Primary authors: M. Dorigo, A. M. Zanetti, M. Rescigno, B. Di Ruzza

Using  CDF II data collected by the Two Track Trigger in the period starting from March 2001 untill April 2008 corresponding to an integrated luminosity of 2.9 fb^-1, we have performed the  first measurement of the polarization amplitudes for the charmless B⁰_s → φφ → [K⁺K^-][K⁺K^-] decay of the B⁰_s meson.
The results are obtained with an unbinned Maximum Likelihood fit to the reconstructed B⁰_s candidate mass and three angular variables in a sample containing approximately 300 signal events.

We have used the same data as in the updated measurement of the B⁰_s → φφ branching ratio (BR update). Details on event selection are reported  in CDF note 10064.  We measure the polarization fractions and the cosine of δ_|| = arg (A_||A₀^*) for B⁰_s → φφ as:

Jump to Motivation, Analysis Description, Fit Results, Systematics, B⁰_s→ J/ψφ Results, Comparison with Theory, List of Approved Plots and Tables.

Motivation

Due to V-A nature of weak interaction and helicity conservation in QCD, f_L >> f_T is naively expected in B decays to two light vector mesons.
This expectation was experimentally confirmed by BaBar and Belle in tree-dominated transitions like B⁰ → ρ⁺ρ^-, B⁺ → ρ⁰ρ⁺  and B⁰ → ρ⁰ρ⁰.
In contrast it was found f_L ≈ f_T in B⁺ → φK^*+ and in B⁰ → φK^*0 for the b → s penguin decays.

This is known as the Polarization Puzzle. Explanations invoking either New Physics or sub-leading corrections to the naive expectation within the Standard Model has been proposed. Updated predictions for the B⁰_s → φφ exists and can now be confronted with experiment.

Analysis Description

In this analysis we look at the untagged time-integrated differential decay rate as a function of three angular variables of the final state decay products. The time integrated polarization fractions are corrected for the expected lifetime difference for the CP-even and CP-odd B⁰_s mass eigenstates  using world average B⁰_s  lifetime and width difference. Since the CDF Two Track Trigger biases the natural decay proper time distribution of the available sample, we study the resulting bias in the polarization measurement with Monte Carlo simulation and assign as a systematic uncertainty the full expected effect. We validate this approach by performing a similar measurement using  B⁰_s→ J/ψφ decays, collected via the same trigger, and comparing results with current experimental information on the polarization in such a decay.

The time-integrated differential decay rate with respect to the final state particle decay angles depends on the three polarization amplitudes (and their relative phase). Neglecting the tiny CP phase in B⁰_s mixing (as expected in the Standard Model) the only interference allowed is between the two CP-even amplitudes A₀ and A_||; hence, the only measurable phase is δ_|| = arg (A_||A₀^*). Thus the rate depends only on three observables (two polarization amplitudes squared |A₀|², |A_|||², and the strong phase  δ_||).   The strength of A_^ can be determined from the normalization condition:

|A₀|² + |A_|||² + |A_^|² = 1

Fig. 1 Helicity frame definition of angular variables for a generic decay to V1 V2 with V1 decaying to particles P1 P2 and V2 to Q1 Q2. We take the K+ as P1 and Q1.	Fig. 2 Transversity frame definition of angular variables.

The fit to the mass and decay product angular distribution is performed in the helicity basis defined as in Fig.1 (in the transversity basis in the case of the  B⁰_s→ J/ψφ decay, Fig. 2). The time of decay is not observed and only the time integrated rate is measured. The time integrated polarization fractions are corrected to t=0 using the PDG 09 averages τ_L=1.408^+0.033_-0.030 ps and τ_H=1.543^+0.058_-0.060  ps, where τ_L,H are the lifetime of the Light and Heavy B⁰_s state respectively. Assuming equal production for B⁰_s and anti-B⁰_s the differential decay rate (including acceptance) can then be written as:

The background model for the angular distribution is a constant in the φ angle and is parameterized as 1+B*cos²θ for the θ₁ and θ₂. This is checked to be adequate using sideband data and the parameter B is determined in the fit. The acceptance A(ω), with ω=(cosθ₁, cosθ₂, φ), is calculated from Monte Carlo simulation and is displayed below. The reconstructed mass for signal events is parameterized with a double Gaussian as:

with parameters k and h fixed from Monte Carlo simulation. The background model for the mass is a simple exponential, e^-b*m, with b a fit parameter. Finally we also fit for the background fraction f_b in the B⁰_s candidate mass fit range  5.2 < m < 5.6 GeV/c².

Fit Projections and Results

Fit results with statistical uncertainties (click to download table)
Correlation matrix (click to download table)

Systematic Uncertainties

Several systematic uncertainties have been studied with Monte Carlo samples of size equal to the data sample and generated with a model including the effect under study. The quoted uncertainty is the shift in the mean value of the fit parameters in 1000 such pseudo-experiments. The largest effect come from the inclusion of a scalar non-resonant component under the φ meson mass peak. This has been studied generating a B⁰_s → φf₀ and a non-resonant B⁰_s → φ(K⁺K^-) sample with branching ratio similar to the equivalent B⁰ decays.  Other important effect are related to the proper time acceptance of the displaced track trigger that introduces a bias in the observed polarization fraction which is dependent on the true value of the B⁰_s width difference ΔΓ. Finally, the effect related to a possible non vanishing CP-violating phase in mixing at a level consistent with the current world average is included.

Cross Check: Fit Projections and Results for B⁰_s→ J/ψφ Decay

Fit results with statistical uncertainties (click to download table)

Comparison of present measurement with SM theoretical prediction

In the following plot we show the measured polarization fractions f₀ versus f_||=|A_|||² within the 68% confidence region (orange area) compared with the expectations of the QCD factorization models (Beneke et al., Datta et al.) and the perturbative QCD (Ali et al.). The cross-bars of the experimental point are statistical and systematic uncertainties added in quadrature; in the QCD factorization cases, f_|| has been set to f_||=(1-f₀)/2 (the dashed line) ±4% (Beneke et al., NPB 774).

List of Public Plots and Tables 

• B⁰_s → φφ fit projection 1:  invariant mass (gif),(pdf)
• B⁰_s → φφ fit projection 2:  φ angle (gif),(pdf)
• B⁰_s → φφ fit projection 3:  cosθ₁ (gif),(pdf)
• B⁰_s → φφ fit projection 4:  cosθ₁ (gif),(pdf)
• pull distributions 1  (gif) (pdf)
• pull distributions 2  (gif) (pdf)
• pull table  (gif) (pdf)
• Fit Results (gif) (pdf)
• Correlation Matrix (gif) (pdf)

• Angular sculpting for B⁰_s → φφ events cosθ₁(gif) (pdf), cosθ₂ (gif) (pdf), φ (gif) (pdf)

   

• |A₀|²  vs |A_|||² confidence region  (gif) (pdf) 
• |A₀|²  vs cos(δ_||) confidence region  (gif) (pdf)
• |A_|||²  vs cos(δ_||)  confidence region  (gif) (pdf)

• P_T for K⁺ in B⁰_s → φφ candidate events (gif) (pdf)
• P_T for K^-  in B⁰_s → φφ candidate events (gif) (pdf)

• B⁰_s→ J/ψφ fit projection 1 :invariant mass (gif),(pdf)
• B⁰_s→ J/ψφ fit projection 2: φ angle (gif),(pdf)
• B⁰_s→ J/ψφ fit projection 3: cosθ(gif),(pdf)
• B⁰_s→ J/ψφ fit projection 4: cosψ (gif),(pdf)

• Angular sculpting for B⁰_s→ J/ψφ events cosθ (gif) (pdf), cosψ (gif) (pdf), φ (gif) (pdf)

• f₀ vs f_|| plot: comparison with theoretical models (gif), (pdf)