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Amplitude analyses of charm meson decays at BESIII
Yu LuInstitute of High Energy Physics, Beijing(On behalf of the BESIII collaboration)
1
Outline
The used dataset.
Amplitude analysis.
Recent results for π·(π ) hadronic decays
1. Amplitude analysis of π·0/π·+ β πΎπππ.
2. Amplitude analysis of π·π + β π+π0π and the first observation of
π·π + β π0 980 π
Summary
2
3
Based on RooFit:
Use RooFit framework to construct the PDF
Use RooMinuit to perform the unbinned likelihood fit
Use RooFitResult to deal with the fit results
Using GPU to accelerate the analysis:
C99 standard for programs in kernel
Connect the RooFit and GPU kernel with OpenCL and C++
Much faster than CPU version. (One or two weeks three hours for a fit to data
in Amplitude analysis of π·0 β πΎβπ+π+πβ )
Amplitude analysis
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PDF construction
π·
π1
π 1
π2
π3
Three-body hadronic decays
π·
π 1
π 2
π1
π2
π3
π4
π·
π1
π 1
π2
π 2 π3
π4
Four-body hadronic decays
Topology A Topology B
π΄π ππ = ππ1 π1 ππ
2 π2 ππ ππ πΉπ1 ππ πΉπ
2 ππ πΉππ·(ππ) π΄π ππ = ππ ππ ππ ππ πΉπ
π ππ πΉππ·(ππ)
Isobar model is used in amplitude analysis:
the total amplitude π ππ is the coherent sum of the sub-amplitudes π΄π ππ ,
π ππ =
π
πππππππ΄π( ππ)
β’ ππ ππ : Propagator of intermediate resonance.
β’ ππ ππ : Angular terms, constructed with covariant tensor formalism (Zemach tensor).
β’ πΉπ ππ : Blatte-Weisskopf barriers 5
Propagator of intermediate resonance
β’ For the resonance π, the GS formula (PRL 21, 244) is considered. In the decay of π·+ β πΎπ
0π+π+πβ, the obviously π β π interference effect should be considered.
π·+ β πΎπ0π+π+πβ
π·0 β πΎβπ+π+πβ
PRD95, 072010
GS RBW
Relative magnitude and phase.
6
Propagator of intermediate resonanceFor resonance π (π0(500)), the Bugg formulais used (PLB 572, 1):
π4π = 1 β16ππ
2
π2 /(1 + π2.8βπ2
3.5 )
All the parameters are fixed to Ref .PLB 598, 149.
For resonance π0 980 and π0(980), the flatte formula is used,
ππ0 π0 π =1
π02βπ2βπ ( π ππ
2ππ).
Here, only two channels coupling (ππ, πΎπΎfor π0 980 and ππ, πΎπΎ for π0(980)) is considered.
The parameters for π0 980 are fixed to Ref. PRD 95, 032002.
The parameters for π0(980) are fixed to Ref. PLB 607, 243.
β’ The πππ (ππ) above is the two-body phase space factor, 2π/π, π is the magnitude of the
daughter momentum in the resonance rest frame.
β’ For other resonances, the RBW is used.
7
Propagator of intermediate resonance
The πΎπ πβπ€ππ£π parameterization:
The LASS model used in the analysis of π·0 βπΎπ
0π+πβ of BABAR (PRD 78, 034023)οΌ
π΄ π = πΉπ πππΏπΉπππΏπΉ + π π πππΏπ πππΏπ ππ2πΏπΉ ,
πΏπΉ = ππΉ + πππ‘β1(1
ππ+
ππ
2),
πΏπ = ππ + π‘ππβ1πΞ ππΎπ
π2 β ππΎπ2 ;
The parameters are fixed to BABAR results.
Scattering term π²πβ (ππππ) term
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Three body
Angular terms The decays of π·0 β πΎβπ+π+πβ and π·0 β πΎβπ+π0π0 allow the decay chains shown as
topologies A and B. The decay of π·+ β πΎπ
0π+π+πβ only allows the decay chain of topology B since the contributions from doubly Cabibbo suppressed decays are negligible with our data statistic.
Four body
β’ The determination of π and π‘ are the same withref. EPJA 16, 537.
β’ The amplitudes with angular momentum > 2 are not considered.
9
Likelihood construction
The signal PDF π(ππ) is given by:
π ππ =π ππ π ππ
2πππ (ππ)
π ππ π ππ2πππ (ππ)
,
where, π ππ is the efficiency function; πππ (ππ) is the n-body phase space. Then the
likelihood is:
πππΏ = πππππ‘π π€π
πππ‘ππππ(ππ) + π
ππππ βπ€ππππ
πππ(ππ),
Signal part Background part
β’ Background sample can be simulated events or sideband events in data.
β’ The normalized integration is performed with MC integration method.
Independent with the floated parameters
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Fit Fraction and the statistical uncertainty
The fit fraction is calculated with MC integration method:
πΉπΉ π = π
πππΆ | π΄π§ ππ |2
ππππΆ |π ππ |2
,
β’ πππΆ is the number of MC sample events.
β’ π΄π§ ππ is either the π§π‘β amplitude ( π΄π§ ππ = ππππππ π΄π ππ ) or the π§π‘β subset
(component) of coherent sum of amplitudes ( π΄π§ ππ = πππππ
ππππππ΄ππππ ).
To obtain the statistical uncertainty, the fit results are randomly modified according to the covariance matrix of the fit result. Under the RooFit framework:
the function: randomizePars() returns the randomly shifted values.
shift parameters
calculate FF
300 times
Fit with a Gaussian
Stat. error = width
11
Fit quality
Binned method:
β’ The global π2 is calculated in a five-dimensional
(two-dimensional) phase space for four-body (three-
body) decays: π2 = π ππ and ππ =ππβππ
exp
βππexp .
β’ ππ and ππexp
is the number of data and simulated
events in ππ‘β cell.
β’ Cells with expected events less than 20 will be
merged with next cell until they satisfy the minimum
number of events criterion.
π·0 β πΎβπ+π+πβ
(PRD95, 072010)
Unbinned method:
A mixed-sample method is used in π·0 β πΎβπ+π0π0 unbinned likelihood fit to determine the fit quality (M. Williams, Journal of Instrumentation 5, P09004 (2010)).
12
Recent results for π·(π ) hadronic decays
Amplitude analysis of π·0/π·+ β πΎπππ:
β’ These decays provide windows to investigate the decay modes π· β ππ and π· β π΄π (π:
pseudo-scalar, π: vector, π΄: axial-vector), which are important in searching for CPV and
learning π·0 π·0mixing.
β’ The amplitude analysis results can also be used in many other experimental measurements:
Branching fraction measurement.
Strong phase determination (Only for π·0)
πΎ angle determination (Only for π·0)
β’ There are seven decays of π·0/π·+ β πΎπππ, previous amplitude analyses for π·0 βπΎβπ+π+πβ, πΎπ
0π+πβπ0, and π·+ β πΎπ0π+π+πβ, πΎβπ+π+π0 have been performed by
Mark III and E691. Both measurements are affected by low statistics.
β’ The results of π·0 β πΎβπ+π+πβ, πΎβπ+π0π0, and π·+ β πΎπ0π+π+πβ are presented here.
13
Result of π·0 β πΎβπ+π+πβ(PRD95, 072010)
Double tag: π·0 β πΎβπ+π+πβ vs π·0 β πΎ+πβ are reconstructed. About 16K events with a purity ~99.4% are selected for amplitude analysis. Model is constructed with 23 amplitudes.
14
Projections of π·0 β πΎβπ+π+πβ(PRD95, 072010)
The two identical π+ are required with: π π1+πβ > π(π2
+πβ)
Points with error bars: data, curves: fit, red histograms: background.
15
Branching fractions for different components (PRD95, 072010)
Stat. uncertainty from FF
Sys. uncertainty from FF
uncertainties related to π΅(π·0 β πΎβπ+π+πβ) in PDG
Significantly improved than previous PDG values.
16
Result of π·0 β πΎβπ+π0π0(BESIII Preliminary)
Double tag: π·0 β πΎβπ+π0π0 vs π·0 β πΎ+πβ are reconstructed. About 5.95K events with a purity ~99% are selected for amplitude analysis. Model is constructed with 26 amplitudes.
Preliminary
Preliminary
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Projections of π·0 β πΎβπ+π0π0(BESIII Preliminary)
Points with error bars: data, red histograms: fit.
PreliminaryPreliminary
18
Branching fractions for π·0 β πΎβπ+π0π0(BESIII Preliminary)
Yields: DT, 6101 Β± 83; ST, 534581 Β± 769. The amplitude analysis results are used to determine the reconstructed efficiency.
First measurement!
Fits to distributions of DT and ST dataοΌ
Preliminary Preliminary
BCM
Beam-Constrained Mass: 2 2
BC beam | |D
M E p Signal: peaks at D mass BCM
19
Result of π·+ β πΎπ0π+π+πβ
(BESIII Preliminary)
Double tag: π·+ β πΎπ0π+π+πβ vs π·β β πΎ+πβπβ are reconstructed. About 4.56K events with a
purity ~99% are selected for amplitude analysis. Model is constructed with 12 amplitudes.
Preliminary
20
Projections of π·+ β πΎπ0π+π+πβ
(BESIII Preliminary)
The two identical π+ are required with: π π1+πβ < π(π2
+πβ).
Points with error bars: data, red histograms: fit, green histograms: background estimated from MC.
21
Branching fractions for different components (BESIII Preliminary)
Stat. uncertainty from FF
Sys. uncertainty from FF
uncertainties related to π΅(π·0 β πΎβπ+π+πβ) in PDG
Preliminary
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Amplitude analysis of π·π + β π+π0π:
β’ Extract the branching fraction of the π-annihilation process involved decay π·π + β π+π.
β’ Improve the precise of π΅(π·π + β π+π0π).
A Cabibbo favored decay with large branching fraction but poor precision (PDG: 9.2 Β± 1.2 %).
β’ Search for the pure π-annihilation decay π·π + β π0 980 π.
The pure π-annihilation decay π·π + β ππ+ has been observed with BF ~0.1%.
Theoretical analyses based on previous measurement show the size of π-annihilation amplitude is about 0.1-0.2 of the size of tree-emission amplitude in the decay modes of π· βππ and ππ.
No experimental or theoretical studies on π-annihilation processes are performed for the decay mode π· β ππ (S: scalar)
(The π-annihilation processes, induced by FSI in charmed decays, can not be calculated with perturbative theory. Determination π-annihilation amplitude with the experimental input is the only method.)
23
Events are selected with double tag:
β’ Tag modes: π·π β β πΎπ
0πΎβ, π·π β β πΎ+πΎβπβ, π·π
β β πΎπ 0πΎβπ0, π·π
β β πΎ+πΎβπβπ0, π·π β β
πΎπ 0πΎ+πβπβ, π·π
β β πβππΎπΎ, and π·π β β πβππ+πβπ
β² .
β’ Signal mode: π·π + β π+π0π.
β’ Multi-variate analysis is performed to suppress the background from fake π.
1239 events are selected with a purity of 97.7 Β± 0.5 % for amplitude analysis.
Result of π·π + β π+π0π (BESIII Preliminary)
Amplitude Significance (π) Phase FF
π·π + β π+π > 20 0.0 (fixed) 0.783 Β± 0.050 Β± 0.021
π·π + β π+π0
ππ 5.7 0.612 Β± 0.172 Β± 0.342 0.054 Β± 0.021 Β± 0.026
π«π+ β ππ πππ π ππ. π π. πππ Β± π. πππ Β± π. πππ π. πππ Β± π. πππ Β± π. πππ
The significances, phases, and FFs for intermediate processes obtained from the amplitude analysis.
The amplitudes agree with the isospin conserved expectation (π΄ π·π + β π0 980 +π0 =
β π΄ π·π + β π0 980 0π+ ) within statistical uncertainty, thus we set the magnitudes of these
two amplitudes to be the same and the phase difference to be Ο. 24
Projections of π·π + β π+π0π (BESIII Preliminary)
Full sampleSub-sample with ππ+π0 >
1.0 GeV/π2
Dalitz plot and projections
Dots with error bar: data; solid: total fit; dashed: π·π + β π+π;
dotted: π·π + β (π+π0)ππ; long dashed:π·π
+ β π0 980 π.
Obvious peaks for two π0 980 mesons!
25
26
Branching fraction measurement (BESIII Preliminary)
DT yield: ππππ Β± ππ
Fit to π΄π +π ππΌ
β’ Dots with error bars: data.
β’ Total fit.
β’ Signal: MC shape convoluted
with a Gaussian.
β’ Background: second-order
Chebychev.
Efficiency is determined with the amplitude analysis result.
π΅πΉ π·π + β π+π0π = 9.50 Β± 0.28π π‘ππ‘. Β± 0.41π π¦π . %
The systematic uncertainty on π΅πΉ π·π + β π+π0π is dominated by π0 and π reconstruction (4%).
Branching fraction measurement (BESIII Preliminary)
Branching fraction This measurement (%) PDG value (%)
π΅πΉ(π·π + β π+π) 7.44 Β± 0.48π π‘ππ‘. Β± 0.44π π¦π . 8.9 Β± 0.9
π΅πΉ(π·π + β π0 980 π) * 2.20 Β± 0.22π π‘ππ‘. Β± 0.34π π¦π . -
π΅πΉ(π·π + β π0 980 +π0) *
π΅πΉ(π·π + β π0 980 0π+) *
1.46 Β± 0.15π π‘ππ‘. Β± 0.22π π¦π . -
*Here, ππ πππ β π πΌ.
For the ππ‘β amplitude, the BF can be calculated with: π΅πΉ π = π΅πΉ π·π
+ β π+π0π πΉπΉ(π)
β’ The π΅πΉ(π·π + β π0 980 +/0π0/+) is larger than the PDG values of π΅πΉ(π·π
+ β ππ+)( 2.4 Β± 0.6 Γ 10β3) and π΅πΉ(π·π
+ β π π) ( 1.3 Β± 0.4 Γ 10β3) by one order of magnitude.
β’ The magnitude for the ratio of the W-annihilation amplitude over tree-emission amplitude |π΄/π| is obtained to be 0.84Β±0.23 in the decay mode of π· β ππ, which is much larger than the level of 0.1-0.2 in the decay mode of π· β ππ (PRD 93, 114010).
β’ The contribution from π-annihilation processes should be taken into considered in investigating the π· β ππ decays.
β’ This provides theoretical challenge in understanding such a large W-annihilation contribution in π· β ππ decays. 27
Summaryβ’ BESIII provides large data samples close to charm related threshold to study the π·(π ) multi-
body hadronic decays.
β’ Amplitude analysis is a powerful method to investigate the multi-body decays and to extract
the physical information.
β’ Three amplitude analysis results for π· β πΎπππ and one for π·π + decay are presented.
β’ Amplitude analyses for π·0 β πΎβπ+π0π0 and π·π + β π+π0π are firstly performed.
With the amplitude analysis result, the π΅πΉ(π·0 β πΎβπ+π0π0) is firstly measured to be
8.98 Β± 0.13π π‘ππ‘. Β± 0.40π π¦π . %.
The precision of π΅πΉ π·π + β π+π0π is improved with a factor of 2.5 to the PDG values.
The decays π΅πΉ(π·π + β π0 980 +π0) and π΅πΉ(π·π
+ β π0 980 0π+) via π0 980 β ππ
are firstly observed and measured to be (1.46 Β± 0.15π π‘ππ‘. Β± 0.22π π¦π .)%
β’ More results for π·(π ) multi-body hadronic decays are coming.
28Thank you!