Background Fighting in Charm Less Two Body Analyses

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BA BAR Analysis Document #346, Version 3

Background ghting in CharmlessTwo-body analyses

J. Ocariz, M. Pivk, L. Roos

A. Hocker, H. Lacker, F. R. Le Diberder

July 22, 2002

Abstract

LPNHE, ParisLaboratoire de lAccelerateur Lineaire, Orsay


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Contents1 Introduction 3

2 Variable Denitions 3

2.1 The CLEO cones and the monomials . . . . . . . . . . . . . . . . . 32.2 Standard topological variables . . . . . . . . . . . . . . . . . . . . . 62.3 New idea: P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Flipped mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Kinematic variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5.1 Angular momentum conservation . . . . . . . . . . . . . . . . 82.5.2 Three Pt variables . . . . . . . . . . . . . . . . . . . . . . . . 82.5.3 Momentum of the fastest lepton . . . . . . . . . . . . . . . . 8

2.6 Super Fox-Wolfram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Variable selection criteria 93.1 Correlation between variables . . . . . . . . . . . . . . . . . . . . . . 103.2 Signal efficiency for xed background efficiency . . . . . . . . . . . . 113.3 Z -Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 MVA evaluation 124.1 Cones vs. monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Charged vs. Neutrals . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Global vs. roe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4 Constructing the best Fisher variable. . . . . . . . . . . . . . . . . . 154.5 Retrainning the cones and {L0 , L2} Fisher . . . . . . . . . . . . . . 174.6 Combining Tagging Categories . . . . . . . . . . . . . . . . . . . . . 194.7 Neural Network performance. . . . . . . . . . . . . . . . . . . . . . . 22

5 Toy Monte Carlo studies 235.1 Expected improvement on the branching ratio statistical error . . . . 245.2 Tagging Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Systematics 276.1 Mode dependence of the MVA . . . . . . . . . . . . . . . . . . . . . 276.2 Tagging Category dependence of the MVA . . . . . . . . . . . . . . . 306.3 Monte Carlo/data comparison from it Breco events . . . . . . . . . . 306.4 Offpeak/onpeak effect . . . . . . . . . . . . . . . . . . . . . . . . . . 326.5 p.d.f Fit defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7 Conclusions: OldFisher NewFisher 33

8 Acknowledgements 33

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1 IntroductionThis document presents a detailed study of background suppression in two-bodyanalyses. The dominant background in selecting B 0 + , B 0 K , B 0 K + K decays is the continuum e+ e qq events. Given the low branching ratiosof those decays and the large amount of qq events, background ghting is indeed amajor issue.

The CLEO experiment has taken advantage of the different topology betweenbackground and signal events to dene a Fisher discriminant based on the energyspatial distribution in the center of mass frame: the jetiness of qq events impliesthat the energy ow is pulled in the direction of the B candidate.

In this note we revisit the CLEO treatment in various ways. An approach basedon continuous variables is explored. Topological variables like thrust and sphericityangles may bring additionnal informations. In section 2, we present the usual andthe new variables that are tested. The selection or the rejection of a given variable isdecided according to several criteria such as its separation power or the correlation

with other variables. The criteria are discussed in section 3. The selected variablesare then combined using either a Fisher discriminant or a neural network. Section4 summarizes the performance of several combinations. Finally, the impact of thebackground suppression on branching ratios measurements and some systematiceffects are studied with toy Monte Carlo simulations and described in sections 5and 6.

2 Variable DenitionsMost of the variables described below use the fact that, in the (4 S ) rest frame,

the topological aspect of the qq events is different from the BB one. The two Bmesons are produced almost at rest in the center-of-mass frame, there is no preferreddirection by their decay products. The BB events are thus spherical . On the otherhand, the light quarks are produced with a signicant momentum and their decayproducts are contained in two more or less collimated back-to-back jets.

In the following denitions, the event tracks (a loose notation for charged tracksas well as neutral energy deposits) are often divided in two subsets: the B candidatedaughters (the B -of-event, or Boe) tracks on the one hand, and the rest-of-event (roe) tracks on the other hand. The charged and neutral particles in the roe aretaken from the GoodTracksAccLoose and GoodNeutralLooseAcc lists. Whennot explicitely notied, the following quantities are computed in the (4 S ) center-of-mass.

2.1 The CLEO cones and the monomials The CLEO cones , introduced by the CLEO collaboration [1] are 9 concentric,

mutually exclusive, cones centered around the Boe thrust axis, dividing half of the solid angle in 9 slices of 10 each. For each cone j , one denes the quantityC j :

C j =roe

i pi ji (|cosi |) (1)

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where pi is the momentum in the (4 S ) center-of-mass of the roe track i,|cosi | is the (positive) cosine of the angle between its momentum axis andthe Boe thrust axis, and ji (cosi ) equals to 1 if ( j 1) 10 < i j 10 ,and 0 otherwhile. The 9 C j are then combined into a single CLEO Cones-Fisher discriminant denoted F cones which reads:

F cones = c0 +9

j =1c j C j (2)

where the Fisher coefficients are given in Table 1

j 0 1 2 3 4c j 0.1957 0.4273 0.5154 0.5304 0.1725 j 5 6 7 8 9c j 0.0287 -0.0495 -0.1173 -0.2780 -0.2150

Table 1: Fisher coefficients for the 9 CLEO Cones [2]

Remark: Although the F cones variable is made of discrete angular slices of the solid angle, the F cones distribution of events appears smooth because themomentum weight spreads the contribution of each cone in a wide numericalrange. However, if one of the cone coefficient c j is zero, the event distribu-tion exhibits a delta function at c0. This is because events for which all theroe tracks fall into this cone will have F cones = c0. Most likely these are singletrack events. In practice, as exemplied by the value of c5 in Table 1, one conecoefficient gets very small but non-zero: as a result a weak momentum de-pendence remains and the event distribution exhibits a peak nearby c0. Sucha feature of the distribution will affect mostly background events since signalevents are less prone to have a single track roe. The effect is not large and itcan to be taken care of, if the tting functions used to describe the p.d.fs allowfor the presence of such a peak. This is illustrated on Fig.1.

The monomials, a set of momentum-weighted sums of the roe tracks akin tothe CLEO cones are dened as [4]:

L j =roe

i pi | cos(i )| j (3)

These variables were considered following the introduction by V. Shelkov [3]

of the second order Legendre polynomial sum

L2 =roe

i pi

12

(3cos2(i ) 1) (4)

which was shown to provide a very similar discriminant power as the 9 cones.There is no obvious reason why L2 should be the optimal continuous extensionof the F cones . Accordingly we examine the potential gain one may obtain byusing the sums

F {L j } = cst + j

l j L j (5)

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0

25

50

75

100

-2 -1 0 1 2

FCones Signal

0

25

50

75

100

-2 -1 0 1 2

FCones Bkg

Figure 1: Exemple of F cones distributions for signal events (left plot) and backgroundevents (right plot). One observes a clear peak located nearby F cones = c0 0.53 (see

text) overshooting over the double gaussian like distribution of background events.

The best set of L j is determined as follow: L0, which is the sum of the partic-ules momenta, is primarily selected because it constitues a shape independentoffset. The next step is to combine L0 with every other L j =0 in a linearcombination dened by the Fisher algorithm. The best signal/backgrounddiscrimination is obtained with the {L0, L2} pair. It was checked that addingfurther L j does not bring signicant extra information. Taking charged andneutrals tracks at once, the {L0, L2} Fisher F {L 0 ,L 2 } is:

F {L 0 ,L 2 } = 0 .5319 0.6023 L0 + 1 .2698 L2 (6)

Using the L j monomials or the Legendre polynomials as Fisher components isa matter of taste: they are linearly related the ones to the others. For example,Eq.(6) can be rewritten:

F {L 0 ,L 2 } = 0 .5319 0.1790 L 0 + 0 .8465 L 2 (7)

The reason why F {L 0 ,L 2 } brings a discriminant power nearly identical to theF cones one can be found in Fig. 2 where it is shown that the contributionof a given track to F cones and to F {L 0 ,L 2 } are very close, for all i values. Aslightly better agreement is obtained by adding L1 to the {L0, L2} pair. Thisis also shown on Fig. 2. As already stated above, the discriminant power is

increased only by a negligible amount by this renement which is thereforenot considered in the following. The improvement achieved by following moreclosely the F cones response is negligible because, although the c j coefficientsare optimized by the Fisher algorithm, the optima are not deeply pronounced:what appears graphically as a signicant change is in fact irrelevant, as far asthe discriminating power is concerned.The choice of a polynomial expansion, being in L j or L j , is more or less arbi-trary: a linear expansion is practical to optimize the (linear) Fisher variable.However, a similar result would be obtained using a (non linear) Gaussianparametrization of the c j ji (|cosi |) function.

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Figure 2: The contribution to F cones as a function of i , for a single track of momentum pi = 1 GeV (i.e c j ji (|cosi |) is indicated by the solid line histogram. The contribution toF {L 0 ,L 2 } of the same track is indicated by the dotted line. The third line which followsmore closely the histogram shows the result of a second-order polynomial t, includingthe linear term L1 absent in F {L 0 ,L 2 }.

Remark: The above remark on the departure from a gaussian shape of theF cones distributions applies also to F {L 0 ,L 2 } , but the effect is weaker. As

shown on Fig.3, the F {L 0 ,L 2 } background distribution cannot be tted perfectlywith a double-Gaussian: an anomaly shows up nearby F {L 0 ,L 2 } 0.52. It isdue to events with an unique track in roe: those for which the roe tracksatisfy | cos()2| 0.6023/ 1.2698 (i.e., 0.8) cluster nearby the Fisheroffset, because the F {L 0 ,L 2 } dependence on the momentum is reduced in thisparticular case.

2.2 Standard topological variablesThe following standard event shape variables are tested:

the thrust [5] computed on the whole event including the B candidate tracks( T hr )or on the roe ( T hr roe ).

cosT : is the cosine of the angle between the thrust axis of the B candidateand the thrust axis of the roe.

the sphericity [6] computed on the whole event including the B candidatetracks ( Sph ) or on the roe (Sph roe ).

cosS : is the cosine of the angle between the sphericity axis of the B candidateand the sphericity axis of the roe.

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0

20

40

60

80

-2 -1 0 1 2

F{L0L2} Signal

0

50

100

-2 -1 0 1 2

F{L0L2} Bkg

Figure 3: Exemple of F {L 0 ,L 2 } distributions for signal events (left plot) and backgroundevents (right plot). One observes a signicant anomaly located nearby F {L 0 ,L 2 } = 0 .5 as

an echo of the F cones peak nearby c0 (see text).

R l and R roel : are the ratio of the Fox-Wolfram moments H l /H 0 [7], computedrespectively on the whole event and on the roe.

2.3 New idea: P The present section is here to record an attempt: it can (should) be skipped by thereader at rst (second!) reading. The variable described here is quite involved, but,at the end, it does not fare better than the (much simpler) other variables, to the

great dismay of its proponent.The monomials dened on Eq.(3) are build upon a sum over the tracks: but a

sum is not obviously the right choice. For instance, the presence of even a singletrack with a momentum closely aligned to the B candidate thrust axis can be ahint for a qq background event. For a variable based upon a sum, this piece of information (if relevant) will be diluted by the presence of tracks away from thethrust axis. This will be less the case for a variable based upon a product over thetracks of 1 | cosi |: then, a single track with |cosi | 1 is capable of inducinga very low value for the product, and hence may trigger the identication of a qqevent.

The contribution of each track to the product can be regulated by a momentum-

dependent exponant f(p). In particular, f(p) should be such that low momentumtracks are practically removed from the product, while the contribution from stiff tracks is enhanced: this means f(p = 0) = 0.

However, the more tracks in roe, the more likely one of them is to satisfy |cosi | 1. Because of that, the product is to be taken only as an intermediate step: oneshould correct it afterwards to account for the number of tracks in the event.

Therefore, we dene the P product

P roe

i(1 | cosi |)f(p i ) (8)

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as an intermediate step toward the discriminating variable actually used, which isdenoted C P :

C P =roe

iP

1f(p i )

j = i

f(p i)f(p i) f(p j)

(9)

Because for signal events |cosi | is uniformly distributed between 0 and 1, it can beshown [8] that, irrespective of the f(p) function, the C P variable is itself uniformlydistributed for signal events, whereas it peaks at zero for background events. Thef(p) function should be adjusted to maximize the discriminant power of C P . A goodestimate for the optimal function is provided by

f(p) =1 2 | cos|1 | cos|

(10)

where the average of |cos| is to be taken over background events of momentum p.It can be taken as different for charged and neutral roe tracks.

2.4 Flipped massThe ipped mass is the invariant mass reconstructed with the ipped momenta withrespect to the B thrust axis [9, 10] of each particule of the whole event ( M flipped )or of the roe M roeflipped .

2.5 Kinematic variables2.5.1 Angular momentum conservation

Additionnal separation can be gained using angular momentum conservation.

| cos(P B , z )| is the cosine of the angle of the B candidate momentum withrespect to the z axis. In BB decays, it follows a sin 2(P B , z) distribution whileit is at in qq events.

| cos(T B , z)| is the cosine of the angle of the B candidate thrust axis withrespect to the z axis. Signal events have a uniform distribution and backgroundevents follow a 1 + cos2(T B , z ) shape.

| cos(S B , z)| is equivalent to |cos(T B , z)| but is dened with sphericity axis.

2.5.2 Three Pt variables

These are three scalar sums over the transverse momenta [9, 10]:

PtScal : computed with the whole event, wrt the event thrust axis

PtScal roe : computed with the roe, wrt the event thrust axis

PtBScal roe : computed with the roe, wrt the B thrust axis

2.5.3 Momentum of the fastest lepton

In order to take advantage of the hard spectrum of lepton momentum in B decays,one denes P fast as the momentum of the fastest lepton in the event. It is set tozero if no lepton is found.

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2.6 Super Fox-Wolfram

The Super Fox-Wolfram moments were introduced by the BELLE collaboration [12]and are dened as:

SF W =l=1 ,4

lRSOl +l=1 ,4

lROOl (11)

with

Rx,yl =H x,ylH 0

(12)

where the upperscripts S and O denote the B candidate tracks and the roe tracksrespectively. H x,yl follows the Fox-Wolfram moments denition, constructed withthe lth order Legendre polynomial P l :

H x,yl =i x,j y

| pi || p j | P l(cosij ) (13)

One can notice that ROOl is strictly equivalent to R roel as dened in 2.2. Thedenition of RSOl is close to a Legendre version of L l (cf. 2.1). Nevertheless, a majordifference is that the B candidate track momenta enter the denition of RSOl :

ROOl =roei= j | pi || p j | P l(cosij )

H 0= R roel (14)

RSOl = i S,j O | pi || p j | P l(cosij )H 0L l (15)

3 Variable selection criteriaThe selection of the MVA variables to be used in the likelihood analysis should bedone using as criteria the nal (statistical systematical) uncertainty it yields for themost cherish measurement we are aiming at. It must take into account the presenceof the other variables used in the analysis. However, for background ghting, it ishardly practical to proceed that way, because one must be using the whole analysis

chain, including massive toy monte-carlo simulations, for each variable (and com-binaison of variables) one is willing to consider. Therefore, simplied estimatorsshould be dened to probe quickly new ideas and rank the variables the ones withrespect to the others. These estimators are bound to be qualitative in nature andthey cannot by themselves asses the nal gain on the measurements. This is doneas a last step, when the choices boiled down enough to make a full study practical.

It is important to keep in mind that the signicative qualitative improvementsone may observe when using more and more involved MVA can very well dwindledown to a negligible when one is down to the full study. Thus, one very importantcriteria, to be applied at the very end, is simplicity!

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3.1 Correlation between variablesThe correlation between two variables x1 and x2 we are considering here is dened,in the usual way, as:

12 = x1x2 x1 x2

( x21 x1 2)( x22 x2 2)(16)

where the sum extends over all the signal or background events retained for thelikelihood analysis. Three comments are in order:

1. The correlation coefficients should be as small as possible when they are refer-ing to one MVA variable and one variable which p.d.f enters in the likelihood.This should be true both for signal events and for background events. This isbecause the likelihood analysis is based upon a product of p.d.fs, including theMVA p.d.f: the product of p.d.fs provides a correct likelihood only insofar thevariables are uncorrelated. But some residual correlations are unavoidable,and one must not seek for strictly zero correlation. Hence some know-how isrequired to judge how large the correlation can be tolerated without hurtingthe nal analysis. The current feeling is that 10% is a reasonable goal:obviously this should be quantied precisely when evaluating the systematicaluncertainties.

2. This correlation does not have to be small when it is refering to two MVA vari-ables. In that case, what matters is that the MVA distinguishes as best aspossible between signal and background. A pair of two correlated variablescan be more powerful than a pair of two uncorrelated variables, provided thesignal and background correlation coefficients are different enough. Because

of that, the MVA 1 ,MVA 2 values are to be considered with caution: they arequoted below only to give a feeling of how degenerate two MVA variables are.

3. The correlation coefficients are obtained from a sum over all the events en-tering the likelihood analysis. For a tight-cut (extended)likelihood analysiskeeping only a signal-box region, and where few background events remain,such a sum makes sense: the signal yield and its CP properties are trulyextracted from the whole sample of events retained. However, for a loose-cut (non-extended)likelihood analysis keeping large side-bands (i.e., a largebackground-box) with the signal-box, such a sum is misleading. This is be-cause a large correlation can arise from the background-box and hence beirrelevant (it would not affect the measurement): one may then be lead to

wrongly reject a variable on the ground that it appears too correlated withothers. Similarly a correlation can be small because it is small in the irrelevant,but highly populated, background-box, although it is large in the relevant, butpoorly populated, signal-box: one may then be lead to wrongly accept a vari-able on the ground that it appears uncorrelated with others, whereas in factit might bring a signicant systematical bias.

The x 1 ,x 2 correlation matrix is shown in g.4.

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Sph roe

R2 roe

cos(P,z)

cos(T,z)

L0L2

cones

S p

h r o e

R 2 r o e

c o s (

P , z

)

c o s (

T , z

)

L 0 L 2

c o n e s 0

0.10.20.30.40.50.60.70.80.91

100-84 12 18

-84100 -1 -10 -17

-1 100 -1 -1

-1 100-17 -16

12 -10 -1 -17100 92

18 -17 -16 92 100

Sph roe

R2 roe

cos(P,z)

cos(T,z)

L0L2

cones

S p

h r o e

R 2 r o e

c o s (

P , z

)

c o s (

T , z

)

L 0 L 2

c o n e s 0

0.10.20.30.40.50.60.70.80.91

100-85 1 -29-11

-85100 -3 3 29 10

1 -3 100 -8 2 5

3 -8 100-17 -23

-29 29 2 -17100 86

-11 10 5 -23 86 100

Figure 4: Correlation coefficient matrices between variables. The signal (background)event matrix is on the left (right).

3.2 Signal efficiency for xed background efficiencyAn analysis can aim for a signicant measurement only if the signal is not drownunder a huge background. For a loose-cut (non-extended)likelihood analysis, thesample of events used in the likelihood is mostly populated with background. De-noting f the fraction of signal in the sample, a good qualitative indicator of thepower of a MVA is given by the signal selection efficiency one is left with when ap-plying a cut on the MVA such that the background selection efficiency is set equal tof, and hence bring down the background to a level comparable with the signal level.In the h + h case f 5% and we thus decided to quote as a qualitative indicatorthe signal efficiency obtained for a background efficiency xed at 5%.

3.3 Z -TransformThis is a two-step change of variable introduced [13] to make easier the choicebetween the different variables by standardizing the shapes of MVA distributionsand to simplify the t of the p.d.f used in the likelihood. No gain of information isobtained by using the Z -Transform , but it is a convenient tool.

If x is the discriminating variable, B (x) and S (x) the distributions for back-ground and signal respectively, we dene the intermediate variable y, and then theZ -variable as follows:

x y =B (x)

B (x) + S (x) Z =

y

0b(y )dy (17)

where b(y ) is the background distribution of the y variable. Stated differently, Z isthe background selection efficiency for y < y .

Introducing the y variable, one removes the shape arbitrariness inherent in allMVA distributions. Under a one-to-one change of variable x x the signaland background distributions S (x) and B (x) transform into S (x ) = S (x)x and

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B (x ) = B (x)x, where the common function x is the derivative of x with respectto x . Being common to both signal and background transformed-distribution, xcan be omitted since it drops out of a likelihood analysis: one is left with the initialdistributions, which demonstrates that the one-to-one change of variable is arbitrary.As a result, one can dramatically change the aspect of a MVA distribution, withoutaffecting by any means its discriminating power: stated differently, the shapes of the MVA distributions may lead to misleading impressions. Taking Fisher as anexemple, a unique set of variables can lead to widely different MVA distributions:changing the constant offset or the overall scale factor are only two examples of harmless changes of variable. However, the y variable is invariant under x xchanges of variable. The fact that x y is itself a harmless change of variablecan be seen readily. Since one can multiply at will the likelihood expression by anevent-dependent factor, if one chooses B (x) + S (x) for this factor, one makes thelikelihood depends only on y. Therefore the y variable contains all the informationneeded to ght against background.

Introducing the Z variable, one removes the shape arbitrariness inherent inthe y distributions. One uses a one-to-one change of variable y y to forcethe background distribution to take a standardized shape. By construction, withy Z , the background distribution B(Z ) is uniformly distributed between zero andone, while the signal distribution S (Z ) peaks at Z = 0: in effect, the latter canbe expressed as S (Z ) = ( 1 y)y 1. The background distributions being identicalfor all MVA , when comparing different MVA, it is only the signal distributionS (Z ), and more precisely its peaking nearby the origin which matters. The mostdiscriminating the variable is, the higher the peak of S (Z ) at zero is.

The software implementation of the Z -Transform is very simple (it is a matterof a dozen lines of code). Figure 5 shows an example of the two-step change of variable.

4 MVA evaluation

A Fisher algorithm is used to combine the discriminant variables. Variables are keptor rejected according to the criteria described in 3. The training of the algorithm isperformed on a signal SP4 Monte Carlo K sample and onpeak mES side-band dataof the Winter 2002 sample (5 .20 < m ES < 5.26GeV/c 2 and | E | < . 150GeV/c )data. The standard selection described in [19] is applied. One is left with 8007signal events and 9677 background events. 4003 events of each category are usedto train the Fisher or the neural network alogorithm. The performance and theFisher distributions are extracted from the remaining events (4004 signal and 5674background). The coefficients of the Fisher combinations of the cones and {L0, L2}given in section 2.1 are recomputed with these samples in order to have a faircomparison between the sets of variables tested in the following. In subsection 4.5,it is shown that this reoptimization does not change signicantly the results.

4.1 Cones vs. monomialsAs mentionned in section 2.1, it is established that the best combination of mono-mials is {L0 , L2}. Table 2 summarizes the performances of {L0, L2} and the CLEO

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0

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0.02

0.03

0.04

0.05

-2 0 2 4

x

0

0.01

0.02

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0.04

0.05

0 0.5 1

s(y)

b(y)

y

10-1

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S(Z)

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Z

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0 0.5 1

s(y)

b(y)

y

10-1

1

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0 0.5 1

S(Z)

B(Z)

Z

Figure 5: The upper plots represent both signal and background x, y, Z distributions of CLEO Cones-Fisher . The bottom plots correspond to the CLEO-Fisher (11 variables:cones + 2 kinematics). The dashed line in the bottom right-hand-side plot reproduces theabove signal Z -distribution of CLEO Cones-Fisher : one observes that CLEO-Fisher is

slightly more discriminating than the CLEO Cones-Fisher .

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which are found to be very similar. This is expected from the fact that the samephysics information is used in both cases. {L0, L2} is preferred since it is continuousand involves only two variables.

variables (S )@ (B) = 5% < Z >CLEO cones 0.338 0.007 0.213 0.004

{L0, L2} 0.344 0.007 0.213 0.004

Table 2: Background efficiency at signal efficiciency of 5% and < Z > for the CLEO cones and {L0, L2}.

4.2 Charged vs. NeutralsIn the B 0 a0 analysis [4], it is shown that distinguishing neutrals and chargedobjects in the evaluation of the cones or the monomials brings a signicant improve-ment in the separation power of the variable.

The study is repeated in the B 0 h+ h analysis. One calculates neutral cones with only the neutral particles of the roe and charged cones taking into ac-count only the charged tracks of the roe. The performance of the 18 variables{cones neut , cones ch } are given in table 3. The comparison with the results of ta-ble 2 and gure 6 show that no signicant improvement is observed in the sig-nal/background separation.

variables (S )@ (B) = 5% < Z >{conesneut , cones ch } 0.341 0.007 0.212 0.004

Table 3: Background efficiency at signal efficiciency of 5% and < Z > average for the 18 variables {conesneut , cones ch }.

In the same way, neutral monomials Lneut j and charged monomials Lch0 are com-puted. It is found that the best L j combination remains {L0, L2} for both thecharged and the neutral cases. As for the cones, {Lneut0 , Lneut2 , L ch0 , L ch2 } does notshow a better separation power than {L0, L2}.

Nick Danielson worked as well on the charged/neutral separation [14] and reachedthe same conclusion.

4.3 Global vs. roeThere is a number of discriminant variables for which the B candidate tracks enterthe denition: the thrust T hr , the sphericity Sph , the ratio of the Fox-Wolframmoments R l , the ipped mass M flipped , PtScal and nally the Super Fox-Wolfram moments. They exhibit two major disavantages:

the validation of the Monte Carlo signal distribution with the fully recon-structed hadronic B sample [16] is not straightforward,

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Figure 6: The left plot shows the Z distribution for the cones computed globally (full) andcomputed with neutrals and charged separetely(dashed). The right one is the backgroundefficiency vs the signal efficiency for the same variables.

because the momenta of the B candidate tracks enters the evaluation of thesevariables, they are correlated with the B candidate substituted mass and E .Figure 7 shows the correlation in the case of R2 and Sph . All the variableslisted above are then rejected from the nal choice. For the same reason, P fastis also rejected.

We use then the above shape variables, but computed on the roe only. They aredenoted below R roe2 , Sph roe , T hr roe ...

4.4 Constructing the best Fisher variable.Starting from {L0 , L2}, we try to improve the signal/background separation byadding other variables:

At a previous stage of our study, combining R roe2 and Sph roe with {L0, L2} didbring some improvement (see for example version #2 of this BAD). With thepresent event selection, the improvement is now negligible: the signal efficiencyat 5% background efficiency goes from 34 .4 0.7% to 35.0 0.7%.

Adding higher order of roe Fox-Wolfram moments ratio does not help.

T hr roe carries the same information as Sph roe . It could have been choosen aswell.

cosS is a very powerful discriminating variable. A cut at |cosS | < 0.8 rejectssignicantly the background (70% rejected) with 80% efficiency on the signal.There is no gain to apply a softer cut and use cosS in the Fisher, while acut at 0 .8 highly reduces the number of events entering the nal t. As cosT carries the same information, it is not used either.

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0.1

0.2

0.3

0.40.5

0.6

0.7

0.8

0.9

1

5.1 5.15 5.2 5.25 5.30.1

0.2

0.3

0.40.5

0.6

0.7

0.8

0.9

1

-0.4 -0.2 0 0.2 0.4

0

0.1

0.2

0.3

0.4

0.5

5.1 5.15 5.2 5.25 5.30

0.1

0.2

0.3

0.4

0.5

-0.4 -0.2 0 0.2 0.4

Figure 7: R2 (top) and sphericity Sph (bottom) as a function of the B candidate substi-tuted mass (left) and E (right).

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The roe transverse momenta PtScal roe and PtBScal roe are correlated to 86%and 90% resp. to {L0 , L2} and do not improve the separation. The sameconclusion is reached with M roeflipped .

Not surprisingly, the kinematic variables |cos(P B , z )| and |cos(T B , z )| (or

|cos(S B , z)|) do improve: the efficiency increases from 37 .8 0.1% for {L0, L2 , Rroe2 ,Sph

roe}to 40.5 0.1% for {L0 , L2, R roe2 ,Sph roe , |cos(P B , z )|, |cos(T B , z )|} . Here again,

the choice between |cos(T B , z )| and |cos(S B , z)| is arbitrary.

In summary, three sets of variables are retained [11]:

the simplest set: {L0, L2},

a basic set of 4 topological variables: base4 = {L0, L2 , R roe2 ,Sph roe },

the most powerful set: var 6 = {L0, L2, R roe2 ,Sph roe , |cos(P B , z)|, |cos(T B , z )|}.Given that the kinematic variables |cos(P B , z )| and |cos(T B , z)| have zero cor-relation with base4, they could be used in the nal likelihood t as well. Inany case, combining them with a Fisher algorithm allows to evaluate the gainfrom these variables in a very simple way.

In table 4 and in the following, these three sets are compared to the cones. Theseresults can be seen in g. 8.


{L0, L2} 0.344 0.007 0.213 0.004base4 0.350 0.007 0.213 0.004var 6 0.371 0.008 0.201 0.004

Table 4: Powers of four Fisher combinations.

4.5 Retrainning the cones and {L0, L2} FisherThe coefficients given in section 2 are the currently last version of NCTwoBodyAnal(see for example NCTwoBodyAnal V00-13-00) used to produce version 13 of thetwo-body ntuples. The cone coefficients were estimated in the early charmless two-

body analysis [17] and remain unchanged. The {L0, L2} coefficients were computedfrom MC sample and offpeak data passing the following cuts: R2 < 0.95, Sph > 0.01and |cosS | < 0.08.

Although the performances are similar (see table 5), for the sake of consistency,it is recommended to switch to the {L0 , L2} Fisher coefficients as calculated in thissection in the future:

F {L 0 ,L 2 } = 0.0015 0.5417 L0 + 1 .4930 L2 (18)

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Figure 8: The left plot shows the Z distribution for the cones, {L0, L2}, base4 and var6.The right one is the background efficiency vs the signal efficiency for the same set of variables.


{L0, L2} 0.335 0.007 0.228 0.004

Table 5: Background efficiency at signal efficiciency of 5% and < Z > for the CLEO cones and {L0, L2} as dened in section 2.

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4.6 Combining Tagging CategoriesTagging being performed using information from the roe, signal-to-background dis-crimination varies signicantly among different tagging samples. Figure 9 shows them es distribution, both for untagged and tagged hh candidates, and Figure 10 shows

the latter splitted according to their category. Table 6 summarises the signal andbackground efficiencies. In particular, candidates tagged via the lepton categoryare mostly pure signal events, while non tagged candidates have the smallest S/Bratio. This fact speaks in favour of a tagging-dependent analysis.

The expected gain from this tagging-dependent analysis can be evaluated ana-lytically. The statistical error (N S ) on the yield, measured from a n-dimensionalML t is

(N S ) = N s< s 2f > (19)where the separation < s 2f > is dened as

< s 2f > = dn x S 2(x)

fS (x) + (1 f )B (x)(20)

with the integration performed on the n variables x1,...x n used on the t, and S (x),B (x) being the n-dimensional PDFs for the signal and background distributions. If several independent ts are performed on uncorrelated samples, the combined yieldhas an error given by

(N S ) = 1i Qi (21)Qi =

2i < s

2f i >

N i(22)

Where the index i refers to each tagging category, i , N i and < s 2f i > the signalefficiency, signal yield and signal-to-background separation within the category,

In this study, we evaluate the quality factors Q i by using an integration overthree discriminant variables: m es , E and F {L 0 ,L 2 }. The signal PDFs are extractedfrom signal + Montecarlo, and background PDFs from onpeak sidebands: 5 .2 0.15 for m es . The PDFsare evaluated both globally, and independenlty for each tagging category, except inthe case of the background lepton category, for which the kaon distribution PDF

is also used to estimate the uncertainty coming from the poor statistics on thebackground lepton tag sample. Table 7 records the functions used to parametrisethe PDFs.

The results of the analytical computation, done using the + signal fractionsobtained in the Moriond 02 analysis, are summarised in table 8. Adding the factorsfor the Lepton+Kaon+NT1+NT2+NoTag categories, we obtain a combined qualityfactor almost 7% larger than the equivalent for a single analysis, thus implying a

3.2% improvement on the N yield precision.Equation 21 shows a three-fold dependence of the expected gain from a combined

analysis:

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Figure 9: Left (right) plot: mes distribution for untagged (tagged) hh candidates.

Figure 10: mes distributions separated by tagging category.

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Variable Signal PDF Background PDFmes Argus Gaussian E Parabola Gaussian

F {L 0 ,L 2 } Bifurcated Gaussian Double Gaussian

Table 7: Used signal and background PDF parametrisations.

sub-sample Q(1) Q(2)

Lepton Tag 0 .1040417 0.1058392(with Kaon PDFs) 0 .1061336

Kaon Tag 0.1978656 0.1961083NT1 Tag 0.0447742 0.0407896NT2 Tag 0.0774646 0.0729409No Tag 0.1211795 0.1301118

Combined 0.5453256 0.5457898Tag-independent 0 .5102391

Table 8: Quality factors. Q(1) quality factors are obtained using category-dependent PDFs; Q(2) estimation uses the single, average PDF for each variable.

The signal fraction f . Figure 12 shows the expected improvement as a functionof f , up to 10%. The computation was performed with the same PDFs as forthe tablated in Table 8. We see that our method brings a sensible improvementfor the case of channels with very low S/B , and is reduced to 1% for K + case (f 0.065).

The differences in the distributions among different categories. Figure 11 showsthe relevant mes , E and F {L 0 ,L 2 } background PDFs splitted by tagging cate-bory. Shape variations are globally marginal, except for the lepton and NT1categories, which are anyway limited by statistics. Table 8 compares the qual-ity factors obtained either by using these category-dependent PDFs or a singleset of PDFs. We conclude that all numbers remain about the same, we seethat, except for lepton tags, the values of quality factors are mostly unsensi-tive to variations in the shapes among categories. As a further cross-check,we have also used the set of PDFs obtained from the Kaon category for thelepton category. Again, the result is essentially unchanged.

The variations of relative signal and background efficiencies among categories,which is the dominant effect. This is a remarkable fact: the expected gain willnot be diluted by the limited knowledge of the individual distributions, evenfor those categories suffering from poor statistics.

4.7 Neural Network performance.A way to take advantage of non linear correlation between variables is to use a NeuralNetwork. We use a multi-layer perceptron neural network [20]. The architecture for

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{L0, L2} 0.344 0.007 0.216 0.004base4 0.353 0.007 0.194 0.004var 6 0.411 0.008 0.171 0.004

Table 10: Powers of four Neural Network outputs.

are changed. The number of events, signal and background, of each specie as well asthe K asymmetries are tted. Also left free in the t are the following parametersfor background PDFs : the shape parameter of the ARGUS substituted massdistribution, the two highest order coefficients of the quadratic parameterization of the E spectrum and the ve parameters of the double Gaussian describing Fisher.

5.1 Expected improvement on the branching ratio sta-tistical errorFor each of the four Fishers (CLEO cones, {L0, L2}, base4 and var6 ), the distri-butions of background and signal are extracted from the onpeak mES side-banddata sample (5 .20 < m ES < 5.26GeV/c 2 and | E | < . 150GeV/c ) and signal MonteCarlo respectively. Standard selection cuts [19] are applied. The signal distributionsare tted with a bifurcated Gaussian, the background ones with a double Gaussian.The same PDFs are used to generate and t the data sample.

variables signal background L R A1 1 1 2 2

CLEO cones 0.277 0.604 0.551 0.619 0.429 0.648 0.208 0.402{L0, L2} 0.181 0.671 0.492 0.762 0.387 0.608 0.206 0.309

base4 0.071 0.777 0.456 0.477 0.382 0.393 0.343 0.648var 6 0.186 0.783 0.535 0.562 0.428 0.470 0.398 0.720

Table 11: Signal and background PDFs parameters for the four Fisher combinations.

Figure 14 shows the statistical error on the tted number of and K sig-

nal events for the four Fishers. The improvement on the error distribution mean,relative to the CLEO cones, is given in table 12. {L0, L2} and base4 show similarperformances. For var6 , a slight improvement is observed, larger in the channeldue to a higher background level. A maximum 2 .3% improvement is reached withvar6 .

5.2 Tagging CategoriesThe aim of this study is to validate the analytical calculation presented in 4.6.Four independent toy MC sets of 250 experiments are performed, with signal and

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0

50

100

150

-2 0 2

SIGNAL CONES

0

100

200

-2 0 2

BKG CONES

0

50

100

150

200

-2 0 2

SIGNAL L0L2

0

100

200

300

-2 0 2

BKG L0L2

0

50

100

150

-2 0 2

SIGNAL L0L2-Sph-R2

0

100

200

300

-2 0 2

BKG L0L2-Sph-R2

0

50

100

150

-2 0 2

SIGNAL L0L2-Sph-R2-cos P,z -cos T,z

0

100

200

-2 0 2

BKG L0L2-Sph-R2-cos P,z -cos T,z

Figure 13: Fisher distributions for the cones, {L0, L2}, base4 and var6 . The bifurcated(double) Gaussian ts are superimposed on signal (background) distributions.

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(N ) CONES

0

25

50

75

100

14 16 18 20

10.95 / 8Constant 98.35 5.571

Mean 16.45 0.3405E-01Sigma 0.6986 0.2521E-01

(N K ) CONES

0

50

100

22 24 26 28

6.938 / 7 Constant 117.1 6.614

Mean 24.14 0.2709E-01Sigma 0.5895 0.1998E-01

(N ) L0L2

0

25

50

75

100

14 16 18 20

13.77 / 8Constant 99.40 5.793

Mean 16.41 0.3359E-01Sigma 0.6864 0.2638E-01

(N K ) L0L2

0

50

100

22 24 26 28

6.973 / 8Constant 120.1 6.769

Mean 24.14 0.2605E-01Sigma 0.5736 0.1924E-01

(N ) L0L2-Sph-R2

0

25

50

75

100

14 16 18 20

10.16 / 10Constant 101.6 5.660

Mean 16.47 0.3074E-01Sigma 0.6735 0.2192E-01

(N K ) L0L2-Sph-R2

0

50

100

22 24 26 28

10.01 / 9Constant 114.0 6.368

Mean 24.18 0.2872E-01Sigma 0.6012 0.2047E-01

(N ) L0L2-Sph-R2-cos P,z -cos T,z

0

25

50

75

100

14 16 18 20

12.60 / 9Constant 99.08 5.575

Mean 16.08 0.3219E-01Sigma 0.6868 0.2334E-01

(N K ) L0L2-Sph-R2-cos P,z -cos T,z

0

50

100

22 24 26 28

3.388 / 9Constant 114.6 6.320

Mean 23.93 0.2802E-01Sigma 0.6043 0.1976E-01

Figure 14: Statistical error on the tted number of (left) and K (right) signal events for the four Fishers. From top to bottom : CLEO cones, {L0 , L2},base4 and var6

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{L0, L2} base4 var6 (N ) 0.2% 0.1% 2.3%(N K ) 0.0% 0.2% 0.9%

Table 12: Relative difference of the error distribution mean with respect to the CLEO cones performance. The uncertainty is 0.3%.

background yields corresponding to the lepton tag category, the kaon tag cat-egory, NT1+NT2+noTag categories and all categories merged, respectively. Only events are generated. All t parameters are xed but the signal and background yields. The K and KK yields are xed to zero. The same PDFs are used inall four sets. The {L0, L2} sher is used.

Table 13 shows the mean of the generated signal and background yields in each of the four sets of experiments, the average tted yields and their statistical error. Thequadratic sum of the uncertainties on the lepton, kaon and NT1+NT2+noTag signalyields is 15.45 0.05, to be compared with 15 .59 0.04 when tting all categoriesat once. The conclusion of this study is that, for the + signal-to-backgroundratio, the statistical uncertainty on yields is improved by about 1%. As this numberis smaller than obtained in section 4.6, the same exercise was repeated for fourdifferent values of the signal fraction, and is summarised on Figure 15. Alreadyfor a S/B ratio just below the + value, the gain in statistical precision exceedsseveral percent. We conclude that the realisitic gain, while slightly diluted withrespect to the analytical expectation, is very substantial for channels with signalfractions just below the + case.

leptons kaons NT1+NT2+noTag Allgenerated N sig 15.5 43.4 65.4 124.3generated N bkg 100.6 2116.9 5856.9 8074.4

tted N sig 15.2 0.3 42.9 0.6 65.4 0.9 124.0 0.9(N sig ) 4.23 0.03 8.73 0.04 12.04 0.03 15.59 0.04

Table 13: Mean of the generated signal and background yields, average tted yields and their statistical uncertainty, for the lepton, kaon, NT1+NT2+noTag category and all categories merged.

6 Systematics

6.1 Mode dependence of the MVAA simple selector based on the DIRC measurement is applied on both tracks of the Bcandidate to split the onpeak side-band data sample in three sub-samples : , K ,KK . In order to get pure subsamples, strong requirements are applied : N > 10and a track is identied as a pion (resp. a kaon) if ( C

)2 ( C

K

)2 > 10 (resp.

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Improvement ( % ) on Statistical Error vs. S/B ratio

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Figure 15: Toy MC estimation of the improvement in the statistical error, as functionof the S/B ratio, by combining tagging-dependent ts. The Moriond S/B fraction was

0.0223.

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( C K

)2 ( C

)2 > 10). 21% of events are classied as , 15% as K and 9%

as KK , while 55% are rejected.A two Gaussian function is used to t the distribution. The K and KK

distributions do not show a strong asymmetric shape at the level of the availablestatistics and are tted with a single Gaussian. The resulting mode dependent PDFsare shown in Figure 16 together with the common background and signal PDFsdescribed in section 5.1.

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-2 0 2

CONES

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-2 0 2

L0L2

-0.04

-0.03

-0.02

-0.01

0

0.010.02

0.03

0.04

-2 0 2

L0L2-Sph-R2

-0.04

-0.03

-0.02

-0.01

0

0.010.02

0.03

0.04

-2 0 2

L0L2-Sph-R2-cos(P,z)-cos(T,z)

Figure 16: Difference of the mode dependent background PDFs with the global PDFs given in table 11. All background PDFs have been normalized to one. In order to show the signal region, the signal PDF is displayed with an arbitrary normalization.

The t is performed in its standard way, i.e. with one background Fisher PDFcommon to all modes, the parameters of the double Gaussian being free.

In histograms of Figure 17, the difference of the t results with the mean of the

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generated Poisson distributions ( N = 124 .3 and N K = 402 .7, DIRC acceptancecorrected yields) is plotted. Table 14 summarizes the average difference on thenumber of and K events. No signicant biaises are observed in the case of the cones, L0L2 and var 6. A 3.3 effect is seen on the yield with the base4Fisher. However, the high purity selector used to split the background sample leadsto the relatively low statistics mode sub-samples and statistical uctuations on thedouble Gaussian of Gaussian parameters are large. One has to wait for more datato conclude on the mode dependance implications on tted yiels.

CLEO cones {L0, L2} base4 var6 N 124.3 0.6 0.8 0.1 0.8 2.6 0.8 0.4 0.7N K 402.7 0.4 1.1 1.0 1.1 0.1 1.1 1.0 1.2

Table 14: Average bias on the tted numbers of events.

6.2 Tagging Category dependence of the MVAAs seen in Fig. 11, the Fisher PDFs vary signicantly among tagging categories.In the standard conguration t, global PDFs, extracted from the distribution onall events of all categories, are used for both signal and background. In orderto evaluate a possible bias due to this approximation, 250 toy experiments wereperformed. The events are generated according to their Elba tagging category PDFand the t is performed in its standard way. Table 15 shows the average bias onthe tted number of and K signal events. Two Fisher combinations are tested: {L0 , L2} and the CLEO cones. The results are similar: one observes a small bias

on N (0.7 to 1.6 ) and a signicant bias on N K (2.4 to 2.8 ).Note that this study was perfomed with the Elba tagger. Difference withintagging category difference seems to be less important with the new Moriond tagger[15].

CLEO cones {L0, L2}N 124.3 0.7 1.0 1.8 1.1N K 402.7 4.3 1.6 3.9 1.6

Table 15: Average bias on the tted numbers of events.

6.3 Monte Carlo/data comparison from it Breco eventsThis section still needs to be updated.

This study relies on the work described in [16]. By comparing the Monte Carloand real data in open charm fully reconstructed neutral B decays, one can extracta linear correction to the signal Fisher distribution from the B 0 K + MonteCarlo.

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0

25

50

75

100

-100 -50 0 50 100

6.052 / 7 Constant 114.8 6.511

Mean -0.6115 0.7877 Sigma 17.21 0.6065

(N( )-124.3) CONES

0

50

100

-100 0 100

7.888 / 7 Constant 120.1 6.616

Mean 0.4081 1.140Sigma 24.56 0.7822

(N(K )-402.7) CONES

0

50

100

-100 -50 0 50 100

3.109 / 6 Constant 118.2 6.475

Mean 0.1270 0.7969

Sigma 16.94 0.5696

(N( )-124.3) L0L2

0

50

100

-100 0 100

7.790 / 9Constant 120.8 6.492

Mean 1.049 1.115

Sigma 24.41 0.7136

(N(K )-402.7) L0L2

0

50

100

-100 -50 0 50 100

1.828 / 7 Constant 119.2 6.587

Mean 2.553 0.7707 Sigma 16.76 0.5614

(N( )-124.3) L0L2-Sph-R2

0

50

100

-100 0 100

9.505 / 9Constant 121.4 6.528

Mean -0.8222E-01 1.100Sigma 24.21 0.7039

(N(K )-402.7) L0L2-Sph-R2

0

50

100

-100 -50 0 50 100

3.019 / 7 Constant 127.6 7.036

Mean 0.3630 0.7050Sigma 15.56 0.5005

(N( )-124.3) L0L2-Sph-R2-cos P,z -cos T,z

0

50

100

-100 0 100

2.309 / 7 Constant 117.5 6.835

Mean 0.9716 1.195Sigma 25.58 1.011

(N(K )-402.7) L0L2-Sph-R2-cos P,z -cos T,z

Figure 17: Difference of the tted numbers of events with the mean of the gener-

ated Poisson distributions. Left: N . Right: N K . From top to bottom : CLEOcones,{L0, L2},base4 and var6

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The possible systematic effect of neglecting such a correction is studied here inthe same way as in 6.1: the linearly corrected signal PDF is used to generate theevents. The t is then performed twice:

with the corrected PDF

with the raw PDF extracted from Monte CarloTable 16 gives the average difference on the tted number of and K events.

Similar behaviour is observed with all Fishers. The kinematical variables cos(Bmom )and cos(BT ) seem to introduce a slightly larger effect.

CLEO cones {L0, L2} base4 var6 N raw N corr 1.52 .xx 1.20 .xx 1.84 .xx .xx .xxN rawK N corrK 2.78 .xx 2.16 .xx 3.39 .xx .xx .xx

Table 16: Average difference of the numbers of events tted with the MC raw PDF and the corrected PDF.

6.4 Offpeak/onpeak effectThis section still needs to be updated.

The systematic uncertainty due to the limited statistical accuracy of the back-ground Fisher parametrization can be evaluated by extracting the PDF on twoindependent data sets. The E and the mES side-bands in the onpeak data canbe used as an alternative to the offpeak data. The toy MC proceeds as usual: the

events are generated following the PDF extracted from offpeak data and the t isperformed with:

the offpeak data PDF

the side-band onpeak data PDF, dened either by | E | > 100 MeV or5.20 GeV/c 2 < m ES < 5.27 GeV/c 2 .

The systematic uncertainty is given by the average difference between the num-bers tted with the offpeak PDF and the numbers tted with the onpeak PDF.Results are summarized in table 17.

CLEO cones {L0, L2} base4 var6

N of f N SB ( E ) 1.2 .xx 1.3 .xx 2.1 .xx .xx .xxN of f K N

SB ( E )K 1.2 .xx 0.6 .xx 0.9 .xx .xx .xx

N off N SB (m ES ) 0.7 .xx 0.5 .xx 0.9 .xx .xx .xxN off K N

SB (m ES )K 0.9 .xx 0.6 .xx 0.7 .xx .xx .xx

Table 17: Average difference of the numbers of events tted with the MC offpeak data PDF and the onpeak data PDF from | E | and mES side-bands.

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6.5 p.d.f Fit defects

7 Conclusions: OldFisher NewFisher

(S ) at (B) = 5% Stat Mode BReco Off-OnCLEO cones:

{L0, L2} + + +base4 + + - -var6 ++ ++ + - -

Table 18: Summary.

8 AcknowledgementsWe wish to thank Ran Liu and Jinwei Wu for the help they provided us in in-corporating the discriminant variables introduced by the Wisconsin group into thestudy presented here. Vasia Shelkov contributions to background ghting have beeninstrumental to our study: we beneted a lot from his constant smiling advices. Fi-nally we would like to extend our warm thanks to Jim Olsen who encouraged us toundertake the dreadful attempt to do better than CLEO did: although we did notsucceed to the extent we hoped for, it was fun and we learned a lot in the process.

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Documents

Background Fighting in Charm Less Two Body Analyses