Dalitz plot analysis in D meson decays Hai-Bo Li IHEP Workshop on Partial Wave Analysis and Dalitz Plot Analysis Beijing, Jan. 25-26 2018/11/19

Rich structures in D three body decays D0 ! K B0 ! K Interference effects More important in D than in B decay (usually!), since there are many resonance below D mass region (scalar)! Or rich final state interaction. 2018/11/19

History of D Dalitz decays PDG2006 by D. Asner Charm three body analyses from several experiments summarized in PDG2006 review by D. Asner. Variety of Physics probed Doubly-Cabibbo suppressed decays CP Violation charm mixing Properties of light mesons Properties pp & Kp S-wave Usual analysis technique for P  3P decays is Dalitz-plot analysis technique 2018/11/19

Dalitz-plot Analysis Technique(1) Want to describe the internal dynamics of D0  ABC (P  3P) decay Daughter 4-momenta: 12 parameters Conservation of 4-momentum: 4 constraints Masses of Decay products: 3 constraints D is spin-0 : 3 orientations uninteresting Decay described by 2 degrees of freedom 3 Lorentz invariants (MAB)2, (MAC)2, (MBC)2 Related by (MD)2 + (MA)2 + (MB)2 + (MC)2 = (MAB)2 + (MAC)2 + (MBC)2 Dalitz plot is (MAB)2 vs (MBC)2 Phase space is “flat” in these variables 2018/11/19

Dalitz-plot Analysis Technique(2) Structure on Dalitz plot is due to Non-uniform reconstruction efficiency Background Internal dynamics of the decay D  ABC Analyze structure on the Dalitz plot to elucidate a broad range of physics topics Phase space “triangular” for massless daughters becomes rounder as Q decreases BKSK+K- 2018/11/19

BESIII Opportunity in Dalitz plot analysis e+e-  (3770)  DD, 18 M D0D0 , 14M D+D- /year 1. D mesons are produced almost at rest; clean events; 2. Flavor tag 3. CP tag 4. 20 modes D  P1P2P3 ( P= p, K) good for DP analysis e+e-  s=4170  DSD*S, 2 M DSD*S /year DS Dalitz plot can also be done at BESIII. Huge C0, C from (2S) and J/ Decays. 2018/11/19

Technical Issues on Dalitz Plot analyses Parameterization of the angular distributions for 3-body Dalitz plots. Parameterization of the angular distributions for 4-body Dalitz plots. Parameterization of the resonant line-shapes. Parameterization of Non-Resonant contributions. Normalization of the amplitudes (analytic? toyMC technique? ) Efficiency. Backgrounds. Mass resolution. Visualization of fit results (mass projections and angular distributions). Estimators of the fit quality (chi^2, likelihood, adaptive binning). How to tabulate Results (fractions, amplitudes and phases). Common systematic errors evaluation. Validations (use of toyMC, of simulated data). 2018/11/19

Dalitz Model ---Parameterization of Dynamics But: very often not just an isolated resonance Mass dependence Behaviour near decay-channel thresholds Strongly overlapping resonances and non-resonant dynamics (e.g. crossed channel) By S. Spanier K matrix Isobar Model starting from 2-Body Scattering LASS parameterizations for S wave Relativistic Breit-Wigner form 2018/11/19

Methodology? We made no recommendation on the type of fit to make, but we do expect that the isobar model will still play a large role in many analyses. – Let’s start there Recommend reading D. Asner’s review hep-ex/0410014 v1, 5 Oct 2004 Isobar Model 2 a b c H d2G FH Fk H a b c k +  ak e ifk / dm2dm2 k bc ab NR Isobars 2018/11/19 From D. Asner

Isobar Model Mk = (-2pq)L YL0(cosq bc) TL(m2) x FH(q,rH) x Fk(m2) Each isobar term can be written as Mk = (-2pq)L YL0(cosq bc) TL(m2) x FH(q,rH) x Fk(m2) (for resonance J=L decay to 0+0) TL(m2) = BWL(m2) = sin dk eidk tan dk = M0G(m)/(M02-m2) G(m) = G0 (p/p0)2L+1 (M0/m) Fk(p)2 Form factor Fk usually taken as Blatt-Weisskopf damping factor. A suggestion – define it to be unity at M0 for each resonance. ! Easier comparison of complex coefficients ak . 2018/11/19

K Matrix Fits  Klaus Peters talk in this meeting Best (only?) method for coupled channel analyses Single pole in single channel -> BW Distant, narrow poles -> sum of BW’s One pole in >1 channel -> Flatte BW Production vector allocates poles produced in three body decays Recommend: D. Asner, hep-ex/0410014 v1, 5 Oct 2004 (again) S U. Chung et al., Annalen Phys.4:404-430,1995 k j  Klaus Peters talk in this meeting 2018/11/19

K matrix parameterization For review see: Partial wave analysis in K matrix formalism, S.U. Chung et al., Annalen Phys.4:404-430,1995 2018/11/19

Resonances in K-Matrix Formalism The Breit-Wigner form is an approximation for isolated resonances far from the opening of thresholds. The K-matrix parameterization of multi-body interaction in terms of well constrained 2-body interactions (isobar model) is a way to build on existing amplitude measurements. The K-matrix can be parameterized as an extension of the relativistic Breit -Wigner form for s-channel resonance dominance. The K-matrix is a good starting point to construct Lorentz invariant, 2-body unitary, analytic (crossing) amplitudes. Rem: In some models even K-matrix poles are interpreted as particles !? 2018/11/19

Interest in K-+ s-wave—E791 hep-ex/0506040 B. Meadows for E791 ~138 % c2/d.o.f. = 2.7 Flat “NR” term does not give good description of data. 2018/11/19 Traditional iso-bar model

E791 D+ ! K-p+p+ PRL 89, 121801 hep-ex/0506040 B. Meadows for E791 ~89 % c2/d.o.f. = 0.73 (95 %) Probability Mk = 797 § 19 § 42 MeV/c2 Gk = 410 § 43 § 85 MeV/c2 The precise line-shape of K* is very important for this analysis. J/ KKp0 can definitely do that at BESIII. 2018/11/19

E791 Model Independent PWA Compare MIPWA & BW Isobar Compare MIPWA & LASS Data Phase Amplitude Phase S-wave P-wave D-wave 2018/11/19

E791 D+ ! p-p+p+ PRL, 86, 770 (2001) E791 collaboration 2018/11/19

D+  p+p-p+ analysis from CLEOc hep-ex/0607069 CLEO collaboration 281 pb-1/1.8M D pairs KS removed Leading fractions shown on the projection plots. 2018/11/19

K-matrix Analysis D0KSp+p- BaBar AS param: 81496 events with 91.5 fb-1(EPS05 paper) Much better fit quality than BW fit with only known resonances Similar quality to BW fit that included extra pp S-wave resonances Compare pp S-wave content CLEO BW Model - 15.1% BW fit with s1, s2 – 30.4% A/S K-matrix fit - 16.2% BaBar Data: K-matrix Model 2018/11/19

Efficiency over the DP Branching fractions: B = N_signal/N_DD* Is the efficiency is constant across the DP? In general, no, there can be significant efficiency variation across the DP, from PID/tracking efficiency. Solution: Signal events and efficiency found bin by bin. However, we need to find the number of bins such that: 1. Not to few: efficiency variation are covered; 2. Not too many bins: Fewer statistics. Look at the variation of <> versus bin area. Efficiency variations from PID/tracking. 2018/11/19

Efficiency on the DP 2018/11/19

The momentum resolution effect Events does not respect dalitz kinematics boundary! Due to the momentum resolution, some signal events fall outside the Kinematics boundary. Since some signals are being cut away, the amplitude/Branching Fraction is being altered! 2018/11/19

Alternative Representation of the Dalitz Plot 2018/11/19

Rectangular Dalitz Plot Homogenization B0 p+p-p0 BaBar Easy to do efficiency correction, and bin by bin fit. 2018/11/19

Efficiency and Resolution on B0p+p-p0 Dalitz Plot Selection efficiency on Dalitz plane Large variations in the Dalitz plot corners Fraction of mis-reconstructed events in Dalitz plane 00 Efficiency roughly flat Fraction of SCF events highly dependent on the Dalitz plot position +– 00 +– –+ –+ Truth-matched mass res.: double Gaussian core: 7÷8 MeV/c2 Truth-matched events Misreconstructed events SCF events: large variations in resolution over the DP  4D-convolution functions introduced G(m’rec, q’rec, m’true, q’true) 2018/11/19

D0-D0 mixing from D0K+p-p0 Decay (1) Charm2006 Event-level tagged Prominent K* peak in DCS Mode 2018/11/19

D0-D0 mixing from D0K-p+p0 Decay (2) Charm2006 Event-level tagged Prominent r peak in CF Mode 2018/11/19

Resonant structures on the DP sensitive to D mixing The resonance amplitudes are different for DCS and CF- there is more sensitivity to mixing In D0  K-p+p0 the main resonance is K- r+ In D0  K+p-p0 the main resonance is K*+ p- The ratio will depend on the point On the DP, full DP analysis will be very sensitive to mixing parameters, especially RM. 2018/11/19

D0K0+- decay & CKM angle γ γ can be measured from the interference between decays with bcus and bucs transitions bu transition Vub bc transition Vcb e i δB -i γ Interference occurs when some final state is accessible by both D0 and D0 Giri-Grossman-Soffer-Zupan: PRD68, 054018 (2003): Final state = Ks0π+π-  Dalitz Plot Analysis 2018/11/19

BESIII Impact on g Simultaneous CP tagged & flavor tagged analysis of D0Ks+- [correlated D’s,(3770)DD, (4140)DD(n)g(m)p0] One can write We will extract as well as in a model independent way. This is exactly what the g analyses need. CLEO-c has enough data in hand to reduce systematic error to 2o with 20 fb-1 at BESIII. 2018/11/19

CP-tagged D KSp+p- decay at BESIII Toy study from A. Bondar BESIII sensitivity A model independent way is proposed by A. Bondar and A. Poluektov hep-ph/0510246 50 ab-1 at super-B will allow a model independent g/f3 measurement at accuracy 2o. However, 10 fb-1 at y(3770/4170) data needed to accompany the analysis @BES-III: 10, 000, KS p+p- ,7,500 p+ p- p0, 1,900 KSK+K- The d(cos(dD)) 2%  10 –20 2018/11/19

Dalitz Fitter--Tools RooFit BaBar ……..many RooFit based fitters Charmfitter Miniut based, BaBar …. Charmfitter 2018/11/19

Summary Learn more experiences from BaBar, Belle, CLEO, FOCUS, E791 ….. Towards Uniformity Discussions with CLEO and BELLE Methodology 2018/11/19