Issues related to proper time resolution, bias, acceptance etc.

Is "unbiased selection" really unbiased?

According to our study, the answer is basically no.

• Have found evidence that efficiency of reconstruction and "unbiased selection" gradually drops with proper time . Need a convincing explanation.
• There is a spike of efficiency at t~0 caused by the BIP significance cut in selection. The is due to wrong inclusion of signal tracks in PV reconstruction. Refitting PV after removing signal tracks useing the PV refit tools we developed is demonstrated to solves the problems. Need to make a proposal to make the procedure standard.

Then should proper time acceptance be modeled as a function of true proper time or measured proper time?

For simplicity of implementation, we prefer acceptance as a function of measured proper time since otherwise it is impossible to do the analytical proper time convolution.

Eventually we should check which is the right way to go. One can check in MC data the bias as a function of true proper time. If the efficiency changes with true proper time, this introduces no biases. If efficiency drops at small measured time, then one can see a positive bias of proper time at small true proper time.

How does proper time resolution depends on proper time?

It has been reported that proper time resolution of Bs->JpsiPhi and other Jpsi channels increases with proper time. We have confirmed this. Need to check if this is due to changes made in proper fitter or not. The new feature of the proper time fitter is that the error of measured mB and its correlation with pB and BV-PV distance is taken into account. The problem is this makes it difficult to use the proper time resolution extracted from either promp Jpsi or tail at negative measured time for Bs->JpsiPhi. Possible way out: if sigma of the proper time pull distribution doesn't depend on proper time itself, then we can calibrate the per event error at time~0 and use the scaling factor for any value of time.

The fitting code should be carefully checked.

How does proper time bias depends on proper time?

There is a significant negative bias at small time. Again this is due to wrong inclusion of signal tracks in PV reconstruction and PV refitting solves the problem.

Should Jpsi mass constraint be used?

The Jpsi mass constraint is not used in Bs vertex fit by default. This can be changed in options. How does this affect B mass resolution, bias and proper time resolution is to be re-examined. The main concern with using Jpsi mass constraint is muon bremsstrahlung.

new method to calibrate proper time error

Assuming Gaussina errors, calibrate vertex errors by splitting PV and SV, see early talk. For Bs veretx, this is equivalent to looking at (JpsiVi-BVi)/sqrt(JpsiViErr**2-BViErr**2) for i=x,y,z. Main issue: No or very loose cut on Bs chi2.

More general method: Expect D=(JpsiVx-PhiVx, JpsiVy-PhiVy, JpsiVz-PhiVz) to follow a 3d Gaussian function

1/(2pi)^(3/2)/sqrt(|V|)exp(-D^T*V^-1*D/2),

where V = K_Jpsi*Cov_JpsiV + K_phi* Cov_PhiV is the covariance matrix of D,

Cov_JpsiV and Cov_JpsiV are input covariance matrixes of Jpsi and phi vertices,

and K_Jpsi and K_Phi are calibration matrixes.

Actually the method works better in control channel Bd->JpsiK* than Bs->Jpsiphi since the errors of phi vertex are so big compared to J/psi veretx that any sensitivity to K_Jpsi is lost. In Bd->JpsiK* the situation is much better. So we can use Bd->JpsiK* to calibrate J/psi vertex errors and use the scaling factors for Bs->JpsiPhi.

Things to be investigated

• selection and trigegr without cutting on Bs veretx chi2 since a Bs chi2 cut removes tails in D distributions. This requires inclusive J/psi trigger.
• extraction of K_Jpsi and K_Phi with background
• non-gaussian errors, which even makes the obtained scaling factors dependent on the used range of D
• variation of scaling factors in phase sapce
• extention of method to track level
• using several gaussians for Jpsi and phi, equivalent to using sum of several gaussians for D

What do we do if the so obatined distributions are not gaussian?

Maybe this means the scaling factors have some dependence on phase space. Once the vairibales on which the scaling factors depend on, we can either parameterize the dependence or simply bin the data. It is found the scalign factors depend on B and J/psi vertex chi2.

A more practical solution is to use several gassian functions to model the resoltuion.

Correlation between mass and proper time

See very early talk.

Another way: add a bias term proportional to mass_B-nominal_mass_B in the resolution model and fit for the coefficient.

Problems to be investigated

AdaptivePVReFitter: compare defining weight as a function of chi2=Cm^T*(Xmeas-Xfit)*Cm and defining weight as a function of chi2_R = Cr^T*(Xmeas-Xfit)*Cr where Cm is the measurement covariance matrix and Cr=Cm-Cfit is the covariance matrix of the reduced residual Xmeas-Xfit. The former is usually used.

Try Adaptive fitter for secondary vertex. This may reduce tails. But this method requires tracks as input and can't use the calibrated jpsi and phi vertex errors. Require calibration of track errors.

(BTW, adaptive method can also be used in primary and inclusive secondary vertex finding, with the latter being interesting for flavour tagging.)