C6 w3(y), fitted C8 w4(y), fitted C10

|Vus|

0.228

0.225

0.222

2

2.5

2

3

s0 [GeV ]

tional implementation results, roughly half of the improved agreement results from the data-based treatment of higher D OPE contributions, and half from the use of the new preliminary BaBar K − π0 ντ branching fraction. The removal of s0 - and w-instability however, results entirely on the data-based D > 4 OPE treatment.

Table 1: Relative wN -weighted us spectral integral contributions in the s0 fit window of the alternate FB FESR implementation. s0 is in GeV 2 . Kπ column entries are the sum of the K − π0 and K¯ 0 π− contributions, Kππ column entries the sum of the K − π+ π− and K¯ 0 π− π0 contributions, and Resid column entries the contributions of the residual mode part of the 1999 ALEPH distribution.

Weight w2 w3 w4

s0 2.15 3.15 2.15 3.15 2.15 3.15

K 0.496 0.360 0.461 0.331 0.441 0.314

Kπ 0.426 0.414 0.446 0.415 0.456 0.411

Kππ 0.062 0.162 0.073 0.182 0.082 0.194

Resid 0.017 0.065 0.019 0.074 0.021 0.081

While significant reductions in the |Vus | error resulting from this new FB FESR implementation are possible through improvements to the low-multiplicity strange exclusive branching fractions [15], uncertainties in the combined, higher-multiplicity 1999 ALEPH “residual mode” distribution, whose weighted spectral integrals have errors of ∼ 25%, are likely to prove an branching fraction, the analysis yields |Vus | = 0.2200(5)th (23)exp , 0.0024 higher than the conventional implementation result obtained with the same experimental input. More work on the branching fraction of this mode is clearly called for.

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important limiting factor in the near future. A competitive determination of |Vus | requires sub-0.5% precision, which requires sub-% precision on the weighted inclusive us spectral integrals. The relative contributions of the lower-multiplicity exclusive modes, as well as that of the residual mode sum, to the inclusive weighted us spectral integrals for the w2 , w3 and w4 FESRs are shown in Table 1 at the lowest and highest s0 of the analysis fit window. The ∼ 25% residual mode errors correspond to ∼ 2% errors on the inclusive us spectral integrals at the lower end of the s0 fit window. A factor of > 2 improvement in the residual mode sum distribution errors is thus needed to make the FB FESR approach fully competitive with kaon-physics-based determinations. A way around this limitation is to switch to a dispersive analysis based on the inclusive us data alone in which weights are chosen that allow lattice data to be used in place of the OPE on the theory side of the dispersion relations. This works as follows [15]. From Eq. (1), the experimental dRus;V+A /ds distribution provides a direct determination of |Vus |2 ρ(s), ˜ with no, even mildly model-dependent, continuum J = 0 subtraction required. The combination ρ(s) ˜ is the spectral function of the kinematic-singularity-free J = 0 and 1 us V+A polarization combination,   2 ˜ us;V+A ≡ 1 − 2 Q Π(J=1) + Π(J=0) , (9) Π us;V+A us;V+A m2τ where Q2 = −s. Choosing weights, WN (s), 1 , 2 (s k=1 + Qk )

WN (s) ≡ N

∞ 0

ds WN (s) ρ˜ us;V+A (s) =

N

k=1

References [1] [2] [3] [4] [5] [6] [7]

[8] [9]

[10] [11] [12] [13]

(10)

which have poles at the N distinct Euclidean locations Q2 = Q21 , · · · Q2N , Q2k > 0, one has, for N ≥ 3, the convergent, unsubtracted dispersion relation 

ρ(s)) ˜ are very small. This goal can be accomplished by keeping all Q2k below ∼ 1 GeV 2 and choosing N large enough. Increasing N, however, increases the errors on the lattice side of Eq. (11) (the level of cancellation in the sum of residues grows with increasing N). The error on |Vus | extracted using Eq. (11) is minimized by optimizing the choice of N and pole locations, subject to these two competing constraints. A preliminary implementation of this approach, showing that contributions from the higher-error, high-s part of the us spectral distribution can be suitably suppressed without blowing up theoretical errors, is described in the H. Ohki’s write-up in these proceedings.

[14] [15] [16] [17]

˜ us;V+A (Q2 ) Π k . (11)  2 2 jk Q j − Qk

˜ us;V+A (Q2 ) on Lattice data is to be used to evaluate the Π k the RHS of this relation. This can be done with good accuracy if all Q2k are kept to a few to several tenths of a GeV 2 . The s ≤ m2τ contribution to the LHS is determinable, up to the unknown factor |Vus |2 , from experimental dRus;V+A /ds data. To control errors on the LHS, the number, N, and locations, Q21 , · · · , Q2N , of the poles, are to be chosen such that contributions from both the region where us data errors are large and the region s > m2τ (where data do not exist and pQCD is used for

[18] [19]

[20]

[21] [22] [23]

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