Study of t Resolution Function

Transcription

1 Belle-note 383 Study of t Resolution Function Takeo Higuchi and Hiroyasu Tajima Department of Physics, University of Tokyo (January 6, 200) Abstract t resolution function is studied in detail. It is used for unbinned maximum likelihood fit of lifetime, BB oscillation and indirect CP analysis. Resolution function is described as convolution of z resolutions for reconstructed and tagging side vertex, smearing of tagging side vertex due to K S and charm daughter contamination, and smearing of t due to kinematic approximation. Functional parameters for the first two contributions are obtained using data while parameters for the last three contributions are obtained using MC. I. MOTIVATION Lifetime and indirect CP analyses for ICHEP2000 are based on event by event double- Gaussian resolution function (brief description is given in section III). Although it is good enough to obtain lifetime at a precision of 0.05 ps, we also noticed that the fit result exhibit discrepancy from the data distribution at the tail region. For example, Fig. shows the lifetime fits for B 0 D + l ν and B D 0 π modes. In this study, origin of the resolution tail is studied in detail to reduce the discrepancy. Contribution to t resolution is divided into detector resolution, smearing due to contamination of tracks from non-primary decays and kinematic approximation described in the following section. II. PROPER-TIME DIFFERENCE RECONSTRUCTION In this section, t reconstruction procedure is described. Fig 2 illustrates the vertex reconstruction of two B decay vertices. Since the B momentum in the Υ(4S) rest frame is small, the decay vertex distance in the beam direction ( z B ) is used to extract the lifetime. The decay vertex of the reconstructed B is obtained using all tracks that are associated with SVD hits to form the B candidate. In the B DX analysis, D 0 and D + decay

2 # 7 $&%(' /0,- )*+ B D*! " [Entries/0.6ps] 0 2 (a) B D 0 π (signal region) 0 B t B (ps) FIG.. Lifetime fit results for (a) B 0 D + l ν and (b) B D 0 π decay modes. Dotted line represent background contribution to the fit. D daughters Reconstructed B IP profile D D 0 Associated B FIG. 2. Illustration for vertex reconstruction of two B decay vertices. π + 2

3 vertices are determined first and B decay vertex is determined using the pseudo D track and the primary track from the B decay. Interaction point constraint is applied during the vertex fit to improve the resolution in the rφ plane. Typical decay vertex resolution of the reconstructed B is approximately 40 µm. The decay vertex of the associated B is determined using remaining tracks that are associated with SVD hits in the event. (TagV routine [] is used.) The expected track error in z direction (σz track ) must be less than 0.5 mm to eliminate badly measured tracks. K S daughter tracks should be rejected since they do not originate from the B primary vertex. Tracks which can form K S with any other track are rejected. The K S daughter tracks are further reduced by eliminating tracks with δz >.8 mm. Here δz is the z distance between the vertex point of the reconstructed B meson and the track position at closest approach to the reconstructed B vertex. The cut is loose enough not to bias the reconstruction efficiency while effectively rejecting K S and badly measured tracks. The possible bias due to the reconstruction procedure is evaluated in the section??. We reject tracks with impact parameter to the interaction point (δr) in rφ plane greater than 0.5 mm to further reduce the K S daughter. These cuts, which rejects many K S and badly-measured tracks, is important: if too loose a criterion is used, the fraction of badly-measured events becomes large, and modeling of the resolution becomes difficult. If the reduced χ 2 (χ 2 /n χ 2 /number of degrees of freedom) of the vertex fit with IP constraint is worse than 20, the track that gives the largest contribution to the χ 2 is removed. This procedure is iterated until the χ 2 /n requirement is satisfied or only one track is left. The efficiency to reconstruct the vertex of the associated B meson is 90%. This method does not properly treat the secondary vertex due to charm decay although it tries to reduce the effect by removing tracks which is inconsistent with the decay vertex. The effect of the secondary charm vertex moves the decay vertex point of the associated B toward charm flight direction. It also degrade the vertex resolution. The mean value shift and resolution of the decay point for the associated B are approximately 20 µm and 70 µm, respectively. Since we do not know production points of neither B mesons, we cannot calculate the proper-time of neither B meson decays. Assuming the B mesons are produced at rest in the Υ(4S) rest frame, the proper-time difference, t B, is approximated as; t B z B, (βγ) Υ p z(υ) m(υ), where z B is the distance of the two B decay vertices in z-axis. The error of the kinematic approximation, σ K, is included in the resolution function. III. RESOLUTION FUNCTION FOR ICHEP In this section, we describe the resolution function used for lifetime and indirect CP analyses for ICHEP2000. The resolution function R SIG ( t) is parameterized by the sum of two Gaussians: R SIG ( t) = ( f tail ) e ( t µ t) 2σ t 2 + f tail 2πσ t 2πσ t tail 2 e ( t µ t tail )2 2(σ tail t )2, 3

4 where f tail is the fraction of the tail part of the resolution function, and σ t, σtail t, µ t and µ t tail are the proper-time difference resolutions and the mean value shift of the proper-time difference for the main part and the tail part of the resolution function, respectively. f tail is determined to be 0.08 ± 0.06 by the lifetime analysis of the B D l ν sample using the same resolution function. A detailed description of the lifetime analysis for B D l ν and B J/ψK samples can be found in the Ref 2. µ t and µ t tail originate from the mean value shift of the z B measurement and discussed below. The proper-time difference resolution σ t (and σtail) t is calculated event by event and a convolution of the z B resolution σ z and the error due to the kinematic approximation ( t B z B, (βγ) Υ = pz(υ) ) σ m(υ) K: σ z σ t = ( ) c(βγ) 2 + σk 2, Υ σtail t = σ tail z ( ) c(βγ) 2 + (σtail K )2. Υ The σ K and σtail K values are determined to be σ K = 0.30±0.03 ps and σtail K = ps using the MC simulation since these parameters are independent of the detector performance. The z B resolution σ z ( σ tail z and σ asc mesons, σ rec z z : ) is calculated from the vertex resolutions of the two B σ z 2 = Sdet(σ 2 z rec ) 2 + (Sdet 2 + Scharm)(σ 2 z asc ) 2 tail )2 (σz rec ) 2 + {(Stail det )2 + (Stail charm ( σ z tail )2 = (S det ) 2 }(σ asc z ) 2 where S charm and Stail charm are scaling factors to account for the degradation of the vertex resolution of the tagging side B meson due to contamination of the charm daughters, and S det and Stail det are the global scaling factors to account for the systematic uncertainties in the vertex resolutions σz rec and σz asc computed from the track helix errors in the vertex-fit. The S charm and Stail charm values are determined to be S charm = and Stail charm = using the MC simulation. The S det and Stail det values must be determined from the data as these parameters depends on the detector performance. S det is determined using D 0 K + π sample. Production point of the the D 0 is obtained from the primary tracks in the same hemisphere as the D 0 candidate with the IP constraint. Distance between the D 0 decay vertex and production vertex in z direction is fit with the same resolution function and known D 0 lifetime to obtain the S det. The S det value is measured to be S det =.00 ± 0.03 from the data and S det =.23 ± 0.02 from the MC simulation using the D 0 sample. Since we find S det =.6 ± 0.03 for B J/ψK MC sample, we fix the S det value to be S det = (.00 ± 0.03) (.6 ± 0.03)/(.23 ± 0.02) = 0.94 ± Stail det is determined to be 2.4 ± 0.6 by the lifetime analysis of the B D l ν sample [2]. A small of fraction of events have a large reduced χ 2 (χ 2 /n, n = number of degrees of freedom). We have found that the vertex error computed from track helix errors in the vertex-fit underestimates the vertex resolution and the vertex with larger χ 2 has worse resolution. In order to take into account this effect, we introduce the effective vertex resolutions σ rec t and σ t asc when χ 2 /n is greater than 3: 4

5 ( σ t rec ) 2 [ + α rec {(χ 2 /n) rec 3}](σ rec ) 2 [ + α asc {(χ 2 /n) asc 3}](σ asc ( σ asc t z ) 2 : (χ 2 /n) rec > 3, z ) 2 : (χ 2 /n) asc > 3, where (χ 2 /n) rec and (χ 2 /n) asc are reduced χ 2 of vertex-fits for the reconstructed and associated B decay vertices, respectively. As mentioned above, the µ t (and µ t tail) originates from the mean value shift of the z B measurements µ z (and µ z tail ): µ t µ z, µ t tail µ z tail. The mean value shifts of the z B, µ z and µ z tail, are caused by the contamination of the charm daughters in the vertex reconstruction of the tagging side B meson and are correlated with the σz asc : µ z (σz asc ) = µ 0 + α µ σz asc, µ z tail(σz asc ) = µ 0 tail + α µ tail σasc z. The µ 0 and α µ values are determined to be µ 0 = 6 ± 4 µm and α µ = 0.7 ± 0.06 using the MC simulation. Since α µ tail is found to be consistent with zero, µ z tail is fixed at the value 60 µm determined from the MC sample. Fig 3 shows the t cal t gen distribution for the MC signal events and the resolution function described here, where t cal and t gen is the calculated and true proper-time differences, respectively. The distribution is well-represented by the resolution function. IV. NEW RESOLUTION FUNCTION We use a sum of two Gaussian to represent the resolution function. We do not avoid introducing third Gaussian function to avoid the inflation of the number of fit parameters in the lifetime fit. In addition, the third Gaussian will not describe the tail well since it is non-gaussain as observed in the Fig.. This choice imposes serious limitation on the parameterization in convolution of several different source of the resolution. Detector resolution, smearing due to displaced charm vertex and kinematic approximation are three main sources considered in the previous resolution function. However, as described in the previous section, they are combined in terms of σ and mean value shift for the main and tail part of the Gaussian function. This method assumes (or forces) that three contributions have the same tail fraction. This assumption does not cause any significant problem because the detector resolution is the dominant contribution to the resolution. However, as indicated in the first section, a problem is observed at the tail region of the distribution. The new resolution function is represented as a convolution of three independent contributions so that each resolution function can be chosen independently. Detector resolution, smearing due to non-primary tracks (K S and charm daughters) and kinematic approximation are considered to represent the resolution function as R( t) = dt k R k ( t t k ) dt s R NP (t k t s )R det (t s ), 5

6 [Entries/0.3ps] t rec t gen (ps) FIG. 3. The t cal t gen distributions for the MC signal events. A fit with the resolution function is superimposed. where R det, R NP and R k are resolution functions for detector resolution, smearing due to non-primary tracks and kinematic approximation. Although the new function looks more complicated than the previous one, it will not necessarily increase the number of fit parameters. Since R NP and R k are independent of the the detector performance, those function can be determined using the MC sample. Each function is described in detail in the following sections. A. Detector resolution As described in the previous section, detector resolution depends on the χ 2 /n of the vertex fit. This is understood as originated from poorly measured tracks. Fit.?? shows the distributions for differences of helix parameters (dρ and dz) between incoming and outgoing cosmic tracks divided by calculated helix errors. Since two reconstructed cosmic tracks originate from one cosmic ray, the difference gives resolution. We observe non-gaussian tail in both distributions. The tail is considered to be due to hard-scattering or misassociation of the SVD hits. We use the main Gaussian part to calibrate track error. When the vertex fit uses a poorly measured track which belongs to the tail part of the distribution, χ 2 /n and resolution of the fit gets worse. We take into account the effect of poorly measured track by making the resolution function dependent on the χ 2 /n of the vertex fit. In the old resolution function, the χ 2 /n dependence of resolution function is introduced by mean of the χ 2 /n dependence of the σ value. However, this is found not to be a good representation of the χ 2 /n dependence. We consider that the the tail part of the track resolution is responsible for the tail part of the resolution function. When the χ 2 /n of the 6

7 vertex fit is worse, the probability that the vertex fit includes poorly measured track is larger, which increases the fraction of the tail part of the resolution function. We fit the z cal z gen distribution as a function of the χ 2 /n of the vertex fit. The following function is used for the fit; ( f tail ) e x 2 2σ main 2 + f tail e 2πσmain 2πStail σ main x 2 2(S tail σ main ) 2. S tail is fixed at a value XXX which is obtained by the fit using all events. Fig.?? shows the fit results for f tail and σ main as a function of the χ 2 /n of the vertex fit. The tail fraction f tail increases by a factor of four with larger χ 2 /n while the σ main is more or less constant as we expected. It is ideal to calculate the tail fraction of the resolution function from the tail fraction of the track error. When χ 2 /n is large, confidence level of the fit is very small for the main part of the track error while confidence level of the fit is relatively large for the tail part of the track error. This naturally introduce the χ 2 /n dependence of the tail fraction. However, it is not practical since many helix parameters and their correlations have to be taken into account. We empirically calculate the confidence level for the tail part of the resolution function by scaling χ 2 /n by a factor of /S tail. Using the confidence levels for the main and tail parts of the resolution function (CL main and CL tail ) as weights for two parts, the tail fraction can be expressed as p CL tail f tail = q. ( p) CL main + p CL tail This expression gives f tail = pq for CL main CL tail (χ 2 /n 0) and f tail = q for CL main CL tail (χ 2 /n ). Fig.?? shows the z cal z gen distribution and the fit results with two kinds of resolution function. Fig.?? (a) shows the results with constant f tail and Fig.?? (b) shows the results with f tail with the above χ 2 /n dependence. Confidence level of the fit is improved by the χ 2 /n dependent f tail. Table?? summarize the parameters for detector resolution function obtained using the MC sample. We introduce global scaling factors for both B meson vertex, Smain data, Sdata tail and q data to account for the difference between the data and the MC as S main = SmainS data main, MC S tail = Stail data SMC tail, p CL tail f tail = q data q, ( p) CL main + p CL tail where Smain MC, SMC tail, p and q are different between the reconstructed and associated B mesons, and are determined using the MC sample while Smain data, Sdata tail and q data are common for both B mesons and determined using the data sample in a similar manner as the previous resolution function. Since the χ 2 /n of the vertex fit for the two B meson vertices is independent from each other, the tail fraction of the resolution are calculated independently. Thus the detector resolution function is represented as convolution of two resolution functions for two B meson vertices as 7

8 R det ( t) = dtr z ( t t; σt rec, Smain rec, Srec tail, f tail rec )R z(t; σt asc, S asc t 2 R z (t; σ, S main, S tail, f tail ) = ( f tail ) 2πSmain σ e 2(S main σ) 2 + f tail 2πStail σ e main, Sasc tail, f asc t 2 2(S tail σ) 2, where σt rec and σt asc are errors calculated from the vertex fits for the two B mesons (reconstructed and associated B mesons), and ftail rec asc and ftail are the tail fraction of the resolution function calculated from the χ 2 /n of the vertex fits for the two B mesons. This resolution function is equivalent to a sum of four Gaussians as 4 R det ( t) = f i e t 2 i= 2πσi σ 2 i, f = ( ftail rec asc )( ftail ), σ = f 2 = ftail rec asc ( ftail ), σ 2 = f 3 = ( ftail rec asc )f f 4 = f rec tail f asc tail, σ 4 = (S rec ) 2 + (S asc mainσ asc t ) 2, mainσt rec (Stail recσrec t ) 2 + (Smainσ asc t asc ) 2, tail, σ 3 = (Smainσ rec t rec ) 2 + (Stail ascσasc t ) 2, (S rec tail σrec t ) 2 + (S asc tail σasc t ) 2. Fig.?? shows z cal z gen obtained from the MC sample with zero charm lifetime. Fig.??(a) also shows a fit to R det ( z) function described here and Fig.??(b) shows a fit to the old resolution function for comparison. Confidence level of the fit is improved by XX%. tail ), B. Smearing due to non-primary tracks The effect of non-primary tracks, mainly K S daughters and charm daughters, on the resolution of the associated B meson vertex are considered in this section. The vertex reconstruction algorithm rejects some of non-primary tracks as described above. Remaining non-primary tracks smear the resolution function and its contribution will be non-gaussian. Fig. 4 shows the z cal z gen distribution for the associated B meson obtained from the MC sample with zero charm lifetime (effect of the charm can be ignored). We observe clear asymmetric non Gaussian tail in the event with K S daughters. We parameterize the effect of the K S daughters as R KS (t) = ( f p f n )δ(t) + f p e t f τp n + e t τn, τ p τ n (f p = 0 for t 0, f n = 0 for t 0). The effect of charm daughters are studied using the MC sample with finite charm lifetime. In this sample, tracks from charm decays are moved by the same amount with the displacement of charm decay vertex from the B decay vertex and z vertex position of the associated B meson is calculated (z NC ). This z value correspond to the value with no charm lifetime. z NC is compared with the z vertex position of the associated B meson without moving charm daughter tracks (z charm ). Fig 5 shows the z charm z NC distribution. This distribution represents the resolution smearing of the charm daughters. The distribution 8

9 W/O K S daughters 20 With K S daughters z cal z gen (µm) z cal z gen (µm) FIG. 4. z cal z gen distributions for events without K S daughters (left) and events with K S daughters (right). is asymmetric and non-gaussian. Convolution of this distribution with Gaussian function causes overall mean value shift which we observe in the overall resolution function. The distribution is represented by a sum of eight exponential function as 4 R charm (t) = ( f i t p f i τ e i t i= τp i p + n τ e i τn i n ), (fp i = 0 for t 0, f n i = 0 for t 0). The fit with this function is superimposed in the Fig z charm z NC (µm) 000 FIG. 5. z charm z NC distribution. Fit result with first three terms of the R charm is also shown. Overall resolution function due to non-primary tracks, R NP (t) is a convolution of R KS (t) and R charm (t) as R NP (t) = dt R charm (t t )R KS (t ). 9

10 Since convolution of two exponential function with lifetimes τ and τ 2 is an exponential function with a lifetime τ τ 2 /(τ + τ 2 ), R NP (t) can be simplified as 5 R NP (t) = ( f i t + τ f i t e i i= τ+ i + + τ e i τ i ), (f+ i = 0 for t 0, f i = 0 for t 0). The first four terms correspond convolution of four terms in the R charm (t) and the delta function in the R KS (t) function while the fifth term corresponds to convolution of four terms in the R charm (t) and the exponential function in the R KS (t) function. Four functions are folded into one function since the τ from the R charm (t) is much smaller than the τ s from the R KS (t) function and negligible. Fraction of the exponential term of the R KS (t) is also very small and dividing it into smaller fractions will not improve the resolution function. C. Smearing due to kinematic approximation t error due to kinematic approximation t B (true t) can be expressed as z B t true = z rec z 0 c(βγ) rec z asc z 0 c(βγ) asc, is studied in this section. t true where z rec and z asc are z vertex positions of reconstructed and associated B mesons, z 0 is z position of the B production point, (βγ) rec and (βγ) asc are βγ of reconstructed and associated B mesons. (βγ) rec and (βγ) asc can be written as (βγ) rec = (βγ) ΥE B + γ Υp B cos θ B m B (βγ) Υ ( + p B cos θ B ), ( E B ) m B (βγ) asc (βγ) Υ ( p B cos θ B ), = (βγ) Υ ( E B m B + p B cos θ B ), where EB, p B and θ B are the energy, momentum and angle from the beam direction for the reconstructed B meson in the CM frame. Using the above equations, t true is calculated as t true = z rec z 0 c(βγ) rec z asc z 0 c(βγ) asc, z rec z 0 ( p B cos θ B ) z asc z 0 ( + p B cos θ B ), = z B z rec + z asc 2z 0 = z B z B + 2(z asc z 0 ) p B cos θ B, p B cos θ B, = t B ( t B + 2 z asc z 0 ) p B cos θ B. 0

11 The second term in the above equation is the origin of the error in this approximation because z 0 cannot be detected. Since z asc z 0 is always positive and its distribution is exponential, the resolution function due to the kinematic approximation is parameterized as R k (t) = e t c k t B 2 c k τ B, 2 c k τ B (t c k t B < 0 for c k > 0, t c k t B > 0 for c k < 0), where c k = (p B cos θ B )/(β Υm B ). When the reconstructed B meson is fully reconstructed, c k can be calculated for each event. When the reconstructed B meson is partially reconstructed, cos θ B is unknown. Since cos θ B distribution follows cos 2 θ B, R k (t) is expressed for partially reconstructed events as R k (t) = d(cos θ) e t c k t B 2 c k τ B ( cos 2 θb 2 c k τ ), c k(cos θ) = p B cos θb B (t c k t B < 0 for c k > 0, t c k t B > 0 for c k < 0). Since this integration cannot be performed easily in a fit, this is approximated as R k (t) = dt ( f k τ τk = τ k 0 + τ k t, τk 2 = τ k 20 + τ k 2 t, σ k = σk 0 + σk t. e t f k τ + e t τ 2 τ2 ) e (t t ) 2 2π Fig 6 shows [ z gen / ] t gen distribution in the MC where z gen is true z B while t gen is true t B. 2σ 2 k,

12 0 3 t < ps < t < 3 ps < t [ z gen /c(βγ) ϒ ] t gen (ps) FIG. 6. [ z gen / ] t gen distribution in the MC for different t regions. REFERENCES [] M. Hazumi and T. Kawasaki, Belle-note 34. [2] T. Nakadaira, T. Higuchi, T. Tomura, H. Tajima and H. Aihara, Measurement of B 0 and B Meson Lifetimes, Belle-note