First Measurement of Polarization Amplitudes
and First Search for CP violation in the B0s →φφ decay

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Introduction and Data Sample

We have performed the  first measurement of the polarization amplitudes and the first search for CP violation in the charmless B0s → 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 B0s → 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

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

|A0|2 + |A|||2 + |Aperp|2 |A0|2 + |A|||2 + |Aperp|2

Due to the V-A nature of weak interaction and to the helicity conservation in QCD, fL >> fT is expected in B decays to two light vector mesons. This expectation was experimentally confirmed by BaBar and Belle in tree-dominated transitions while it was found fL ≈ fT in B+ → φK*+ and in B0 → φK*0 where the b → s penguin transition is involved. This is known as the Polarization Puzzle.
Explanations invoking either New Physics or sub-leading corrections to the Standard Model predictions have been proposed. The B0s → φφ decay proceeds via a b → s penguin transition representing a very interesting decay channel in this experimental and theoretical scenario.

Analysis Description

We look at the untagged time-integrated 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 B0s mixing (as expected in the Standard Model), the rate depends only on three observables: two polarization amplitudes squared |A0|2, |A|||2, and the strong phase  δ||.   The strength of Aperp can be determined from the normalization condition |A0|2 + |A|||2 + |Aperp|2 = 1.

The fit to the mass and decay product angular distributions is performed in the helicity basis and we define ω=(cosθ1, cosθ2, φ).

Fig: 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.


The decay time is not observed and only the time integrated rate is measured. Assuming equal production of B0s and anti-B0s mesons, the differential decay rate, including the acceptance A(ω), can then be written as:

and for the B0s → φφ decay we have:

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 fb is determined from the fit in the B0s candidate mass range  5.2 < m < 5.6 GeV/c2. 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 CP-even and CP-odd B0s mass eigenstates  using the 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 account for it as a systematic uncertainty.
We validate this approach by performing a similar measurement using the   B0s→ J/ψφ decays, collected via the same trigger, and comparing the obtained results with the current experimental information on the polarization of this decay.

Fit Projections and Results


Table: Fit results with statistical uncertainties Table: Correlation matrix


Systematic Uncertainties

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 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.  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 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.

Polarization results

We measure the polarization amplitudes and the cosine of δ|| = arg (A||A0*) for B0s → φφ by performing an unbinned Maximum Likelihood fit to the reconstructed B0s candidate mass and the three angular variables. The results are:

|A0|2 = 0.348±0.041 (stat) ±0.021 (syst)
|A|||2 = 0.287±0.043 (stat) ±0.011 (syst)
|Aperp|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 performed a similar measurement on the B0s→ J/ψφ→ [μ+μ-][K+K-] decays collected with the same trigger selections as the B0s→ φφ sample. The polarization results we obtain are consistent with the CDF published ones [1] measured on a larger statistics data sample.

Comparison of present measurement with SM theoretical prediction

In the following plot we show the measured polarization fractions f0 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 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).

Search for CP violation

The B0s → φφ data sample statistics does not allow us to investigate the mixing induced CP-violation. However, a class of CP-violating effects which can reveal the presence of NP are the Triple Products (TP) correlations [3]. TP's are defined as: TP= p*(q1 x q2) where p is a momentum, and both q1 and q2 can be either spins or momenta of the decay particles. Triple products are odd variables under time reversal (T), therefore they constitute potential signals of CP violation. The TP's asymmetry is defined as:
ATP = Γ(TP>0) - Γ(TP<0)

Γ(TP>0) + Γ(TP<0)

where Γ is the decay rate of the process in question. Most of these TP's asymmetries are expected to be small in the SM, but can be enhanced in the presence of NP in the decay. In the untagged case the TP asymmetries are proportional to the so-called "true" TP's asymmetry, that is a true CP violating effect. In what follows, for shortness, we refer to them as TP only.

Analysis Description

In the B0s → φφ decay, there are two Triple Products: the first, TP1, is proportional to Im(A||*Aperp) and the second, TP2, allow us to access Im(A0*Aperp). TP1 can be probed through the observable u=cosφsinφ. The asymmetry on u, Au, is proportional to the asymmetry of TP1, and is defined as: Au=(N+- N-)/(N++ N-), where N+ (N-) are the number of events with u>0 (u<0).
In a similar way, we define an asymmetry Av for the variable v=(sinφ for cosθ1cosθ2>0 and sin(-φ) for cosθ1cosθ2<=0 ), which is proportional to the asymmetry of TP2.
The u and v distributions are shown in the following figures for side-bands subtracted signal events:

Side-bands subtracted distribution of u.

Side-bands subtracted distribution of v.

The asymmetry Au and Av are evaluated through an unbinned Maximum Likelihood fit to the B0s mass candidates.
First, we consider the N candidates in the range [5.2; 5.6] GeV/c2 and we split this sample into two subsamples made of N+ and N- events, according to the sign of u (or v) of the events. Using a large sample of MC data generated with uniform angular distribution, we check that the detector acceptance and the reconstruction requirements are not introducing artificial asymmetries in the u and v distributions. No bias was found in the splitting procedure within a 0.2% accuracy.

Fit Results

We fit the invariant mass distribution of each subsets in order to extract the signal yields and measure the signal asymmetry. No asymmetry is found in the background events. The fit projection are:

Fig: Mass distribution of N+ (left) and N- (right) subsamples with fit projections overlaied for u asymmetry.

Fig: Mass distribution of N+ (left) and N- (right) subsamples with fit projections overlaied for v asymmetry.

Several sources of systematic uncertainities are studied using MC simulation. We estimate the effect of changing the signal and the combinatorial background parameterization model. Potential contamination of some decays, such as B0 → φK*, B0s → φf0 and B0s → φK+K-, are taken into account as well.

In conclusion, we measure the asymmetries of two T-odd helicity angles functions and the final results are:

Au = -0.007±0.064 (stat) ±0.018 (syst)
Av = -0.120±0.064 (stat) ±0.016 (syst)

The first asymmetry, Au, is well consistent with 0 within experimental uncertainties; the second one, Av, is 1.8 sigma from 0 considering both statistical and systematic uncertainties. These asymmetries constrain the size of two T-violating true Triple Products asymmetries of the B0s →φφ decay expected null in the SM.


  1. The CDF collaboration. Measurement of Lifetime and Decay-Width Difference in B0s →J/ψφ Decays. arXiv:0712.2348v2 [hep-ex],2007;
  2. An overview of the theoretical papers on the polarization are reported here;
  3. A.Datta and D.London. Triple-Product Correlations in B→V1V2 Decays and New Physics. International Journal of Modern Physics A, 19:2505, 2004; A.Datta, M.Duraisamy, D.London. Searching for New Physics with B-Decay Fake Triple Products. arXiv:1103.2442, 2011.

List of Public Plots and Tables 

CP violation: