Observation and Polarization Measurements of B+- ->phi K_1+- and B+- ->phi K_2*+-

With the full $\babar$ data sample of 465 million BaBar pairs, we observe the decays B+- ->phi(1020) K_1(1270)+- and B+- ->phi(1020) K_2*(1430) +-. We measure the branching fractions (6.1+-1.6+-1.1)\times 10^{-6} and (8.4+- 1.9+- 0.9)\times 10^{-6} and the fractions of longitudinal polarization 0.46(+0.12-0.13)(+0.03-0.07) and 0.80(+0.09-0.10)+-0.03, respectively. We also report on the B+- ->phi K_0*(1430)+- decay branching fraction of (7.0+-1.3+-0.9)\times10^{-6} and several parameters sensitive to CP violation and interference in the above three decays. Upper limits are placed on the B+- decay rates to final states with phi and K_1(1400)+-, K*(1410)+-, K_2(1770)+-,or K_2(1820)+-. Understanding the observed polarization pattern requires amplitude contributions from an uncertain source.

10 −6 and several parameters sensitive to CP violation and interference in the above three decays. Upper limits are placed on the B ± decay rates to final states with ϕ and K1(1400) ± , K * (1410) ± , K2(1770) ± , or K2(1820) ± . Understanding the observed polarization pattern requires amplitude contributions from an uncertain source. PACS numbers: 13.25.Hw,13.88.+e,11.30.Er Measurements of polarization in rare vector-vector B meson decay, such as B → ϕK * [1,2], have revealed an unexpectedly large fraction of transverse polarization and suggested contributions to the decay amplitude which were previously neglected. Decays to other excited spin-J kaons K decay has three complex amplitudes A Jλ , which describe the three helicity states λ = 0, ±1, except when J = 0. The expected hierarchy of the A Jλ amplitudes |A J0 | 2 ≫ |A J+ | 2 ≫ |A J− | 2 is sensitive to the (V − A) structure of the weak interactions with the left-handed fermion couplings [3,4,5], and therefore is sensitive to physics beyond the standard model. For example, tensor or scalar interactions would violate |A J0 | 2 ≫ |A J+ | 2 and the right-handed fermion couplings would violate |A J+ | 2 ≫ |A J− | 2 [3]. Strong interaction effects could change these predictions as well, but were originally expected to be small [3].
However, all previous studies have been limited to the two-body K * J → Kπ decays, thus considering only the spin-parity K * J states with P = (−1) J . In this paper we report the measurement with the three-body final states K ( * ) J → Kππ which include P = (−1) J+1 mesons such as K 1 and K 2 . We complement these measurements with the two-body K ( * ) J final states in the B ± decays and report polarization in the ϕK 1 (1270) ± and ϕK * 2 (1430) ± final states which have not been seen before. We also search for other final states with ϕ and K * 0 (1430) ± , K 1 (1400) ± , K * (1410) ± , K 2 (1770) ± , or K 2 (1820) ± .
We use data collected with the BABAR detector [6] at the PEP-II e + e − collider. A sample of (465 ± 5) × 10 6 Υ (4S) → BB events was recorded at the the e + e − centerof-mass energy √ s = 10.58 GeV. Momenta of charged particles are measured in a tracking system consisting of a silicon vertex tracker with five double-sided layers and a 40-layer drift chamber, both within the 1.5-T magnetic field of a solenoid. Identification of charged particles is provided by measurements of the energy loss in the tracking devices and by a ring-imaging Cherenkov detector. Photons are detected by a CsI(Tl) electromagnetic calorimeter.
We search for B ± → ϕK ( * )± J decays using three final states of the K ( * )± J decay: K 0 S π ± , K ± π 0 , and K ± π + π − , where K 0 S → π + π − and π 0 → γγ. We define the two helicity angles θ i as the angle between the direction of the K or K + meson from K * → Kπ (θ 1 ) or ϕ → K + K − (θ 2 ) and the direction opposite to the B in the K * or ϕ rest frame. The normal to the three-body decay plane for K ( * ) J → Kππ is chosen as the analyzer of the K ( * ) J polarization instead of the direction of K from K * J in the two-body decays. We define H i = cos θ i .
We identify B meson candidates using two kinematic variables: is the four-momentum of the B candidate in the e + e − center-of-mass frame. We require m ES > 5.25 GeV and |∆E| < 0.1 GeV (or 0.08 GeV for K ( * )± J → K ± π + π − ) GeV. We also require the invariant masses to satisfy 1.1 < m Kπ < 1.6 GeV, 1.1 < m Kππ < 2.1 GeV, and 0.99 < m K + K − < 1.05 GeV. To reject the dominant e + e − → light quark-antiquark background, we use the angle θ T between the thrust axis of the B-candidate decay products and that of the rest of the event requiring | cos θ T | < 0.8, and a Fisher discriminant F which combines event-shape parameters [7].
To reduce combinatorial background in the mode K * ± J → K ± π 0 , we require H 1 < 0.6. When more than one candidate is reconstructed (7.6% of events with K 0 S π ± , 2.9% with K ± π 0 , and 14.6% with K ± π + π − ), we select the one whose χ 2 of the charged-track vertex fit combined with χ 2 of the invariant mass consistency of the K 0 S or π 0 candidate, is the lowest. We define the bquark flavor sign Q to be opposite to the charge of the B meson candidate.
We use an unbinned extended maximum-likelihood fit [1] to extract the event yields n j and the probability density function (PDF) parameters, denoted by ζ and ξ, to be described below. The index j represents the event categories, which include continuum background and several B-decay modes. In the B ± → ϕK * ± J → (K + K − )(Kπ) topology, the following event categories are considered: ϕK * 2 (1430) ± , ϕ(Kπ) * ± 0 , and

contribution includes both a nonresonant component and the
1820) ± , a nonresonant ϕK ± π + π − , and f 0 K 1 (1400) ± contributions. In the latter topology, the mode ϕK 2 (1770) ± is also considered in place of ϕK 2 (1820) ± . In all cases, the modes with f 0 model can account for a possible broad non-ϕ (K + K − ) contributions under the ϕ.
The extended likelihood is L = exp (− n j ) L i . The likelihood L i for candidate i is defined as L i = j,k n k j P k j (x i ; ζ, ξ), where P k j is the PDF for variables The flavor index k corresponds to the value of Q, that is P k j ≡ P j × δ kQ . The ζ are the polarization parame-ters, only relevant for the signal PDF. The ξ parameters describe the background or the remaining signal PDFs, which are left free to vary in the fit for the combinatorial background and are fixed to the values extracted from Monte Carlo (MC) simulation [9] and calibration B → Dπ decays in other cases.
The signal PDF for a given candidate i is a joint PDF for the helicity angles and resonance mass, and the product of the PDFs for each of the remaining variables. The helicity part of the signal PDF is the ideal angular distribution from Ref. [10], multiplied by an empirical acceptance function G(H 1 , H 2 ). In the B → ϕK 1 or ϕK 2 parameterization, the additional kinematic parameters for the decays K ± J → K ± π + π − (such as r 1 , r 2 , and r 02 in Ref. [10]) are modeled using the sequential two-body decay chains [5]. A relativistic spin-J Breit-Wigner amplitude parameterization is used for the resonance masses [5,11], and the J P = 0 + (Kπ) * ± 0 m Kπ amplitude is parameterized with the LASS function [8]. The nonresonant ϕK ± π + π − contribution is modeled through sequential K * (892)π → Kππ decay, while the decay Kρ → Kππ is considered in the systematic uncertainty studies. We use a sum of Gaussian functions for the parameterization of ∆E, m ES , and F .
The interference between the J = 2 and 0 (Kπ) ± contributions is modeled with the term 2Re(A 20 A * 00 ), with the three-dimensional angular and m Kπ parameterization. We allow an unconstrained flavor-dependent overall shift (δ 0 + ∆δ 0 × Q) between the LASS amplitude phase and the tensor resonance amplitude phase. The polarization parameters ζ include the fractions of longitudinal polarization f L = |A J0 | 2 /Σ|A Jλ | 2 in several channels, δ 0 , and ∆δ 0 . Similar interference between the K 1 (1270) ± and K 1 (1400) ± contributions is allowed in the study of systematic uncertainties but is not included in the nominal fit due to observed dominance of only one mode and therefore unconstrained phase of the interference.
Since the K * 2 (1430) ± meson contributes to all three K 0 π ± , K ± π 0 , and K ± π + π − final states and (Kπ) * ± 0 contributes to two Kπ final states in this analysis, we consider the total L as a product of three likelihoods constructed for each of the three channels. The corresponding yields in different channels are related by the relative efficiency. We fit the yields in each charge category k independently and report them in the form of the total yield n j = n + j + n − j and direct-CP asymmetry A CP = (n + j − n − j )/n j . The combinatorial background PDF is the product of the PDFs for independent variables and is found to describe well both the dominant quark-antiquark background and the background from random combinations of B tracks. We use polynomials for the PDFs, except for m ES and F distributions which are parameterized by an empirical phase-space function and by Gaussian functions, respectively. Resonance production occurs in the background and is taken into account in the PDF.
We observe nonzero B ± → ϕK 1 (1270) ± and B ± → ϕK * 2 (1430) ± yields with significance (excluding systematic uncertainties in parentheses) of 5.0(5.3)σ and 5.5(6.0)σ, respectively. The combined ϕK 1 (1270) ± and ϕK 1 (1400) ± significance is 5.7(6.4)σ. The significance is defined as the square root of the change in 2 ln L when the yield is constrained to zero in the likelihood L. We have tested this procedure with the generated MC samples and account for a small observed deviation from the one-dimensional χ 2 statistical treatment.
In Table I, results of the fit are presented, where the combined results are obtained from the simultaneous fit to the three decay subchannels. In the branching fraction calculations we assume K 2 → K * 2 (1430)π and B(K * (1410) → K * π) = 0.934 ± 0.013 [5]. The signal is illustrated in the projection plots in Figs. 1 and 2, where in the latter we enhance either the ϕK 1 (1270) ± signal (left) or the ϕK * 2 (1430) ± signal (right). The nonresonant K + K − contribution under the ϕ is accounted for with the B 0 → f 0 K 1 category and its yield 7 ± 16 is consistent with zero. Similarly, the nonresonant category ϕKππ yield is 148 ± 54 with statistical errors only.
We vary those parameters in ξ not used to model combinatorial background within their uncertainties and derive the associated systematic errors. Interference between the K 1 (1270) ± and K 1 (1400) ± is one of the dominant systematic uncertainties on both yields and is modeled with simulated samples. We take the flavordependent reconstruction efficiency into account in the study of asymmetries. The biases from the finite resolution of the angle measurement, the dilution due to the presence of false combinations, and other imperfections in the signal PDF model are estimated with MC simulation. Additional systematic uncertainty originates from possible B background, where we estimate from MC simulation that only a few events can fall in the signal region.
The ϕK 2 (1770) ± yield is not considered in the nominal fit due to large correlation with ϕK 2 (1820) ± . But we substitute the K 2 (1820) resonance for the K 2 (1770) I: Results: the reconstruction efficiency εreco; the total efficiency ε, including the daughter branching fractions [5]; the number of signal events nsig; significance S; fraction of longitudinal polarization fL; the branching fraction B; and the flavor asymmetry ACP . The branching fraction B(B ± → ϕ(Kπ) * ± 0 ) refers to the coherent sum |Ares + Anon-res| 2 of resonant and nonresonant J P = 0 + Kπ components [8] and is quoted for mKπ < 1.6 GeV, while the B(B ± → ϕK * 0 (1430) ± ) is derived from it by integrating separately the Breit-Wigner formula of the resonant |Ares| 2 Kπ component [8] without mKπ restriction. When several subchannels contribute, yield and efficiency are quoted for each subchannel. The 90% confidence level upper limit on B is quoted with the central values and errors in parentheses. The insert shows two interference parameters δ0 and ∆δ0 for ϕK * 2 (1430) ± and ϕ(Kπ) * ± 0 . The ϕK2(1770) ± yield is not considered in the nominal fit and the value indicated with † is obtained with these ϕK2(1820) ± yield constrained to zero. The systematic errors are quoted last.
In summary, we have performed an amplitude analysis and searched for CP -violation with the B ± → ϕK ( * )± J decays which include significant K 1 (1270) and K * 2 (1430) contributions. Our results are summarized in Table I. The polarization measurement in the vector-tensor B ± decay is consistent with our earlier measurement in the B 0 → ϕK * 2 (1430) 0 decay [1] and with the naive expec-tation of the longitudinal polarization dominance. However, our first measurement of polarization in a vectoraxial-vector B meson decay indicates a large fraction of transverse amplitude, similar to polarization observed in the vector-vector final state B → ϕK * (892) [1,2]. Both measurements indicate substantial A 1+1 (or still possible A 1−1 for vector-axial-vector decay) amplitude from an uncertain source. Among potential sources are penguin annihilation, electroweak penguin, QCD rescattering, or physics beyond the standard model [3].
We are grateful for the excellent luminosity and machine conditions provided by our PEP-II colleagues, and for the substantial dedicated effort from the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and kind hospitality.  , ∆E (f), H1 (g), and H2 (h) for the signal ϕK * 2 (1430) ± and ϕ(Kπ) * ± 0 candidates combined. The step in (g) is due to selection requirement H1 < 0.6 in the channel with π 0 . Data distributions are shown with a requirement on the signal-to-background probability ratio calculated with the plotted variable excluded. The solid (dotted) lines show the signal-plus-background (combinatorial background) PDF projections, while the dashed lines show the full PDF projections excluding ϕK ± 1 (left) or ϕK * 2 (1430) ± (right).