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 B^{0}_{s} → φφ → [K^{+}K^{}][K^{+}K^{}]
decay of the B^{0}_{s} meson.
The results are obtained with an unbinned Maximum Likelihood fit to the
reconstructed B^{0}_{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 B^{0}_{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_{}A_{0}^{*}) for B^{0}_{s} → φφ as:
A_{0}^{2}  =  0.348±0.041  (stat)  ±0.021  (syst) 
A_{}^{2}  =  0.287±0.043  (stat)  ±0.011  (syst) 
A_{^}^{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) 
A_{0}^{2}  =  0.534  ±0.019(stat. only) 
A_{}^{2}  =  0.220  ±0.025(stat. only) 
Jump to Motivation, Analysis Description, Fit Results, Systematics, B^{0}_{s}→ J/ψφ Results, Comparison with Theory, List of Approved Plots and Tables.
the transverse amplitude fraction  f_{T}  =  A_{}^{2} + A_{^}^{2} 


A_{0}^{2} + A_{}^{2} + A_{^}^{2}  
and  
the longitudinal amplitude fraction  f_{L}  =  A_{0}^{2} 


A_{0}^{2} + A_{}^{2} + A_{^}^{2} 
Due to VA 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 treedominated transitions like
B^{0} → ρ^{+}ρ^{}, B^{+} → ρ^{0}ρ^{+} and
B^{0} → ρ^{0}ρ^{0}.
In contrast it was found
f_{L} ≈ f_{T} in
B^{+} → φK^{*+} and in B^{0} → φK^{*0}
for the b → s penguin decays.
This is known as the Polarization Puzzle. Explanations invoking
either New Physics or subleading corrections to the naive expectation
within the Standard Model has been proposed. Updated predictions for
the B^{0}_{s} → φφ exists and can now be confronted
with experiment.
In this analysis we look at the untagged timeintegrated 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 CPeven and CPodd B^{0}_{s }mass eigenstates _{ }using 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 assign as a systematic uncertainty the full expected effect. We validate this approach by performing a similar measurement using B^{0}_{s}→ J/ψφ decays, collected via the same trigger, and comparing results with current experimental information on the polarization in such a decay.
The timeintegrated 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 B^{0}_{s} mixing (as expected in the Standard Model) the only interference allowed is between the two CPeven amplitudes A_{0} and A_{}; hence, the only measurable phase is δ_{ }= arg (A_{}A_{0}^{*}). Thus 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_{^} can be determined from the normalization condition:
A_{0}^{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 B^{0}_{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 B^{0}_{s} state respectively. Assuming equal production for B^{0}_{s} and antiB^{0}_{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*cos^{2}θ for the θ_{1} and θ_{2}. This is checked to be adequate using sideband data and the parameter B is determined in the fit. The acceptance A(ω), with ω=(cosθ_{1}, cosθ_{2}, 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 B^{0}_{s }candidate mass fit
range 5.2 < m < 5.6 GeV/c^{2}.
Fit results with statistical uncertainties (click to download table)  
Correlation matrix (click to download table) 
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 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. 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 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.
Fit results with statistical uncertainties (click to download table) 
f_{L}[%]  f_{T}[%]  
CDFII experimental result 2.9 fb^{1}  34.8  ±4.1(stat.)±2.1(syst.)  65.2  ±4.1(stat.)±2.1(syst.)  
QCD factorization (2009)  34  ±28  66  ±28  A. Datta, D. London, J. Matias, M. Nagashima and A. Szynkman, Finalstate Polarization in B^{0}_{s} Decays. arXiv:hepph/0802.0897v2 
QCD factorization 1.a (2007)  43  ±0^{+61}_{34}  57  ±0^{+61}_{34}  M. Beneke, J. Rohrerand and D. Yang, Branching fractions, polarization and asymmetries of B→VV decays. Nuclear physics B,vol. 774(Issues 13):pgs.64101,9 July 2007 or arXiv:hepph/0612290v2 
QCD factorization 1.b (2007)  48  ±0^{+26}_{27}  52  ±0^{+26}_{27}  idem 
QCD factorization 2  86.6  13.4  X. Li, G. Lu and Y. Yang, Charmless B→VV decays in QCD Factorization. Phys. Rev. D 71, 019902(E) (2005) 

NAIVE factorization  88.3  11.7  idem  
NLO EWP 1  86.3  13.7  D. Du and L. Guo, Electroweak penguin contributions in charmless B→VV decays beyond leading logarithms J.Phys.G 23, 525.(1997), 

NLO EWP 2  86.3  13.7  idem  
Perturbative QCD (2002)  61.9  ^{+3.6+2.5+0}_{3.23.30}  38.1  ^{+3.6+2.5+0}_{3.23.30}  A. Ali, G. Kramer, Y. Li, C. Lu, Y. L. Shen,
W. Wang and Y. Wang, Charmless nonleptonic B^{0}_{s} decays to PP, PV and VV final state in the pQCD approach. Phys. Rev. D 76, 074018 (2007) 
In the following plot we show the measured polarization fractions f_{0} 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 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).
Angular sculpting for B^{0}_{s} → φφ events cosθ_{1}(gif) (pdf), cosθ_{2 }(gif) (pdf), φ (gif) (pdf)
Angular sculpting for B^{0}_{s}→ J/ψφ events cosθ (gif) (pdf), cosψ _{ } (gif) (pdf), φ (gif) (pdf)
f_{0} vs f_{} plot: comparison with theoretical models (gif), (pdf)