Introduction to Dalitz Plot Analysis - University of Warwick

Introduction to Dalitz Plot AnalysisTim GershonUniversity of Warwick479.WE-Heraeus-SeminarPhysics at LHCb26 April 2011Tim GershonIntroduction to Dalitz Plot Analysis

What Is a Dalitz Plot?●Visual representation of●the phase-space of a three-body decay– involving only spin-0 particles– (term often abused to refer to phase-space of any multibody decay)●Named after it's inventor, Richard Dalitz (1925–2006):●“On the analysis of tau-meson data and the nature of the tau-meson.”– R.H. Dalitz, Phil. Mag. 44 (1953) 1068– (historical reminder: tau meson = charged kaon)●For scientific obituary, see– I.J.R. Aitchison, F.E. Close, A. Gal, D.J. Millener,– Nucl.Phys.A771:8-25,2006Tim GershonIntroduction to Dalitz Plot AnalysisImage credit: Mike Pennington

“Equilateral representation”scattering regionsAtCConnection between scattering anddecay processes provides a solidtheoretical ground for describing somehadronic effects – “Watson's theorem”BDMandelstam variablesABuDCACDsABCDTim GershonIntroduction to Dalitz Plot AnalysisBImage credit: Mike Pennington3

In case anyone is wondering ...● Previous plot shows pp→π–0 π 0 π 0 fromCrystal Barrel experiment (LEAR)●C.Amsler et al., EPJC 23 (2002) 29●Highly symmetrized because 3 identicalbosons in the final state●symmetric under interchange of any 2 π 0Tim GershonIntroduction to Dalitz Plot Analysis4

“Right-angled representation”Removes some of the symmetry … but easier for most software to handle(Most decays that we are interested in are not so symmetric anyway)Retains & heightens intuitive connection between visualisation and kinematics0 → 1 2 3M m 1m 2m 3Lorentz invariant phase-spaceof three-body decaydΩ ~ ds 12ds 13Tim GershonIntroduction to Dalitz Plot Analysis5

Dalitz plots as visualiser of kinematics● Illustration for D→K Sπ + π –●●●●green & blue: K*(892) (vector)cyan & magenta : K 2*(1430) (tensor)yellow : ρ(770) (vector)red : f 0(980) (scalar)●but main advantage of Dalitz plots is ability toexploit inference between different resonancesImage credit: Tom LathamTim GershonIntroduction to Dalitz Plot Analysis6

Some examples of Dalitz plotsD 0 → K sπ + π –BaBar PRL 105 (2010) 081803D s + → π + π – π +BaBar PRD 79 (2009) 032003K*,interferencef 0Tim GershonIntroduction to Dalitz Plot AnalysisNote symmetries7

Some examples of Dalitz plotsB + → D – π + π +Belle PRD 69 (2004) 112002B + → K + K + K –Belle PRD 71 (2005) 092003D 2* Φ,nonresonantTim GershonIntroduction to Dalitz Plot AnalysisNote symmetries8

Why are we so interested in Dalitz plots?●●●–Condition for DCPV: |A/A|≠1– Need A and A to consist of (at least) two parts– with different weak (φ) and strong (δ) phasesOften realised by “tree” and “penguin” diagramsA = ∣T∣e i T − T i∣P∣e P − P A = ∣T∣e i T T i∣P∣e P P A CP = ∣A∣2 −∣A∣ 2∣A∣ 2 ∣A∣ 2 =2 ∣T∣∣P∣sin T− P sin T− P∣T∣ 2 ∣P∣ 2 2 ∣T∣∣P∣cos T− Pcos T− PExample: B→Kπ(weak phase difference is γ)Tim GershonIntroduction to Dalitz Plot Analysis9

Why are we so interested in Dalitz plots?●●●–Condition for DCPV: |A/A|≠1Need A– and A to consist of (at least) two partsProblem with two-body decays:– with different weak (φ) and strong (δ) phases● 2 observables (B, A CP)Often realised by “tree” and “penguin” diagrams● 4 unknowns (|T|, |P|, δ T– δ P, φ T– φ P)A = ∣T∣e i T − T i∣P∣e P − P A = ∣T∣e i T T i∣P∣e P P 2 ∣T∣∣P∣sin T− P sin T− PA CP = ∣A∣2 −∣A∣ 2∣A∣ 2 ∣A∣ 2 =∣T∣ 2 ∣P∣ 2 2 ∣T∣∣P∣cos T− Pcos T− PExample: B→Kπ(weak phase difference is γ)Tim GershonIntroduction to Dalitz Plot Analysis10

Direct CP violation in B→Kπ● Direct CP violation in B→Kπ sensitive to γtoo many hadronic parameters ⇒ need theory inputNB. interesting deviation from naïve expectation“Kπ puzzle”A CP K − =−9.8 1.2 −1.1% A CP K − 0 =5.0±2.5%A CP=−14.8±2.8%HFAG averagesBABAR PRD 76 (2007) 091102 & arXiv:0807.4226; also CDFand now LHCb!(results not in average yet)Could be a sign of new physics …… but need to rule out possibility of largerthan expected QCD correctionsBelle Nature 452 (2008) 332Tim GershonIntroduction to Dalitz Plot Analysis11

Dalitz plot analysis●●●Amplitude analysis to extract directly information relatedto the phase at each Dalitz plot positionMost commonly performed in the “isobar model”●●●sum of interfering resonanceseach described by Breit-Wigner (or similar) lineshapes,spin terms, etc.fit can be unbinned, but has inherent model dependenceAlternative approaches aiming to avoid modeldependence usually involve binning●partial wave analysisTim GershonIntroduction to Dalitz Plot Analysis12

Pros and cons of Dalitz plots●●Pros●●●Cons●More observables (B & A CPat each Dalitz plot point)Using isobar formalism, can express total amplitude as coherent sum ofquasi-two-body contributionsAm 122 ,m 232 =∑ rc rF rm 12– where c r& F rcontain the weak and strong physics, respectively– n.b. each c ris itself a sum of contributions from tree, penguin, etc.Interference provides additional sensitivity to CP violationNeed to understand hadronic (F r) factors2 ,m 2 23●– lineshapes, angular terms, barrier factors, ...Isobar formalism only an approximation● Model dependenceTim GershonIntroduction to Dalitz Plot Analysis13

Model includes:● K ∗ 0 (892)π+, K *0 2(1430)π +●(Kπ) 0 ∗ π + (LASS lineshape)B + →K + π + π – BaBar PRD 78 (2008) 012004See also Belle PRL 96 (2006) 251803● ρ 0 (770)K + , ω(782)K + , f 0(980)K + , f 2(1270)K + , χ c0K +●f X(1300)K + , phase-space nonresonantTim GershonIntroduction to Dalitz Plot Analysis14

Model includes:● K ∗ 0 (892)π+, K *0 2(1430)π +●(Kπ) 0 ∗ π + (LASS lineshape)B + →K + π + π – BaBar PRD 78 (2008) 012004See also Belle PRL 96 (2006) 251803● ρ 0 (770)K + , ω(782)K + , f 0(980)K + , f 2(1270)K + , χ c0K +●f X(1300)K + , phase-space nonresonantTim GershonIntroduction to Dalitz Plot AnalysisEvidence for direct CP violationBut significant model dependence15

B + →K + π + π – BaBar PRD 78 (2008) 012004Evidence for direct CP violationModel includes:But significant model dependence● K ∗ 0 (892)π+, K *0 2(1430)π +● (Kπ) ∗ 0π + Huge potential for LHCb ...(LASS lineshape)… but need to reduce model dependence● ρ 0 (770)K + , ω(782)K + , f 0(980)K + , f 2(1270)K + , χ c0K +●f X(1300)K + , phase-space nonresonantTim GershonNabis-related stuffSee also Belle PRL 96 (2006) 25180316

Sources of model dependence●●●Lineshapes●coupled channels, threshold effects, etc.Isobar formalism●●“sum of Breit-Wigners” model violates unitarityproblem most severe for broad, overlapping resonances– even talking about “mass” and “width” for such states is not strictlycorrect (process dependent) – can only be defined by pole positionNonresonant contributions●●such terms are small for D decays, but are found to be large forsome B decays (not well understood why)interference with other (S-wave) terms can lead to unphysicalphase variationsTim GershonIntroduction to Dalitz Plot Analysis17

Are methods used for D decayDalitz plots also valid for B decays?D→K – π + π 0 B→K – π + π 0Same modelas D decayTim GershonIntroduction to Dalitz Plot AnalysisImage credit: Brian MeadowsD Dalitz ploton same scale18

How to address modeldependence?●●Model independent methods●●●Dalitz plot anisotropy (“Miranda” method)binned D Dalitz plots for measurement of γ in B→DK (andcharm mixing & CP violation)partial wave analysisMore robust (but still model dependent) methods●●●K-matrix approachuse of scattering data to constrain phase variation (Watson'stheorem) – e.g. LASS shape for (Kπ) S-waveinput from theory (chiral symmetry, dispersion relations)Tim GershonIntroduction Nabis-related to Dalitz Plot stuff AnalysisMore details in talks of Christoph Hanhart andLeonard Lesniak in parallel session on Thursday19

“Miranda” procedurea.k.a. Dalitz plot anisotropyPRD 80 (2009) 096006Toy model (using B + →K + π + π – )Without CP violation With CP violationGaussianNot Gaussian●●●Good model-independent way to identify CP violation● could be sufficient to identify non-SM physics in, e.g., charm decaysConstant (DP independent) systematic asymmetries can be accounted forCan isolate region of the Dalitz plot where CP violation effects occurTim GershonIntroduction to Dalitz Plot AnalysisBut does not provide quantitative measure of weak phase20

Example partial wave analysis:D s+→K + K – π + (BaBar)Plot m(K + K - ),, weighting events by factorsY L0(cos θ ΚΚ )/ε to obtain “moments“”φ(1020)~101K Events96% purityK* 0 (892) π +D s+θ KKTim GershonIntroduction to Dalitz Plot Analysis21

Example partial wave analysis:D s + →K + K – π + (BaBar)(Approximately) model-independent informationon the KK S-wave magnitude and phaseAmbiguity in φ SPresolvedby knowledge of φ(1020)phase variationSeems processindependent?From K + K -cross channelTim GershonIntroduction to Dalitz Plot Analysis22

Quasi-model-independent partialwave analysis● Pioneered by E791 (B.Meadows) in D + →K – π + π +●Describe S-wave by complex spline (many freeparameters)● Example: D s + →π – π + π + from BaBarMagnitudePhase (radians)f 0(980)Symmetrized Dalitz plot13K eventsf 0(980)f 0(980)Tim GershonIntroduction to Dalitz Plot Analysis23

ππ S-wave comparisonData points from BaBarTim GershonIntroduction to Dalitz Plot Analysis24

ππ S-wave comparisonData points from BaBarTim GershonIntroduction to Dalitz Plot AnalysisPrediction from theory:Kaminski et al. PRD77:054015,200825

Extracting weak phases from Dalitz plots●Many methods exist in the literature●some have been used by the B factories●●– most results are statistically limited– in some cases model uncertainties are an issuestill plenty of room for good new ideasExamples (there are many more!)● Snyder-Quinn method to measure α from B → π + π – π 0● GGSZ/BP method to measure γ from B ± → DK ± with D → K Sπ + π –● Measurement of charm oscillation parameters using D → K Sπ + π –●Various methods to measure γ from three-body charmless B decays(B {u,d,s}→ πππ, Kππ, KKπ, KKK)●Penguin-free measurements of β & β sfrom Dπ + π – & DK + K – , respectivelyTim GershonIntroduction to Dalitz Plot Analysis26

B→π + π – π 0 – B factory results●Results from●Belle, 449 M BB pairs: PRL 98 (2007) 221602, PRD 77 (2008) 072001●BaBar, 375 M BB pairs: PRD 76 (2007) 012004Tim GershonWeak Introduction Phases to Dalitz from Plot Dalitz Analysis Plots ρ + π – ρ – π + ρ 0 π 027

B→π + π – π 0 – B factory results●Results from●Belle, 449 M BB pairs: PRL 98 (2007) 221602, PRD 77 (2008) 072001●BaBar, 375 M BB pairs: PRD 76 (2007) 012004Contour from B→π + π – π 0 onlyTim GershonWeak Introduction Phases to Dalitz from Plot Dalitz AnalysisPlotsIncluding also information onB + →ρ + π 0 and B + →ρ 0 π +28

B ± → DK ± with D → K Sπ + π –Interference between b → u and b → c amplitudes when D is reconstructed–in final state common to D 0 and D 0 provides sensitivity to γModel (f D(m + 2 , m – 2 )) taken from measurements of |f D| 2 using flavourtagging D* decays – model dependencePRL 105 (2010) 121801 PRD 81 (2010) 112002Tim GershonWeak Introduction Phases to Dalitz from Plot Dalitz AnalysisPlots29

B ± → DK ± with D → K Sπ + π ––Solution – bin the Dalitz plot and use ψ(3770) → DD events (CLEOc,BES) to measure per-bin phasesPRD 68, 054018 (2003), EPJ C 47, 347 (2006); EPJ C 55, 51 (2008)(unusual bin shapes to attempt to optimise sensitivity)c i, s imeasured by CLEOPRD 82, 112006 (2010)Tim GershonWeak Introduction Phases to Dalitz from Plot Dalitz AnalysisPlotsFirst model independent measurementof γ in this mode by Belle30

Looking ahead●●I hope I have already convinced you that there is greatphysics potential in Dalitz plot analysisBut there are also considerable challenges●●theoretical (hadronic effects, etc.)experimental– Dalitz plot dependent effects:– Efficiency– Background– Mass resolution– Misreconstruction effects …● I leave the details to the parallel session speakers, but ...Tim GershonWeak Introduction Phases to Dalitz from Plot Dalitz AnalysisPlots31

Motivation for “Les Nabis”●From I. Bigi, “Hadronization — the Unsung Hero rather than the alleged Villainin the Tale of CP Violation”, proceedings of Hadron '05●“We expect confidently that New Physics surfaces at the TeV scale. Yet wehave to aim beyond ‘merely’ establishing the existence of New Physics – ourgoal has to be to identify its salient features. The discovery potential in Bdecays is essential, not a luxury. Yet due to the past ‘unlikely’ success of theCKM description one cannot count on massive manifestations of New Physics.Therefore we need high accuracy both on the experimental and theoretical sidein heavy flavour studies. This requires a better quantitative understanding ofhadronization to exhaust the discovery potential in B decays.”●“The expertise required to attain an essential goal – namely to exhaust thediscovery potential in heavy flavour transitions by harnessing low-energyhadronization – does exist or can be acquired with no need for a breakthrough.However it tends to reside in a community all too often disjoint from the heavyflavour community – this has to change!”Tim GershonIntroduction to Dalitz Plot Analysis32

Erm, What is “Les Nabis”?●●●●A group of experimentalists and theorists, drawn from bothparticle/heavy flavour and nuclear/hadronic physicscommunities, set up to address issues in Dalitz plotanalysis with the aim of obtaining both qualitative andquantitative information on weak phases and CP violationTheory:●I. Bigi, S. Gardner, C. Hanhart, B. Kubis, T. Mannel, U.-G.Meißner, J. Pelaez, M. PenningtonExperiment:●I. Bediaga, A. Bondar, A. Denig, T. Gershon, W. Gradl, B.Meadows, K. Peters, U. Wiedner, G. Wilkinson(This is the list on the Nabis web page – apologies for any omissions)Tim GershonIntroduction to Dalitz Plot Analysis33

OK, but why is it called “Les Nabis”?●From the manifesto (yes, really):●“Les Nabis” means “the Prophets” in Hebrew, and this name had beenadopted by a group of post-impressionist artists who set the pace forpaintings and graphics in France in the 1890s. With some of usharbouring a similar sense of modesty as those artists, we found a reincarnationof “Les Nabis” an appropriate motto for our group and itsefforts. This is further strengthened by the fact that one work by a centralmember of the original “Les Nabis” seems to show a (somewhatartistically enhanced) image of a Dalitz plot – it is “The Talisman” by PaulSérusier. We have chosen it as our logo.Tim GershonIntroduction to Dalitz Plot Analysis“The Talisman”, Paul Sérusier34

Summer school●●School on Amplitude Analysis in Modern Physics:from hadron spectroscopy to CP phasesAugust 1-5, 2011 at Physikzentrum Bad Honnef, Germany●http://www2.fz-juelich.de/ikp/workshops/LesNabis/index.shtmlTim GershonIntroduction to Dalitz Plot Analysis35

Summary●●●●●Dalitz plot analyses provide promising methods tomeasure weak phases and CP violationMany attractive features …… but significant complications due to modeldependenceNeed progress on several fronts– Understand better (ππ), (Kπ), (KK), (Dπ), (DK) systems– “Nonresonant” contributions and 3-body unitarity– Methods to combat model-dependence– Nabis initiative set up to try to address thisMany new possibilities opening up with LHCbTim GershonIntroduction to Dalitz Plot Analysis36

For more details, please come tothe parallel sessionThursday 28 April 2011Tim GershonIntroduction to Dalitz Plot Analysis37