The first observation of $\tau^{\pm} \to \phi K^{\pm} \nu$ decay

We present the first measurement of tau-decays to hadronic final states with a $\phi$-meson. This is based on 401.4 fb$^{-1}$ of data accumulated at the Belle experiment. The branching ratio obtained is $B(\tau^{\pm}\to\phi K^{\pm}\nu) = (4.06\pm 0.25\pm 0.26)\times 10^{-5}$.


INTRODUCTION
While hadronic τ ± decays with a φ meson in the final state are valuable to investigate QCD at a low mass scale, they have never been observed due to their small branching fractions. The decay τ ± → φK ± ν is Cabibbo suppressed and further restricted by its small phase space, while the decay τ ± → φπ ± ν is suppressed by the OZI rule although it is Cabibbo allowed. Taking into accounts differences in Cabibbo mixing effects and phase space relative to the Cabibbo allowed transition τ ± → K 0 * K ± ν, the branching fraction for τ ± → φK ± ν is estimated to be B(τ ± → φK ± ν) = 2 × 10 −5 [1]. On the other hand, the vector dominance model predicts B(τ ± → φπ ± ν) = (1.20 ± 0.48) × 10 −5 [2].
CLEO searched for these decays using 3.1 fb −1 of data taken on Υ(4S) resonance. They set upper limits of B(τ ± → φK ± ν) < (5.4 − 6.7) × 10 −5 and B(τ ± → φπ ± ν) < (1.2 − 2.0) × 10 −4 at the 90% confidence level [1]. Here we report the first measurement of τ ± → φK ± ν and τ ± → φK ± π 0 ν decays. These results are based on a data sample of 401.4 fb −1 corresponding to 3.6 × 10 8 τ + τ − pairs collected near the Υ(4S) resonance with the Belle detector at the KEKB asymmetric-energy e + e − (3.5 on 8 GeV) collider [3]. The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel thresholdČerenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter comprised of CsI(Tl) crystals (ECL) located inside a super-conducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside of the coil is instrumented to detect K 0 L mesons and to identify muons (KLM). The detector is described in detail elsewhere [4]. Two inner detector configurations were used. A 2.0 cm beampipe and a 3-layer silicon vertex detector was used for the first sample of 157.7 fb −1 , while a 1.5 cm beampipe, a 4-layer silicon detector and a small-cell inner drift chamber were used to record the remaining 243.7 fb −1 [5].
The detection of φ-mesons relies on the φ → K + K − decay (B = (49.1 ± 0.6)%); the final evaluation of the signal yield is carried out using the K + K − invariant mass distribution.
Transverse momentum (p t ) for a charged track is required to be larger than 0.06 GeV/c in the barrel region (−0.6235 < cos θ < 0.8332, where θ is the polar angle opposite to the incident e + beam direction in the laboratory frame) and 0.1 GeV/c in the endcap region (−0.8660 < cos θ < −0.6235, and 0.8332 < cos θ < 0.9563). The energies of photon candidates are required to be larger than 0.1 GeV in both regions.
To select τ -pair samples, we require four charged tracks with zero net charge and a total energy of charged tracks and photons in the center-of-mass (CM) frame less than 11 GeV. Furthermore, the missing momentum in the laboratory frame is required to be greater than 0.1 GeV/c and its direction to be within the detector acceptance, where the missing momentum is defined as the deficit of sum of the observed momentum vectors from that of the initial e + e − system. We also require cos θ CM thrust−miss < −0.6 to reduce qq backgrounds, where θ CM thrust−miss is an opening angle between the thrust axis and the missing momentum in CM frame. The event is subdivided into 3-prong and 1-prong hemispheres according to the thrust axis in the CM frame. These are referred to as the signal and tag side, respectively.
In order to remove the dominant qq background, we require that the lepton probability P µ/e be greater than 0.1 and that the invariant mass of the particles on the tag side be less than 1.8 GeV/c 2 (≃ m τ ). Similarly, we require that both kaon daughters of the φ candidate have kaon probabilities P K > 0.8. The effective mass of the signal side must be less than 1.8 GeV/c 2 . Here P ℓ is the likelihood that a charged particle is of type ℓ (ℓ = µ or e or K or π), defined as P ℓ = L ℓ /(L ℓ + L x ), where L ℓ and L x are the likelihoods of the particle for ℓ and other species hypotheses, respectively, determined from responses of the relevant detectors. We allow at most one photon on the tag side to take account of initial state radiation, while requiring no extra photons on the signal side.
Candidates φ mesons are reconstructed using oppositely charged kaons within the barrel and forward endcap region. To suppress combinatorial backgrounds from other τ decays and qq processes, we require that the φ momentum be greater than 1.5 GeV/c in the CM frame. After these requirements the remaining contributions from B 0B0 , B + B − , Bhabha, µ pair and two photon backgrounds are negligible.
To separate φK ± ν from φπ ± ν, the remaining charged track is required to satisfy the same kaon identification criteria as the φ daughters. The τ τ and qq contributions are reduced by requiring the opening angle (θ CM φK ) between φ and K ± in the CM frame to satisfy cos θ CM φK > 0.92 and the momentum of the φK ± system in the CM frame must be greater than 3.5 GeV/c. For φπ ± ν, we require the charged track to be identified as a pion, P π > 0.8, and the opening angle between φ and π ± in the CM frame to satisfy cos θ CM φπ < 0.98. Fig.1(a) shows the K + K − invariant mass distribution after all τ ± → φK ± ν selection requirements. While there are two possible K + K − combinatorial contributions from K ± K ∓ K ± on the signal side in forming a φ meson; all combinations are included. Nonresonant backgrounds are mostly attributed to τ ± → K + K − π ± ν events with B = (1.55 ± 0.07) × 10 −3 . Very small contributions are expected from qq processes.

SIGNAL AND BACKGROUND EVALUATION
The detection efficiencies ǫ for τ ± → φK ± ν, φK ± π 0 ν and φπ ± ν are evaluated, as listed in Table I, from MC using KKMC [6], where the V − A interaction is assigned on the vertices and the final hadrons decay according to non-resonant phase space. The signal τ ± → φK ± ν detection efficiency is ǫ φKν = (1.82±0.01)%, including the branching fraction of φ → K + K − . Signal yields are evaluated by a fit to a p-wave Breit-Wigner (BW) distribution, convoluted with a Gaussian function (of width σ) to account for the detector resolution. The φ width is fixed to be Γ φ = 4.26 MeV (PDG value [7]), but σ is allowed to float. From fits to  signal MC, the φ mass detector resolution is found to be, σ = 1.07 ± 0.03 MeV/c 2 , which is consistent with the fit result to the data of 1.2 ± 0.3 MeV/c 2 .
The K + K − invariant mass distributions for τ ± → φK ± ν and φπ ± ν candidates are fitted with a p-wave BW distribution plus a linear and a second order polynomial background function, respectively, as seen in Figs. 1(a) and (b). The obtained signal yields are N τ →φKν = 573 ± 32 and N τ →φπν = 753 ± 84.
MC studies show that only τ ± → φπ ± ν, τ ± → φK ± π 0 ν and qq samples yield significant contributions peaking at the φ mass. The contribution of other backgrounds is less than 0.01% and can be neglected. The number of τ ± → φπ ± ν events in the data is already evaluated. Other contributions are estimated below.
In order to evaluate the branching fraction and background contribution from τ ± → φK ± π 0 ν, we select π 0 (→ γγ) candidates and combine them with φK ± ν combinations that satisfy the requirements listed above. The signal yield is estimated by fitting its K + K − invariant mass distribution with a p-wave BW distribution plus a linear background function, as shown in Fig.2. The resulting yield is 8.2 ± 3.8φKπ 0 ν events in data. With the detection efficiency of ǫ φKπ 0 ν = (0.395 ± 0.007)% evaluated by MC and the produced number of τ τ 's, N τ τ = 401.4 (fb −1 ) ×0.892 (nb) = 3.58 × 10 8 , we obtain a branching fraction of with a systematic uncertainty of 6.9%, which is described later. Using this we estimate the contamination of τ ± → φK ± π 0 ν events in the τ ± → φK ± ν signal as N cont τ ± →φK ± π 0 ν = (6.8 ± 3.1) events, given the cross-feed rate of τ ± → φK ± π 0 ν to the τ ± → φK ± ν signal to be (0.327 ± 0.006)%, as listed in Table I. From the MC (731.1 fb −1 ) study, we find the contamination of qq is N qq = 12.1 ± 4.5. To take into account the uncertainty in φ production in qq MC, we compare MC results with enriched qq data by demanding that the effective mass of the tag side be larger than 1.8 GeV/c 2 . With this selection, the background is qq dominated and the other backgrounds are negligible. The yield in data is 262 ± 21 events and 117.4 ± 9.9 events for qq. We then scale the above estimate by the factor, f = 2.23 ± 0.47, and obtain the qq contamination, N cont qq = 14.8 ± 3.5 events.

RESULTS
The peaking backgrounds described above, τ ± → φK ± π 0 ν and qq, are subtracted from the signal yield, N τ ± →φK ± ν = (573 ± 32) − (6.8 ± 3.1) − (14.8 ± 3.5) = 551.4 ± 32.3 events. To take into account cross-feed between τ ± → φK ± ν and τ ± → φπ ± ν due to particle misidentification (K ↔ π), we solve the following simultaneous equations, where B φKν and B φπν is the branching fraction for τ ± → φK ± ν and τ ± → φπ ± ν, respectively. ǫ's are the detection efficiencies listed in Table I. ǫ φπν φKν is the efficiency for reconstructing τ ± → φK ± ν as τ ± → φπ ± ν while ǫ φKν φπν is the efficiency for reconstructing τ ± → φπ ± ν as τ ± → φK ± ν. The resulting branching fraction for τ ± → φK ± ν is where the uncertainty is calculated with only statistical ones of N φKν and N φπν . The uncertainty in the detection efficiencies, ǫ's, will be taken into account in the systematic error. B φπν is obtained by the same way as but this is not the final branching fraction for the decay since the unknown contamination of τ → φ(nπ)ν (n ≤ 5) decays still must be subtracted. Systematic uncertainties are estimated as follows: Evaluation uncertainties of the integrated luminosity, τ τ cross-section and trigger efficiency are 1.4%, 1.3% and 1.1%, respectively. Track finding efficiency has an uncertainty of 4.0%. Uncertainties in lepton and kaon identification efficiencies and fake rate are evaluated, respectively, to be 3.2% and 3.1% by averaging the estimated uncertainties depending on momentum and polar angle of each charged track. To evaluate the systematic uncertainty of fixing Γ φ in the BW fit, we calculate the change in the signal yield when Γ φ is varied by ±0.05 MeV (the uncertainty quoted by PDG) [7]: The uncertainty is 0.2%. The branching ratio for φ → K + K − gives an uncertainty of 1.2% following the PDG [7]. The backgrounds from N φK ± ν and N φπ ± ν have uncertainties of 0.3% and 0.4%, respectively. The signal detection efficiency ǫ φKν has an uncertainty of 0.5%. A total systematic uncertainty of 6.5% is obtained by adding all uncertainties in quadrature. The resulting branching fraction is then B(τ ± → φK ± ν) = (4.06 ± 0.25 ± 0.26) × 10 −5 .
The systematic uncertainties for τ ± → φK ± π 0 ν are similar to those for τ ± → φK ± ν. The main differences are on the trigger efficiency of 0.3%, the detection efficiency ǫ φKπν of 1.8% and the π 0 detection efficiency of 1.7%. Those provide a total systematic uncertainty of 6.9%, and the branching fraction of Finally, we examine a possible resonance state that intermediates the final φK ± hadronic system. CLEO [1] assumed a resonance having a mass of 1650 MeV/c 2 with a width of 100 MeV/c 2 in the evaluation of their detection efficiency for τ ± → φK ± ν, however no signal was found. We generate a resonant MC with the KKMC simulation program. The weak current is generated with a V − A form while the φK ± system is assumed to be produced from a 2-body decay of a resonance. In Fig. 3(a), the φK ± mass distribution for data is compared to MC; the combinatorial background is subtracted using the K + K − sideband. Fig. 3(b) shows the φ's angular distribution in the φK ± rest frame (cos α), where the momentum direction of φK ± in the laboratory frame (P (φ + K)) is taken as the reference axis. It indicates an isotropic distribution in the φK ± system. For both the invariant mass and angular distributions of φK ± system, the phase space MC reproduces the signal distribution well. On the other hand, the 1650 MeV/c 2 state, indicated by the dotted histogram in Fig. 3(a), clearly cannot account for the entire signal. Assuming resonant production, the best agreement with the data is found for a mass and a width of ≃1570 MeV/c 2 and ≃150 MeV/c 2 , respectively, as shown by the dot-dashed histogram. However, since the shape of the resonant MC is similar to the phase space distributed MC, it is inappropriate to look for the intermediate resonance state with Γ ∼ O(100MeV) in this narrow mass range of ∼250 MeV/c 2 . In fact, the phase space distribution (the open histogram) agrees well with data. MeV/c 2 , respectively. In MC, the branching ratio of 4 × 10 −5 is assumed. (b) φ's angular distribution in the φK ± rest frame, where the direction of P (φ + K) in the laboratory frame is taken as the reference axis.