Dalitz plot analysis in D meson decays HaiBo

Dalitz plot analysis in D meson decays Hai-Bo Li IHEP Workshop on Partial Wave Analysis and Dalitz Plot Analysis Beijing, Jan. 25 -26 2021/9/6 1

Rich structures in D three body decays Interference effects More important in D 0 ! K B 0 ! K D than in B decay (usually!), since there are many resonance below D mass region (scalar)! Or rich final state interaction. 2021/9/6 2

History of D Dalitz decays PDG 2006 by D. Asner Charm three body analyses from several experiments summarized in PDG 2006 review by D. Asner. l Variety of Physics probed – Doubly-Cabibbo suppressed decays – CP Violation – charm mixing – Properties of light mesons – Properties & K S-wave l Usual analysis technique for P 3 P decays is Dalitz-plot analysis technique l 2021/9/6 3

Dalitz-plot Analysis Technique(1) §Want to describe the internal dynamics of D 0 ABC (P 3 P) 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 2021/9/6 4

Dalitz-plot Analysis Technique(2) Structure on Dalitz plot is due to 1) Non-uniform reconstruction efficiency 2) Background 3) Internal dynamics of the decay D ABC Ø Analyze structure on the Dalitz plot to elucidate a broad range of physics topics Phase space “triangular” for massless daughters Ø becomes rounder as Q decreases B KSK+K- 2021/9/6 5

BESIII Opportunity in Dalitz plot analysis Ø e+e- (3770) DD, 18 M D 0 D 0 , 14 M D+D- /year 1. D mesons are produced almost at rest; clean events; 2. Flavor tag 3. CP tag 4. 20 modes D P 1 P 2 P 3 ( P= , K) good for DP analysis Ø e+e- s=4170 DSD*S, 2 M DSD*S /year DS Dalitz plot can also be done at BESIII. Ø Huge C 0, C from (2 S) and J/ Decays. 2021/9/6 6

Technical Issues on Dalitz Plot analyses o Parameterization of the angular distributions for 3 -body Dalitz plots. o Parameterization of the angular distributions for 4 -body Dalitz plots. o Parameterization of the resonant line-shapes. o Parameterization of Non-Resonant contributions. o Normalization of the amplitudes (analytic? toy. MC technique? ) o. Efficiency. o. Backgrounds. o. Mass resolution. o. Visualization o. Estimators o. How of fit results (mass projections and angular distributions). of the fit quality (chi^2, likelihood, adaptive binning). to tabulate Results (fractions, amplitudes and phases). o. Common systematic errors evaluation. o. Validations 2021/9/6 (use of toy. MC, of simulated data). 7

Dalitz Model ---Parameterization of Dynamics But: very often not just an isolated resonance • Mass dependence By S. Spanier • Behaviour near decay-channel thresholds • Strongly overlapping resonances • and non-resonant dynamics (e. g. crossed channel) K matrix Isobar Model starting from 2 -Body Scattering LASS parameterizations for S wave Relativistic Breit-Wigner form 2021/9/6 8

Methodology? l. 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 v 1, 5 Oct 2004 2 Isobar Model d 2 dm 2 bc / a H b c ab NR 2021/9/6 From D. Asner + ak e k if H k a k b FH Fk c Isobars 9

Isobar Model l Each isobar term can be written as Mk = (-2 pq)L YL 0(cosq bc) TL(m 2) x FH(q, r. H) x Fk(m 2) (for resonance J=L decay to 0+0) TL(m 2) = BWL(m 2) = sin dk eidk tan dk = M 0 G(m)/(M 02 -m 2) G(m) l. Form = G 0 (p/p 0)2 L+1 (M 0/m) Fk(p)2 factor Fk usually taken as Blatt-Weisskopf damping factor. –A suggestion – define it to be unity at M 0 for each resonance. ! Easier comparison of complex coefficients ak. 2021/9/6 10

l K Matrix Fits Best (only? ) method for coupled channel analyses k –Single pole in single channel -> BW –Distant, narrow poles -> sum of BW’s –One pole in >1 channel -> Flatte BW j Production vector allocates poles produced in three body decays l. Recommend: Klaus Peters talk in this meeting D. Asner, hep-ex/0410014 v 1, 5 Oct 2004 (again) S U. Chung et al. , Annalen Phys. 4: 404 -430, 1995 2021/9/6 11

K matrix parameterization For review see: Partial wave analysis in K matrix formalism, S. U. Chung et al. , Annalen Phys. 4: 404 -430, 1995 2021/9/6 12

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 !? 2021/9/6 13

Interest in K- + s-wave—E 791 hep-ex/0506040 B. Meadows for E 791 ~138 % c 2/d. o. f. = 2. 7 Flat “NR” term does not give good description of data. 2021/9/6 Traditional iso-bar model 14

E 791 D+ ! K- + + hep-ex/0506040 B. Meadows for E 791 PRL 89, 121801 ~89 % c 2/d. o. f. = 0. 73 (95 %) Probability Mk = 797 § 19 § 42 Me. V/c 2 k = 410 § 43 § 85 Me. V/c 2 2021/9/6 The precise line-shape of K* is very important for this analysis. J/ KK 0 can definitely do that at BESIII. 15

E 791 Model Independent PWA Compare MIPWA & BW Isobar Amplitude Phase D-wave P-wave S-wave Phase Compare MIPWA & LASS Data 2021/9/6 16

E 791 + D ! + + PRL, 86, 770 (2001) E 791 collaboration 2021/9/6 17

D+ + - + analysis from CLEOc hep-ex/0607069 CLEO collaboration 281 pb-1/1. 8 M D pairs KS removed Leading fractions shown on the projection plots. 2021/9/6 18

K-matrix Analysis D 0 KS + - Ba. Bar AS param: 81496 events with • 91. 5 fb-1(EPS 05 paper) • Much better fit quality than BW fit with only known resonances • Similar quality to BW fit that included extra S-wave resonances • Compare S-wave content • Ba. Bar Data: K-matrix Model • CLEO BW Model - 15. 1% • BW fit with s 1, s 2 – 30. 4% • A/S K-matrix fit - 16. 2% 2021/9/6 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. 2021/9/6 20

Efficiency on the DP 2021/9/6 21

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! 2021/9/6 22

Alternative Representation of the Dalitz Plot 2021/9/6 23

Rectangular Dalitz Plot Homogenization B 0 + - 0 Ba. Bar 2021/9/6 Easy to do efficiency correction, and bin by bin fit. 24

Efficiency and Resolution on B 0 + - 0 Dalitz Plot Selection efficiency on Dalitz plane 0 0 + – – + Truth-matched events Fraction of misreconstructed events in Dalitz plane Large variations in the Dalitz plot corners Efficiency roughly flat Fraction of SCF events highly dependent on the Dalitz plot position Truth-matched mass res. : double Gaussian core: 7÷ 8 Me. V/c 2 + – 0 0 – + Misreconstructed events SCF events: large variations in resolution over the DP 2021/9/6 4 D-convolution functions introduced G(m’rec, q’rec, m’true, q’true) 25

D 0 -D 0 mixing from D 0 K+ - 0 Decay (1) Charm 2006 Event-level tagged l Prominent K* peak in DCS Mode l 2021/9/6 26

D 0 -D 0 mixing from D 0 K- + 0 Decay (2) Charm 2006 Event-level tagged l Prominent r peak in CF Mode l 2021/9/6 27

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 D 0 K- + 0 the main resonance is K- + –In D 0 K+ - 0 the main resonance is K*+ - The ratio will depend on the point On the DP, full DP analysis will be very sensitive to mixing parameters, especially RM. 2021/9/6 28

D 0 K 0 + - decay & CKM angle γ γ can be measured from the interference between decays with b cus and b ucs transitions b u transition b c transition Vcb e i δB e -i γ Vub Interference occurs when some final state is accessible by both D 0 and D 0 Giri 0 + Grossman-Soffer-Zupan: PRD 68, 054018 (2003): Final state = Ks π π Dalitz Plot Analysis 2021/9/6 29

BESIII Impact on g § Simultaneous CP tagged & flavor tagged analysis of D 0 Ks + [correlated D’s, (3770) DD, (4140) DD(n)g(m) 0] § 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 2 o with 20 fb-1 at BESIII. 2021/9/6 30

CP-tagged D KS + - decay at BESIII CP=+1 CP=-1 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/f 3 measurement at accuracy 2 o. However, 10 fb-1 at y(3770/4170) data needed to accompany the analysis @BES-III: 10, 000, KS + - , 7, 500 + - 0, 1, 900 KSK+KThe d(cos(d. D)) 2% 10 – 20 2021/9/6 31

Dalitz Fitter--Tools Roo. Fit Ba. Bar ……. . many Roo. Fit based fitters Charmfitter Miniut based, Ba. Bar …. Charmfitter 2021/9/6 32

Summary Learn more experiences from Ba. Bar, Belle, CLEO, FOCUS, E 791 …. . Towards Uniformity Discussions with CLEO and BELLE l l Methodology 2021/9/6 33