July, 2006 Selected Topics on Open Charm Physics from CLEO-c 1
Selected Topics on Open Charm Physics at CLEO-c
Batbold SanghiPurdue University
(CLEO Collaboration)
K-
-
e+
K+
(3770)D0 D0
D0K+-, D0K-e+ Main topics:
Overview the CLEO-c experiment and its physics program
Absolute Hadronic D0 and D+ Branching Fractions
Preliminary Results for Absolute Branching Fractions and Form Factor Measurements in and eKD /0 eKD 00
/
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 2
CLEO-c and the CKM Matrix
Recent status
Status withtheory errors reduced)
The CKM matrix provides the only mechanism for CP violation in the SM
An important goal of flavor physics is to measure and (over)constrain the parameters in the CKM matrix (4 parameters) to test the SM
Non-perturbative hadronic effects limit our ability to extract fundamental parameters from experimental measurements
CLEO-c provides unique measurements in the charm sector that test theory and help reduce hadronic uncertainties
CLEO-c tested theory can then be applied to B decay processes to extractCKM matrix elements (especially Vub and Vtd)
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 3
An example of a test of Lattice QCD
Measure form factors in D l at CLEO-c
Theories of Strong Interactions (LQCD)
validate LQCD calculations for form factors
use LQCD to extract Vub from Bl
222
2)(~
)(qfV
dq
lDdcd
222
2)(~
)(qfV
dq
lBdub
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 4
An example of a test of Lattice QCD
m is well measured
But |Vtd| from m has a large uncertainties from fB
measure fD in D l
validate theoretical calculations
fB ~fB (LQCD)/ fD (LQCD) fD
Use fB to extract |Vtd|
Theories of Strong Interactions (LQCD)
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 5
CLEO-c impact
CLEO-c + Lattice QCD +B factories
CLEO-c CLEO-c + Lattice QCD +B factories + ppbar
Vub/Vub 15%lB
lD
Vcd/Vcd 7%lD
Vcs/Vcs =16%l
B D
Vcb/Vcb 5%
Bd Bd
Vtd/Vtd =36%
Bs Bs
Vts/Vts 39% Vtb/Vtb 29%
Vus/Vus =1%
l
Vud/Vud 0.1%ep
n
tb
W
D hadronic branching fractions(Analysis by Cornell, Purdue and CMU)Including: and
D semileptonic B’s and form factors in
(Analysis by Purdue and SMU, my main thesis topic)
I will focus on two CLEO-c analyses that have impact on Vcd, Vcs, Vub and Vcd:
D and Ds leptonic branching fraction
eKD / KD0 KD
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 6
The CLEO-c detector
Over 20 years of data taking at
Y(4S)
An event taken at (4S)
KDKD
DDee
,
,)3770(
The main components of the CLEO-c detector were developed for B physics at the Y(4S).
Minor modifications Replaced silicon with 6 layer inner drift chamber B field 1.5 T 1.0 T
P/P = 0.6% at 1GeV
E/E = 2% at 1GeV
= 5% at 100MeV
Excelent electron and hadron ID
Advantages at (3770) Pure DD, no additional particle Low multiplicity High tagging efficiency
)3770(
Started data taking in the fall of 2003
Note
: Log
S
cale
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 7
Three generations of CLEO-c analyses at the (3770):
Oct-03 through Jan-04: Luminosity = 56 pb 1
all results are published(D hadronic branching fraction)
Sep-04 through Apr-05: Luminosity = 225 pb1
most analyses are on-going(D semileptonic B’s and form factors)
Future running: projected total Luminosity = 750 pb1
CLEO-c is also collecting data above the DsDsbar
productionthreshold (goal 750 pb1) and lower energies at the (2S).
CLEO-c data samples
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 8
Absolute Hadronic D0 and D+ Branching Fractions
Introduction and Overview of the Analysis Measurements of Absolute Hadronic Branching Fractions Summary
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 9
Overview of Technique
ijjiDDij BBND 2
2
42 i
ii
ii
iDD
ji
ij
ij
jiDD D
SNor
D
SSN
e+ e
0D
0D
K+
+
K-
iiDDi BNS 2
XSingle tagged D
Double tagged D
e+ e
0D
0D
K+
ii
i
i
iii
ij
j
j
iji S
DBor
S
DB
2
Use 3 D0 modes and 6 D+ modes
K-+, K-+,0, K-+ ,+ -
K-+ +, K-+ +0, Ks+ , Ks+ 0, Ks+ 0, Ks++ + , K-K+ +
Reference modes D K+ and K++ normalize many B measurements from other experiments.
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 10
Overview of Technique
22candidatebeambc PEM
Determine separately the and yields 18=2(3+6) single tags(ST) and 45(=32 + 62) double tag yields(DT)
In a combined 2 fitter (physics/0503050), we extract 9 branching ratios and and yields : Include both statistical and systematic errors (with correlations): All experimental inputs treated consistently. Efficiency, cross-feed, background corrections performed directly in fit. Some systematic errors for and completely cancelled
Branching fractions are independent of L and cross-sections. The main variables used in the reconstruction are:
DD
00 DDN DD
N
1/ jiij ijjiDD DSSN /~
00 DDN
DDN
candidatebeam EEE
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 11
Yield Fits Unbinned ML fits to MBC (1D for ST, 2D for DT)
Signal function includes ISR, (3770) line shape, beam energy smearing, and detector resolution.
Signal parameters from DT fits, then apply to ST. Background: phase space (“ARGUS function”).
D and D yields and efficiencies separated.c
MBC (log scale) for ST modes:
ISR &
bea
m e
nergy
DT signal shape
Detectorresolution
All D0 DT
2484±51
All D+ DT1650±42
Two dimensional fit allows to separate ISR and beam energy spread (causes correlated
shifts in the mass of the two D’s) Detector resolution (uncorrelated among these
D’s )
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 12
Systematic uncertainties
Dominant error: MC simulation of tracking, K0
S, and 0 finding efficiencies: Correlated errors among all particles of
a given type add up quickly. Missing mass technique measure syst
errors by comparing data and MC: Fully reconstruct entire event, but
deliberately leave out one particle. Fraction of MM peak where the last
particle is found = efficiency.
Example: K efficiency from D0 K+
≈ 91% in fiducial volume
K found (MC)
K not found (MC)
K-+0, etc.
Source Uncertainty (%)
Tracking/K0s/0 0.7/3.0/2.0
Particle ID 0.3 /1.3 K
Trigger < 0.2
E cut 1.0—2.5 per D
FSR 0.5 ST / 1.0 DT
(3770) width 0.6
Resonant substructure
0.4—1.5
Event environment 0.0—1.3
Yield fit functions 0.5
Data processing 0.3
Double DCSD 0.8
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 13
Fit Results (PRL 95,121801) Precision comparable to PDG-04.
Statistical errors: ~2.0% neutral, ~2.5% charged from total DT yields.
(systematic) ~ (statistical). Many systematic errors are
measured in data and will be improved with time.
Our MC simulation includes FSR Using efficiencies without FSR
would lead to lower B.
NDD includes continuum and resonant production.
The CLEO-c measurement is the single most precise measurement for every
mode
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 14
Comparisons with other measurementsB
(D0 K
-+)
B(D
+ K-
++)
Other direct meas.
Overall C.L
25.9%
Reasonable agreement with PDG for all modes Measurements and errors normalized to PDG.
PDG numbers are correlated among modes PDG global fit includes ratios to K-+ or K-++.
No FSR corrections in PDG measurements Our measurements are also correlated (through statistics
and efficiency systematics).
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 15
Results for D cross sections Using a measurement of the luminosity of the data sample (55.8/pb), we obtain
nbDDee )07.007.060.3(00 nbDDee )10.007.079.2(
nbDDee )10.039.6( 17.008.0
)024.0776.0( 014.0
006.000
DDee
DDee
CLEO-c inclusive: nbhadronee )08.038.6()3770( 41.030.0
PRL 96, 092002
Our cross sections are in good agreement with BES [Phys.Lett. B 241, 278(1990) ] and higher than those of MARKIII [Phys.Rev.Lett. 60, 89 (1988) ]
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 16
Absolute Branching Fractions and Form Factor Measurements in
and
Introduction and Overview of the Analysis Measurements of Absolute Branching Fractions Measurements of Form Factors Summary
eKD /0 eKD 00/
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 17
Introduction Semileptonic decays are an excellent laboratory to study
Weak physics QCD physics
Gold-plated modes are P P semileptonic transitions as they are the simplest modes for both theory and experiment: Cabibbo favored : Cabibbo suppressed :
Main goals of the analysis: Measure efficiency-corrected absolutely-normalized decay rate distributions and
form factors Measure form factor parameters to test LQCD and model predictions
We analyze both D0 and D+ decays. By isospin invariance .
.
This is a nice cross check and adds statistics to improve statistical precision.
eDeDeKDeKD
00
00
,,
eDeD
eKDeKD00
00
2
)(
,)(24
)((
22
222
3)(
2
)(2
2
eMqwhere
qfPVG
dq
eKDd KcdcsF
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 18
Overview of the analysis Reconstruct one of the two D’s in a hadronic
decay channel. It is called a tagging D or a tag. Two key variables in the tagging D reconstruction are:
Reconstruct from the remaining tracks and showers the observable particles in the final state of a semileptonic decay.
Define an observable that can be used to separate signal and background as
where Emiss and Pmiss are the missing energy and momentum in the event, approximating the neutrino E and P. The signal peaks at zero in U.
Branching fractions are obtained as
22candidatebeambc PEM
candidatebeam EEE
K-
-
e+
K+
(3770)D0 D0
D0K+-, D0K-e+
tagMsemilep
semilepU
tagtagM
signalsemilep
Usemilep
bcbcN
N
N
NB
/
/
missmiss PEU
Obtained from Fits to U
Obtained from Fits to Mbc
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 19
D0 and D+ tag yields in 281/pb of DATA
~30% event tagging efficiency
~20% event tagging efficiency
Examples of Mbc for tag modes in the data
)281(/163~
:1
pbK
tagsDAll
)281(/308~
:1
0
pbK
tagsDAll
eventsK
KD
51~
0
eventsK
KD
98~
00
eventsK
KD
23~
000 eventsK
KD
80~
eventsK
KD
24~
0
eventsK
KD
16~
0
Tagging provides a beam of D mesons allowing semileptonic decays to be
reconstructed with no kinematic ambiguity
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 20
Measurements of
Absolute Semileptonic Branching Fractions
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 21
Fits to U in 281 pb-1 of Data for
Main backgrounds for
Main backgrounds for
Electron fakes from kaons
eDeKKD
0
0*0 )( eKD0
eD0
eDeKD
)( 00
0
eKD /0
eKD 0
N~7000
eD 0
N~700
missmiss PEU
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 22
Fits to U in 281 pb-1 of Data for
eKD 0
eKD 00 /
Main backgrounds for
Main Backgrounds for
eKKD )( 000*
eKKD )(0*
eKD 0
eKKD )( 000*
eD 0
N~2900
eKD 0
N~290 eD 0
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 23
Preliminary Results for BFs
)(024.0024.1)(
)(0
0
stateKD
eKD
)0.075(stat0.975 )(2
)(0
0
eD
eD
eKD 0 eD 0
eKD 0 eD 0
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 24
eKD 0 eD 0
Reasonable agreement
Systematically limitedStatistically limited
Comparisons with other experiments and projections for 750 pb-1
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 25
Measurements of
Semileptonic Form Factors
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 26
Two Fitting Methods: Fit A and Fit B
Fit A is a fit to efficiency-corrected and absolutely-normalized d/dq2 distributions. This fit is a good match for CLEO-c data as the q2 resolution is excellent. Fit A is our primary fit as the main goal of our analysis is to obtain d/dq2 and f+(q2).
Fit B is a fit to the observed decay rate according to a procedure described in D.M.Schmidt, R.J.Morrison and M.S.Witherell in Nucl. Instr. and Meth. A328 547(1993). The technique makes possible a (multidimensional) fit to variables modified by experimental acceptance and resolution. This method has been used by CLEO several times before, for example, to measure form factor ratios in ce and BD*l.
Both fitting methods were tested using large Monte Carlo samples. Two fits provide cross-checks.
The observed decay rate is related to the true decay rate in the following way:
in terms of Acceptance and Smearing functions. The fit has to take into account both effects. We have developed and tested two types of fits.
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 27
2q
d
d
D0→K-e+ν
2q
d
d
D0→π-e+ν
700 eventsS/B ~40/1
CLEOIII(Y(4S): q2 ~ 0.4 GeV2
CLEO-c((3770)): q2 ~ 0.012GeV2
q2 ~ 0.012GeV2
q2 ~ 0.011GeV2
Raw q2 distributionq2 resolution
q2 resolutions and Raw q2 distributions
Note the background in blue
7000 eventsS/B > 300/1
D0→π-e+ν
D0→K-e+ν
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 28
Efficiency corrected and absolutely normalized decay rates (DATA)
D0→K-e+ν D+→Kse+ν
D0→π-e+ν D+→π0e+ν
Subtracting background and applying efficiency corrections (matrices) we find absolute decay rates in bins of q2 (The bin width is equal q2
max/10, the last bins for D0e+
and D+0 e+ are 2 and 3 times wider):
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 29
Efficiency corrected and absolutely normalized decay rates (DATA)
These rates can be fit to any form factor model w/o knowing CLEO acceptance and resolution
The spectra on the last slide are tabulated here:
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 30
Form Factor Models
Simple pole model :
Modified pole model (BK) [Phys.Lett.B 52, 478,417(2000) ] :
Series parameterization [.Becher and R.Hill, hep-ph/0509090]
ISGW2 [Phys.Rev.D 52,2783,(1985) ] :
;
/1
022
2
poleMq
fqf
;/1
1
/1
02
)*(22
)*(2
2
sDsD MqMq
fqf
2
22max
22
121
rqf
)()(
)0(/11
0
2
22)(
2
2
k
q
KD aFdq
qdf
f
mm
k
kk qza
qqPqf )0,(
0,
1 222
2
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 31
D0 K- e+ 7000 events
D0 - e+ 700 events
C-R Prediction Our Fits
anddqMqFM
MqFI
whereNI
M
polepole
pole
pole
212
22
,,
,1
The efficiency of fits is tested using the Cramer-Rao inequality:
The fitter is consistent with being fully efficient.
The fitter is consistent with being unbiased.
Mpole (GeV)
datastat0.1
Example:
The fitting techniques were tested by making ensembles of fits to mock data samples with the number of signal events equal to the expected number of events in the data. We have tested:
Fits for all 4 form factor models simultaneous fits to isospin conjugate modes fit with two free parameters [f+(0)Vcs and a form factor shape parameter]
Tests of Fit A and B
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 32
Example of a fit (DATA)
D0→K-e+ν D+→Kse+ν
D0→π-e+ν D+→π0e+ν
Modified Pole (BK) Model:
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 33
By isospin invariance:
eDeD
eKDeKD00
00
2
)()(
)()(22
22
00
00
qfqf
qfqfeDeD
eKDeKD
2/1
3)(2
2)( /
))((~)(
iKi
icdcs P
q
eKDqfV
The q2 spectra for isospin conjugate modes are consistent.
The plots show:
f +(q
2)V
cd D0→πe+ν
D+→π0e+ν
q2 (GeV2)
D+→KSe+ν
D0→Ke+ν
f +(q
2)V
cs
q2 (GeV2)
DATA Cross Check 1
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 34
Cross check 2: Hadron & Electron Spectra & W Helicity
D0→K-e+ν D+→Kse+ν D0→π-e+ν D+→π0e+ν
HadronMomentum
ElectronMomentum
cosW
Quantities that are not constrained in the fit are well described
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 35
Systematic Uncertainties for Form Factor Shape Parameters
PK(G
eV)
Systematic uncertainties that are independent of q2 (ex: tag Mbc fit function) do not change the decay rate shape and hence have a negligible contribution to the shape parameter uncertainty
Pe(
GeV
)
eff
Lepton momentum vs q2
q2 (GeV)
q2 (GeV)
Kaon ID efficiency
PK ~ 100MeV Few events
This correlation is not as strong as the hadron momentum correlation
Kaon momentum vs q2
Systematic effects correlated with the hadron (K/KS//0) momentum, change the decay rate distribution and lead to modest systematic uncertainties
Our studies indicate that the total systematic uncertainty is much smaller than the statistical uncertainty for each semileptonic mode
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 36
Fit results with two parameters The shape parameters for modified pole, simple pole model and series
parameterization with two parameters :
The normalization parameter for modified pole model and series parameterization with two parameters :
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 37
First measurements of form factors for the D+ modes;
CLEO-c is the most precise for D→πe+
D → K e+ νD → π e+
ν
Comparison with Other Measurements
GeVMsD
112.2* GeVM
D010.2*
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 38
First measurements of form factors for the D+ modes;
CLEO-c is the most precise for D→πe+
D → K e+
νD → π e+ ν
Comparison with Other Measurements
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 39
Confidence levels for fits results with 2 parameters
The confidence levels for fits with 2 parameters :
Which parameterization does the data prefer? The confidence levels for all parameterizations are comparable, as the functional forms for the parameterization are similar and the shape parameters are not fixed. However, the CLEO-c data exclude the ISGW2 (K/) , pole (K) and modified pole (K) parameterizations when the shape parameters are fixed to the physical values.
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 40
Data vs. physical basis for shape parameters
#1: ISGW2 Output Physical value for ISGW2
D0 K- e+ 1.567 ± 0.045 1.131
D0 - e+ 1.997 ± 0.114 1.410
D+ K0 e+ 1.476 ± 0.070 1.131
D+ -0e+ 1.806 ± 0.209 1.410
Form factor shape parameters in the data for ISGW2 are inconsistent with the model predictions
#2: Pole
#3: Modified Pole
Output Physical value for pole
D0 K- e+ 1.94 ± 0.04 2.112
Output Physical Value for effective pole
D0 K- e+ 0.26 ± 0.06 ~1.75
Because the data do not support the physical interpretation of these three parameterizations we use the series parameterization
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 41
Fit results with 3 parameters Our main form factor shape and intercept results are for the series parameterization :
)0,()0,(10,
222
2122
02 qzbqzbqqP
bqf
The series is expected to converge rapidly, so only the 1st few terms are expected to be measurable: we test for three
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 42
The fit results for 2 and 3 parameters are consistent with each other;
Noticeable improvement for 2 for D→Ke+ν with 3 parameters;
The modes do not show this trend as they lack the statistics to probe the third term in the expansion;
For D → Ke+ν the 3rd term b2 is a order of magnitude larger than b1. This cannot be interpreted as a lack of convergence if the series because both are consistent with zero indicating that the data does not yet have the sensitivity to determine three parameters simultaneously.
Interpretation
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 43
Comparison Between Parameterizations
D0→K-e+ν D+→Kse+ν
D0→π-e+ν D+→π0e+ν
--- Simple pole
--- Modified Pole --- Series with 2 par Series with 3 par Data
Data and Fit results are normalized to the fit results for the series parameterization with 3 parameters.
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 44
Plotted LQCD results (blue) are recent results of FNAL+MILC unquenched three flavor LQCD [C. Aubin et al., PRL 94 011601 (2005)] Lattice systematic uncertainties
dominate:
The green lines are our fits to CLEO-c data
The dashed lines show 1 (stat+syst) regions
.07.004.050.0;07.003.073.0)0(
:)(
f
KeDLQCD
.07.004.044.0;06.003.064.0)0(
:)(
f
eDLQCD
DATA FIT
LQCD
DATA FIT
LQCD
Form Factors as a Stringent Test of LQCDVcd = 0.22380.0029(CKM unitarity, i.e Vcd = Vus)
Vcs = 0.97450.0008(CKM unitarity)
eKD0
eD0
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 45
In these plots, the central values for our projections are equal to the central
values from the LQCD results
The anticipated precision for a larger 750 pb1 data sample to be collected in the future
D0→K-e+ν
D0→π-e+ν
Projections for and f+(0)
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 46
Vcs(d) and f+(0) determination Using
Vcs = 0.97450.0008 (CKM unitarity)
Vcd = 0.22380.0029 (CKM unitarity, i.e Vcd = Vus)
eD0
Using LQCD results [C. Aubin et al., PRL 94 011601 (2005)]: eKD0
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 47
Summary for D semileptonic studies
I have shown preliminary results for DK/ e+ branching fractions and form factor measurements from the 280/pb data sample collected at (3770). Results of this analysis include: the most precise branching fraction measurements for these decays
the most precise or first measurements of form factors for these modes
the most precise or first measurements of the efficiency corrected and absolutely normalized decay rates
a stringent test of LQCD calculations of semileptonic form factors
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 48
Thank you
In summary, CLEO-c provides:
unique input to test LQCD, the theory capable of solving strongly couple field theory equations, and
input to other experiments that help improve their measurements
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 49
Fit A: a 2 fit to efficiency corrected d/dq2
A brief description of the procedure for making Fit A:
Create an N x N efficiency matrix, where N is the number of q2 bins Invert the efficiency matrix Measure raw background subtracted q2 distributions Use the inverted efficiency matrix to obtain efficiency-corrected and absolutely-normalized d/dq2 (or the form
factor)
We make fits for form factor parameters to efficiency-corrected and absolutely-normalized d/dq2 (or the form factor), using the 2 fitter which includes both statistical and systematic errors (with correlations) :
bin migrations, background uncertainty, and efficiency corrections.
The low number of events in the high q2 bins can lead to biases in 2 fits, we find that the
- the decay rate
ij
jest
jcorreffij
iest
icorreff NNVNN 12
jcorreffN j
estN ijV
Bias, if any, is SMALL [~0.10(stat. data) ]
- the decay rate estimated from a form factor
- the correlation matrix
July, 2006 Selected Topics on Open Charm Physics from CLEO-c 50
Efficiency Matrices
eKD0
eKD 0
eD0
eD 0
Full efficiency matrix for
Efficiency matrices in a truncated form for
10 bins
10 bins
9 bins
8 bins
We use 10 q2 bins for and . For and we use 9 and 8 bins, respectively. The last bin for these two modes are two or three times wider than other bins.
eKD0 eKD 0 eD0
eD 0
Do not need to read these tables