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A number of factors contribute to the resolution of the reconstructed value of Δz, and hence to that of the com- puted value of ΔtΔz/βγ. The experimental resolution R(δt, σΔt), as a function ofδt= Δt−Δttrueand the uncer-

tainty on Δt,σΔt, can be accounted for when measuring

time-dependent CP asymmetry parameters by convolut- ingR(δt, σΔt) withf_±Phys(Δt), giving

FPhys ± (Δt) = ∞ −∞ fPhys ± (Δttrue)R(δt, σΔt)dΔttrue, =f_±Phys(Δt)⊗R(δt, σΔt). (10.4.1)

Therefore, one can replacef_±PhyswithF_±Physin Eqs (10.3.3) and (10.3.4) to obtain the corresponding equations that account for both dilution and resolution effects. Factors contributing to the resolution of Δtinclude:

– Btagvertex resolution, which is a combination of track-

ing effects and, for a sub-sample of Btag mesons, the

finite lifetime ofDmesons;

– BCP vertex resolution, which is a superposition of track-

– resolution of the measurement of the boost factorβγ determined from the energy of thee+_ande− _beams.

It is important to understand the Δt resolution in detail as this is of a similar magnitude to the average separation between theBCP andBtagproper decay times. Thus, this

resolution has a significant effect on the extraction ofS andC from a time-dependent analysis.

Different approaches are used to understand resolu- tion effects at theBFactories.BABARadopts a paramet- ric approach to describe the Δtresolution, whereas Belle characterizes resolution effects according to their physical source. Both approaches work well and provide a good de- scription of resolution for use in time-dependent analyses. The nominalBABARΔtresolution function has a triple Gaussian form, where the mean μi and width si of the

two central Gaussian components are scaled byσΔton an

event-by-event basis. The three Gaussians are denoted by Gi, wherei= core, tail, and outlier, in order of increasing

width. The resolution function is given by

Rsig(δt, σΔt) =fcoreGcore(δt, μcoreσΔt, scoreσΔt) +

ftailGtail(δt, μtailσΔt, stailσΔt) +

foutlierGoutlier(δt, μoutlier, soutlier).

(10.4.2) The parametersstail,soutlierandμoutlierare set to 3.0, 8.0

ps and 0.0 ps, respectively, and the other parameters are determined from reference samples of fully reconstructed B meson decays as described in Section 10.6. The tail width was determined from MonteCarlo simulated data, and the outlier mean was taken as unbiased, with a width varying from4−12ps. The mean of this range was taken as the nominal value for soutlier. As the physical tagging

categories forBABARhave different purities and dilutions, the values ofμiandsifor the core Gaussian contribution

to the resolution function depend on the flavor category of an event. This difference is taken into account when analyzing data. For early analyses, each of theBABARfla- vor tagging categories had a separate value forμcore and

score; in later iterations, the distinction was only made

between Lepton and non-Leptontagging categories. For

BABARdata,scoreis typically 1.01±0.04(1.10±0.02) for Lepton(non-Lepton) events.

The Belle Δt resolution function (Tajima, 2004) ac- counts for four different physical effects

– Btagvertex resolution,

– B_CP vertex resolution,

– shift in the Btag vertex position resulting from sec-

ondary tracks from charm meson decays, and

– kinematic approximation that theBmesons are at rest in the center-of-mass frame.

TheBtagandBCP vertices are described by (i) a Gaus-

sian resolution function in the case of multi-track vertices, and (ii) a sum of two Gaussians in the case of single-track vertices. The widths of these Gaussians are scaled by the uncertainty on the reconstructed vertex being described. The resolution function resulting from non-prompt tracks

associated with a decay in flight of charm mesons is de- scribed by the sum of a delta function and exponentials. The kinematic approximation is described by a resolution function dependent on the polar angle of Btag as recon-

structed in the center-of-mass frame of reference. Given that aBCP orBflav candidate is fully reconstructed, and

decays opposite theBtagin the center-of-mass frame of ref-

erence, whereas theBtagmay not be, the polar angle of the

Btagcandidate is determined from the fully reconstructed

BCP orBflav decay. The physical time dependencef_±Phys

is convoluted by each of these resolution functions in turn in order to obtain the resultantF_±Phys.

Figure 10.4.1 shows thef_±PhysandF_±Phys distributions for S = 0.7 and C = 0.0, where both dilution and res- olution effects are considered. The distribution f_±Phys is smeared out considerably as a result of experimental reso- lution when computingF_±Phys. The effect of dilution serves to reduce the reconstructed asymmetry betweenB0_{- and}

B0_{-tagged events. This can be seen as a reduction in the}

asymmetry between F+ and F− in comparison with the

true distributionsf+andf−.

t (ps) Δ -10 -5 0 5 10 Arbitrary scale t (ps) Δ -10 -5 0 5 10 Arbitrary scale

Figure 10.4.1.Distributions of (top)f_±Phys(Δt) withS= 0.7, andC= 0.0 for (solid)B0

- and (dashed)B0

-tagged events for perfectly reconstructed decays, and (bottom) the correspond- ing distributionsF_±Phys after taking into account typical dilu- tion and resolution effects.

10.5 Modeling the

Δ_t

distribution for