Information for Combination in Global Fits

This webpage provides information on how to combine this result with other information in global fits.

Let's first clarify briefly how we treat nuisance parameters: We use a "Projection Method": Our confidence region in the DGs-betas space is the projection of a multidimensional confidence region in the larger space that includes all nuisance parameters. In other words, we exclude a specific value of (DGs, betas) only if it can be exluded for any assumed values of the nuisance parameters (within 5-sigma from their nominal values). To do this in practice, we evaluate for each (DGs-betas) point the most conservative p-value for our observed profile-LR statistics, out of a number of possible choices for the values for nuisance parameters.

We provide for combinations in global fits:
(1) Profile Likelihood: Values of the observed prof-LR in our data, sampled on a grid in the (DGs, betas) space (this statistics does not explicitly contains the values of nuisance parameters)
(2) Probability Distribution: The "most conservative" probability distribution of this statistics, over the possible variations in the nuisance parameters.

(1) Profile Likelihood:
The profile likelihood is given as a 40x20 histogram. There are 40 bins in beta_s and 20 bins in delta_gamma. Beta_s range is from -pi/2+pi/80 to pi/2+pi/80. Delta_gamma range is from -0.7 to 0.7.
For each point on the grid the values of beta_s and delta_gamma are fixed to the center of the bin and the the likelihood function is maximized with respect to all other (nuisance) parameters. The obtained value is L_ij corresponding to bin (i,j). The absolute maximum of the likelihood function is obtained by maximizing with respect to all nuisance parameters and beta_s and delta_gamma as well. The obtained value is L_max. Each bin of the histogram is filled with the difference -2*ln(L_ij) - ( -2*ln(L_max) ), where "ln" means "natural log".
The likelihood has two equal absolute minima corresponding to the exact invariance of the likelihood under the simultaneous transformation:

The likelihood maximization program (MINUIT) will in general find the minimum closest to the starting values of the floating parameters.

To remove this exact symmetry we box delta_para in [0, pi]. The resulting profile likelihood in beta_s-delta_gamma plane is shown as 2D likelihood profile with phi_para in (0, pi): (eps), (png).

One can alternatively box delta_para in [pi, 2pi]. The resulting profile likelihood in beta_s-delta_gamma plane is shown as 2D likelihood profile with phi_para in (pi, 2pi): (eps), (png).

As expected from first princiles, the likelihood contours corresponding to the two ways of boxing phi_para are symmetric w.r.t. the simultaneous transformation (2): The final likelihood profile in the beta_s-delta_gamma plane is obtained by taking the lowest minimum at each (beta_s, delta_gamma) point and it is shown as this 2D likelihood profile: (eps), (png).

The final result is quoted in terms of confidence regions obtained using likelihood ratio ordering with frequentist inclusion of systematic uncertainties. That is, we do exclude a specific value of DG and beta_s only if this value can be excluded for any possible value of all other nuisance parameters within 5-sigma of their nominal values.

The confidence region is evaluated for phi_para boxed in [0, pi] and shown as 2D Feldman-Cousins confidence regions with phi_para in (0,pi): (eps), (png).

The confidence contours corresponding to phi_para boxed in [pi, 2pi] is obtained by reflecting above contours according to transformation (2) and shown as 2D Feldman-Cousins confidence regions with phi_para in (pi, 2pi): (eps), (png).

The final confidence region is obtained by choosing between above two p-values the larger p-value for each (beta_s, delta_gamma) point: 2D Feldman-Cousins confidence regions: (eps), (png).

The nuisance parameters are:
- Bs lifetime
- Decay time resolution scale factor
- Bs mass
- Bs mass resolution scale factor
- Two polarization amplitudes
- Two strong phases
- Bs oscillation frequency (delta_ms) gaussian constrained to 17.77 +/- 12 ps-1
- Opposite side tagger dilution scale factor (gaussian constrained to value determined from B+/- and B0 -> J/Psi K* )
- Opposite side tagger signal efficiency
- Opposite side tagger background efficiency
- Opposite side tagger background asymmetry
- Same side tagger dilution scale factor (gaussian constrained to value determined in Bs mixing analys)
- Same side tagger signal efficiency
- Same side tagger background efficiency
- Same side tagger background asymmetry
- 6 parameters that describe the background angular distributions
- Signal fraction
- Slope of the mass background distribution
- 6 parameters that describe the background decay time distribution
- 10 parameters that parameterize the decay time resolution distributions for signal and background

(2) Probability Distribution:
In order to provide ingredient (2) the authors of the analysis run several tests, and found that we are in a situation where the distribution of prof-LR, while not a 2-dof-chi2, is very nearly independent of the value of (delta_gamma, beta_s). We are therefore providing a single distribution, valid for all values of (delta_gamma, beta_s). For reference, both the distribution obtained with nominal values for the nuisance parameters, and the most conservative distribution obtained by random sampling 16 alternative choices are provided. See this Figure.

As can be seen from this plot, showing the tail integrals, all distributions have longer tails than a chi2, shown for reference in red (they actually overlap quite nicely with a chi2 distribution if the horizontal scale is scaled linearly by an appropriate factor).
The effect on the results is not negligible: if you use the most conservative of these 16 distributions, that has a 90%-point at LR~6, to be compared with the nominal chi2-value of 4.6. (In deriving our 90% confidence regions, we have used an estimated 2.3% of additional tail in the "most conservative" distribution with respect to the distribution for nominal nuisance parameters (shown in black)).

The folowing data files are made available:
  • integrated_likelihood_ratio.root
  • plot_integrated_likelihood_ratio.C
  • profile_likelihood.root
  • plot_profile_likelihood.C
  • rootlogon.C

    Some simple scripts are available to open histograms and make plots. You can download the above files, and in your directory start Root and execute:

    root [0] .x plot_profile_likelihood.C
    you will see the profile likelihood.
    root [1] .x plot_integrated_likelihood_ratio.C
    you will see the tail integral of the LR distribution (black) and the corresponding curve including systematics (dotted red).