Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 50 (2017)

Determination of |Vus | from inclusive τ decay with lattice HVPs†∗2∗1,∗2andH.
H.OhkiOhki∗2∗2T.
T. Izubuchi,Izubuchi∗1,andrelative spectral contributions (N=3)

Precise deteminations of the CKM matrix element|Vus | are important for ﬂavor physics, which is relevant to the search for new physics beyond the standard model (SM) in particle physics. We propose a
new method to determine |Vus | from inclusive strange
hadronic τ decay data and hadronic vacuum polarization functions (HVPs) computed on the lattice. The
basic idea is to consider generalized dispersion relations
for the experimentally observed us V+A inclusive distributions. In the SM, the diﬀerential distribution of τ
decay, dRV /A;us /ds, is related to the spectral function

10.1

0.001

(1)

ρus (s)ωN (s)ds =a)

0.8

1

K
Kπ- + −K π π
0 − 0K π πresidualpQCD

0.01

0.001

0.2

0.4

0.6
2
C[GeV ]

0.81Fig. 1. Ratio of each spectrum integral.

Res Πus (−Qk )ωN (−Qk ) (2).

kΓK[τK], lattice fK
τ us V+A, N=3
τ us V+A, N=4
τ us V+A, N=50.235

Condensed from a paper by authors in preparation.
Physics Department, Brookhaven National Laboratory
RIKEN Nishina Center
We thank RBC-UKQCD collaboration and Kim Maltman
for providing lattice HVP and experimental data.

- 137 -0.23|Vus|

Here, Πus (Q2 ) is the HVP deﬁned in space-like momentum −Q2 . The idea is to use the lattice QCD
data for Πus (Q2 ) on the RHS in Eq. (2) to comparewith the LHS, which can be evaluated up to s = m2τ
from the experimental dRus;V +A /ds distribution withan unknown |Vus |2 . The pQCD result is used to approximate the contribution from s > m2τ . We use
ωN (s) with uniformly spaced poles, which is characterized by N , and the center value of Q2 , C in theinterval, and the nearest-neighbor spacing is ﬁxed toΔ = 0.2/(N −1) [GeV2 ]. In Fig. 1, we plot some example values for spectrum contributions given in the LHSof Eq. (2). Increasing N or decreasing C can suppress
the contributions from high-s modes that have larger
errors, so that we can adjust the “inclusiveness”, thatis, the impact of the high multiplicity modes.
For the RHS in Eq. (2), we use the lattice HVPs
measured near the physical quark masses on the 2 + 1
ﬂavor Mobius domain wall fermion ensembles by theRBC and UKQCD collaborations.a) Here, Π(Q2 ) is
obtained from the Fourier transformation of the lattice
HVPs (Π(t)) in the coordinate space.
∗20.6
2C[GeV ]0.1

where yτ = s/m2τ , ωτ (y) = (1 − y)2 (1 + 2y), ωL (y) =2y(1 − y)2 , and SEW is a known short-distance electroweak correction. Using a weight function ωN (s) =1(s = −Q2k < 0), we obtain the followingΠN(s+Q2k )k=1
dispersion relation,
 ∞N


22

∗10.4

1dRus;V /A
12π 2 |Vus |2 SEW=dsm2τ
0× ωτ (s)ρ0+1(s)−ω
(y)ρ
(s),L
τ
us;V/Aus;V /A†

0.2

relative spectral contributions (N=4)(J)

ρus;V /A (s) by0KKπ- + −K π π0 − 0K π πresidual
pQCD

0.01

0.225
0.220.215
00.2

0.40.6
2C [GeV ]0.8

1Fig. 2. Result of |Vus | as a function of C for N = 3, 4, and
5. The result obtained from K is also shown.

Figure 2 shows our result for |Vus |. Our best value
is |Vus | = 0.2241(14)exp (13)th , from the result with
C = 0.7 [GeV2 ] with N = 4. The theoretical error
is comparable with the experimental one, and the total error is less than the previous inclusive τ decay
determinations. b) Moreover, the result is consistentwith those obtained from analyses of kaon physics and3-family CKM unitarities.
Reference1) P. A. Boyle et al., Int. J. Mod. Phys. Conf. Ser.
35, 1460441 (2014) doi:10.1142/S2010194514604414[arXiv:1312.1716 [hep-ph]].
b)

For a recent study of the inclusive τ decay by using the ﬁnite
1..
energy sum rule, see Ref.1)