B-Mixing Fit Method Studies

Last Update: 29-May-04 (modification log)

Index

Toy MCs Used in Studies

All MC's have generator-level only information and are stored on karthur as TMB_trees. Results are shown for toy MC's generated with mixing on and for mixing off as listed below. See Conclusions in Fit Studies section for discussion of why this matters.

26/03/04 and before: Mixing ON - x_s = 25
16/04/04 and after: Mixing OFF

Mixing is regenerated with the desired Δm for data used in the analysis as part of the root code.

Decay Chain	Code Ver.'s	Macros	Events/Files	Param's
Mixing ON Dataset: 21-Jan-04
B_s(bar) --> D_s⁺ μ^- v(bar) D_s⁺ --> φ π⁺ φ --> K⁺ K^-	o Reco: p14.03.01 o runjob: v05-00-17 o cards: v00-04-06 o EvtGen: p14-br-04	o Generation o d0_mess o user.dec	o Total: 125,000 o Files: 500 o Evts/File: 250	o tau = 1.463 ps o x_s = 25
Mixing OFF Dataset: 17-Apr-04
B_s(bar) --> D_s⁺ μ^- v D_s⁺ --> φ π⁺ φ --> K⁺ K^-	o Reco: p14.03.01 o runjob: v05-00-17 o cards: v00-04-06 o EvtGen: p14-br-04	o Generation o d0_mess o user.dec	o Total: 125,000 o Files: 500 o Evts/File: 250	o tau = 1.463 ps o x_s = 0

Notes

Versions
- runjob: /data/d0usr/products/mc_runjob/NULL
- cardfile: /d0dist/dist/packages/cardfiles
- EvtGen: /data/d0dist/dist/packages/EvtGen
d0_mess Cuts
- at least one B_s(bar)
- at least one μ^- with:
  - Pt(μ) > 1.5 GeV
  - |η(μ)| < 3.5
  Note: this muon is not required to come from the B_s(bar). See below for an explanation of the effects of this oversight.
Parameters
- lifetime: /data/d0dist/dist/packages/EvtGen/p14-br-04/pdt.table
- Δm: /data/d0dist/dist/packages/EvtGen/p14-br-04/DECAY.DEC
- x_x = 25 (default) from src/EvtDecayTable.cpp

B_s --> D_s μ v Properties

Definitions

In the following I use the following definitions. All use generator-level information.

Lxy = xy distance from PV to Bs decay point
P(meas) = Pt(Ds + mu)
K = Pt(meas) / Pt(Bs)
ct0(true) = Lxy M(Bs) / Pt(Bs)
λ = ct0(meas) = Lxy M(Bs) / Pt(meas)
where occasionally the average K-factor <K> is used in the definition of t0(meas).

Kinematics & K-Factor Plots

Some plots of generated event properties are given below.

Bs Kinematics (30-Apr-04)
Bs-->Dsmu K-Factor (30-Apr-04)
Note that the effect of Pt cuts on the final state particles in the B_s decay is to narrow the K-factor distribution. The following cuts are used in the above plots.
- Pt(mu) > 2 GeV
- Pt(pi) > 0.7 GeV
- Pt(K+) > 0.5 GeV
- Pt(K-) > 0.5 GeV

True vs Measured Proper Time

Comparison of t0(true) with t0(meas) (λ) as a function of t0(true) (30-Apr-04)

Note (to remind Hal): using generator-level information, t0(meas) - t0(true) = t0(true) [1 - K] / K
Since t0 and K are uncorrelated, the width of the difference distribution will be smaller at smaller t0(true).

t0(meas)-t0(true) for all generated Bs --> Ds mu nu events
t0(meas)-t0(true) for Bs --> Ds mu nu events passing the cuts described in the K-factor section

Summary:

	RMS of t0(meas)-t0(true) (ps)
t0(true) range (ps)	All Bs --> Ds mu nu Decays	Decays passing Cuts
all	0.70	0.49
0 - 0.4	0.10	0.06
0.4 - 0.8	0.28	0.15
0.8 - 1.2	0.44	0.24
1.2 - 999	0.98	0.67

Effects of K-Factor and Mis-Tag on Fits

Fit methods are tested by splitting the data up into multiple trials and performing event-by-event log-Likelihood fits on each trial. Average fit parameters and RMS's over all trials are reported.

Conclusions & Questions

Biases in fit parameters due to use of <K>. (30-Apr-04)
Note: these biases are strongly correlated with the width of the K-factor distribution. This is reduced if kinematic cuts are placed on the final state tracks.

Fit Type	K-Factor	τ(fit)-τ(inp) /RMS	Δm(fit)-Δm(inp) /RMS	P(fit)-P(inp) /RMS
τ logL	<K>	+1.05	---	---
	K-PDF	-0.20	---	---
Δm_s logL	<K>	+2.35	+0.10	+10.27
	K-PDF	-0.46	+0.02	+0.07
Asym Δm_s χ²	<K>	---	+0.51	+10.55
	K-PDF	---	+0.45	+0.50

Mismatches between fit parameter RMS's and average fit errors.
There is no clear pattern in these mismatches. However, a few points can be made.
- Mismatches are larget for the Δm parameters.
- Using <K> in the fits gives average Δm fit errors that tend to be significantly smaller than the RMS of the returned fit values.
- The mismatch between fit errors and fit parameter RMSs is also present in the K-PDF fits, but is much reduced. In these fits, however, the distribution of returned fit parameters is not very Gaussian.

Comparison of evt-by-evt logL fits and fits to Asymmetry histograms:
The logL fits tend to return smaller spreads in Δm values but have larger mismatches between RMS and fit error averages. Examples for fits using the K-PDF are given below.

Param	event logL	Asym Histo
Δm_s: RMS / <Δm>	0.053	0.066
Δm_s: (RMS-<&error>)/RMS	0.347	0.162
Δm_d: RMS / <Δm>	0.069	0.080
Δm_d: (RMS-<&error>)/RMS	-0.029	-0.108

Effects of Mis-Tag and K-Factor on mixing distributions

Using Mixing OFF Set: 29-Apr-04

Δ(ms): Plots of N(evts) vs ct0 or λ Unmixed Events | Mixed Events
Δ(md): Plots of N(evts) vs ct0 or λ Unmixed Events | Mixed Events
Guide to Plots
- a) plot vs. ct0(true) -- Prob(mis-tag) = 0
- b) plot vs. ct0(true) -- Prob(mis-tag) = 0.2
- c) plot vs. λ -- Prob(mis-tag) = 0
- d) plot vs. ct0 -- Prob(mis-tag) = 0
- e) plot vs. λ -- Prob(mis-tag) = 0.2

Lifetime Fits Tested

All fits use Mixing OFF sample and all B_s(bar) decays (including photons) unless otherwise noted.

χ² fit on ct0 histogram
log-L fit to true ct0
log-L fit to λ using <K>
log-L fit to λ using convolution over K-factor histogram

Date	Fit No.	Trials Evts/trial	<tau(fit)> (ps) ±rms	<Error(fit>> ±rms
	Input		1.463
29/04/04	1) - ct0 histo	1 25000	1.452	0.004 χ² = 120/88
29/04/04	2) - ct0 true o Distrib's o Fit Param's	500 250	1.454 ±0.092	0.092 ±0.006
29/04/04	3) - <K> o Distrib's o Fit Param's	500 250	1.585 ±0.116	0.100 ±0.007
29/04/04	4) - K-PDF o Distrib's o Fit Param's	500 250	1.443 ±0.099	0.100 ±0.007

Mixing Fits Tested

log-L fit to true ct0-distrib & Prob(mis-tag) = 0
log-L fit to λ-distrib using <K>
log-L fit to λ-distrib using convolution over K-factor histogram
χ² fit to mixing asymmetry (N_unm-N_mix/N_tot) vs ct0(true)
χ² fit to mixing asymmetry (N_unm-N_mix/N_tot) vs λ using <K>
χ² fit to mixing asymmetry (N_unm-N_mix/N_tot) vs λ using convolution over K-factor histogram

Date	Fit No.	Trials Evts/trial	<τ> (ps) ±rms	<Err> ±rms	<Δm> (ps-1) ±rms	<Err> ±rms	<P(mis)> ±rms	<Err> ±rms
	Input		1.463		15		0.2 or 0
29/04/04	1) - ct0 true o Distrib's o Fit Param's o most fits have error matrix uncert. ~ 10%	100 1250	1.454 ±0.044	0.041 ±0.001	15.003 ±0.015	0.014 ±0.002	-0.006 ±0.013	0.006 ±0.002
29/04/04	2) - <K> o Distrib's o Fit Param's	100 1250	1.585 ±0.052	0.045 ±0.001	15.071 ±0.709	0.116 ±0.063	0.426 ±0.022	0.020 ±0.000
29/04/04	3) - K-PDF o Distrib's o Fit Param's	100 1250	1.442 ±0.046	0.044 ±0.001	15.020 ±0.802	0.524 ±0.239	0.203 ±0.045	0.046 ±0.003
29/04/04	5) - A(true) o Fit 6 periods o Distrib's o Fit Param's	100 1250	-----	-----	15.000 ±0.033	0.048 ±0.002	0.023 ±0.014	0.028 ±0.000
29/04/04	5) - A(<K>) o Fit 6 periods o Distrib's o Fit Param's	100 1250	-----	-----	15.652 ±1.275	0.481 ±0.236	0.411 ±0.020	0.025 ±0.000
29/04/04	6) - A(K-PDF) o Fit 6 periods o Distrib's o Fit Param's	100 1250	-----	-----	15.452 ±1.014	0.850 ±0.374	0.222 ±0.044	0.058 ±0.003
	Input		1.463		0.5		0.2 or 0
30/04/04	1) - ct0 true o Distrib's o Fit Param's	100 1250	1.454 ±0.044	0.041 ±0.001	0.503 ±0.013	0.015 ±0.003	-0.005 ±0.007	0.004 ±0.001
30/04/04	2) - <K> o Distrib's o Fit Param's	100 1250	1.585 ±0.052	0.045 ±0.001	0.439 ±0.094	0.030 ±0.005	0.214 ±0.014	0.015 ±0.000
30/04/04	3) - K-PDF o Distrib's o Fit Param's	100 1250	1.443 ±0.046	0.044 ±0.001	0.505 ±0.035	0.036 ±0.003	0.200 ±0.014	0.016 ±0.000
30/04/04	4) - A(true) o Fit 2 periods o Distrib's o Fit Param's	100 1250	-----	-----	0.499 ±0.015	0.021 ±0.001	0.004 ±0.008	0.024 ±0.000
30/04/04	5) - A(<K>) o Fit 2 periods o Distrib's o Fit Param's	100 1250	-----	-----	0.446 ±0.037	0.038 ±0.004	0.222 ±0.014	0.020 ±0.000
30/04/04	6) - A(K-PDF) o Fit 2 periods o Distrib's o Fit Param's	100 1250	-----	-----	0.460 ±0.037	0.041 ±0.004	0.214 ±0.014	0.021 ±0.000

Archives of Previous Studies

26-Mar-04