First Measurement of the Polarization Amplitudes of the
0s → φφ decay

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 B0s → φφ → [K+K-][K+K-] decay of the B0s meson.
The results are obtained with an unbinned Maximum Likelihood fit to the reconstructed B0s 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 B0s → 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 B0s → φφ as:

|A0|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)

and the resulting polarization fractions are:
fL = 0.348±0.041 (stat) ±0.021 (syst)
fT = 0.652±0.041 (stat) ±0.021 (syst)

We report also a similar measurement  performed on B0s→ J/ψφ→ [μ+μ-][K+K-] decays collected with the same trigger selections as the B0s→ φφ sample:

|A0|2 = 0.534 ±0.019(stat. only)
|A|||2 = 0.220 ±0.025(stat. only)

which are consistent with the published ones (arXiv:0712.2348v2 [hep-ex],Phys.Rev.Lett.100:121803, 2008).

The results have been approved by CDF as of March 4th, 2010.
2010. The measurement is described in CDF note 10120.

Jump to Motivation, Analysis Description, Fit Results, Systematics, B0s→ J/ψφ Results, Comparison with Theory, List of Approved Plots and Tables.


The B0s → φφ is a P → VV (Pseudo-scalar to Vector Vector) decay whose differential decay rate is determined by  three independent amplitudes corresponding to different polarizations: one longitudinal (A0) and two transverse polarizations with spins parallel (A||) and perpendicular (A^) to each other.
With these amplitudes we can define two amplitude fractions:
the transverse amplitude fraction fT = |A|||2 + |A^|2

|A0|2 + |A|||2 + |A^|2
the longitudinal amplitude fraction fL = |A0|2

|A0|2 + |A|||2 + |A^|2

Due to V-A nature of weak interaction and helicity conservation in QCD, fL >> fT 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 fL ≈ fT 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 B0s → 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 B0s mass eigenstates  using world average B0 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  B0s→ 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 B0s 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  B0s→ 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 τL=1.408+0.033-0.030 ps and τH=1.543+0.058-0.060  ps, where τL,H are the lifetime of the Light and Heavy B0s state respectively. Assuming equal production for B0s and anti-B0s 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 fb in the B0s 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 B0s → φf0 and a non-resonant B0s → 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 B0s 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 B0s→ J/ψφ Decay


   Fit results with statistical uncertainties (click to download table)


Comparison of present measurement with SM theoretical prediction

  fL[%]   fT[%]    
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,
Final-state Polarization in B0s Decays.
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 1-3):pgs.64-101,9 July 2007 or arXiv:hep-ph/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+0-3.2-3.3-0 38.1 +3.6+2.5+0-3.2-3.3-0 A. Ali, G. Kramer, Y. Li, C. Lu, Y. L. Shen, W. Wang and Y. Wang,
Charmless nonleptonic B0s 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 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).

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