First Measurement of Polarization Amplitudes 
We have performed the first measurement of the polarization amplitudes and the first search for CP violation in the charmless B^{0}_{s} → φφ → [K^{+}K^{}][K^{+}K^{}] decay, using CDF II data collected by the Two Track Trigger in the period March 2001 → April 2008 which corresponds to an integrated luminosity of 2.9 fb^{1}.
The B^{0}_{s} → φφ selected decay sample contains approximately 300 signal
events and is the same as in the
branching ratio measurement. Details on events selection are reported in
CDF Public
Note 10064.
Polarization Amplitudes Measurement 
f_{T}  =  A_{}^{2} + A_{perp}^{2}  f_{L}  =  A_{0}^{2}  



A_{0}^{2} + A_{}^{2} + A_{perp}^{2}  A_{0}^{2} + A_{}^{2} + A_{perp}^{2} 
Due to the VA nature of weak interaction and to the helicity conservation in
QCD, f_{L} >> f_{T} is expected in B decays to two light vector mesons.
This expectation was experimentally confirmed by BaBar and
Belle in treedominated transitions
while it was found
f_{L} ≈ f_{T} in
B^{+} → φK^{*+} and in B^{0} → φK^{*0}
where the b → s penguin transition is involved.
This is known as the Polarization Puzzle.
Explanations invoking either New Physics or subleading corrections to the
Standard Model predictions have been proposed. The B^{0}_{s} → φφ
decay proceeds via a b → s penguin transition representing a very interesting
decay channel in this experimental and theoretical scenario.
We look at the untagged timeintegrated differential decay rate as a function of three angular variables of the final state decay products. This rate depends on the three polarization amplitudes (and their relative phase). Neglecting the tiny CP phase in B^{0}_{s} mixing (as expected in the Standard Model), the rate depends only on three observables: two polarization amplitudes squared A_{0}^{2}, A_{}^{2}, and the strong phase δ_{}. The strength of A_{perp} can be determined from the normalization condition A_{0}^{2} + A_{}^{2} + A_{perp}^{2 } = 1.
The fit to the mass and decay product angular distributions is performed in the helicity basis and we define ω=(cosθ_{1}, cosθ_{2}, φ).
Fig: Helicity frame definition of angular variables for a generic decay to V_{1} V_{2} with V_{1} decaying to particles P_{1}, P_{2} and V_{2} to Q_{1}, Q_{2}. We take the K+ as P_{1} and Q_{1}.

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 distribution is a simple exponential
and the background fraction
f_{b} is determined from the fit in the B^{0}_{s }candidate mass
range 5.2 < m < 5.6 GeV/c^{2}.
The background model for the angular distributions is parameterized
using the sideband data. The acceptance A(ω) is calculated from Monte Carlo
simulation.
The time integrated polarization fractions are corrected for the expected
lifetime difference for the CPeven and
CPodd B^{0}_{s }mass eigenstates _{ }using
the world average B^{0}_{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 account for it as a systematic uncertainty.
We validate this approach by performing a similar
measurement using the B^{0}_{s}→ J/ψφ decays,
collected via the same trigger, and comparing the obtained results with the current
experimental information on the polarization of this decay.
Table: Fit results with statistical uncertainties  Table: Correlation matrix 
Several systematic uncertainties have been studied with Monte Carlo samples with statistics similar to the our data sample and generated with a model that includes the systematic effect under study. The quoted uncertainty is the shift in the mean value of the fit parameters in 1000 generated pseudoexperiments. The largest effect come from the inclusion of a scalar nonresonant component under the φ meson mass peak. This has been studied generating a B^{0}_{s} → φf_{0} and a nonresonant B^{0}_{s} → φ(K^{+}K^{}) sample with branching ratio similar to the equivalent B^{0} decays. Another important effect is 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 B^{0}_{s }width difference ΔΓ. Finally, the effect related to a possible non vanishing CPviolating phase in mixing at a level consistent with the current world average is included.
A_{0}^{2}  =  0.348±0.041  (stat)  ±0.021  (syst) 
A_{}^{2}  =  0.287±0.043  (stat)  ±0.011  (syst) 
A_{perp}^{2}  =  0.365±0.044  (stat)  ±0.027  (syst) 
cosδ_{}  =  0.91^{+0.15}_{0.13}  (stat)  ±0.009  (syst) 
f_{L}  =  0.348±0.041  (stat)  ±0.021  (syst) 
f_{T}  =  0.652±0.041  (stat)  ±0.021  (syst) 
In the following plot we show the measured polarization fractions f_{0} versus f_{}=A_{}^{2} with 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.) [2]. The crossbars of the experimental point are statistical and systematic uncertainties added in quadrature; in the QCD factorization cases, f_{} has been set to f_{}=(1f_{0})/2 (the dashed line) ±4% (Beneke et al., NPB 774).
Search for CP violation 
A_{TP}  =  Γ(TP>0)  Γ(TP<0) 


Γ(TP>0) + Γ(TP<0) 
Sidebands subtracted distribution of u. 
Sidebands subtracted distribution of v. 
A_{u}  =  0.007±0.064  (stat)  ±0.018  (syst) 
A_{v}  =  0.120±0.064  (stat)  ±0.016  (syst) 
Angular sculpting for B^{0}_{s}→ J/ψφ events cosθ (gif) (pdf) (eps), cosψ _{ } (gif) (pdf) (eps), φ (gif) (pdf) (eps)