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Dalitz Plot Analysis Techniques Klaus Peters Ruhr Universität Bochum Cornell, May 6, 2004

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2 Klaus Peters - Dalitz Plot Analysis Techniques Overview Introduction and concepts Dynamical aspects Limitations of the models Technical issues

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3Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques What is the mission ? Particle physics at small distances is well understood One Boson Exchange, Heavy Quark Limits This is not true at large distances Hadronization, Light mesons are barely understood compared to their abundance Understanding interaction/dynamics of light hadrons will improve our knowledge about non-perturbative QCD parameterizations will give a toolkit to analyze heavy quark processes thus an important tools also for precise standard model tests We need Appropriate parameterizations for the multi-particle phase space A translation from the parameterizations to effective degrees of freedom for a deeper understanding of QCD

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4Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Goal For whatever you need the parameterization of the n-Particle phase space It contains the static properties of the unstable (resonant) particles within the decay chain like mass width spin and parities as well as properties of the initial state and some constraints from the experimental setup/measurement The main problem is, you don‘t need just a good description, you need the right one Many solutions may look alike but only one is right

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5Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Initial State Mixing - p p Annihilation in Flight p p Annihilation in Flight scattering process: no well defined initial state maximum angular momentum rises with energy Heavy Quark Decays Weak Decays B 0 3 π D 0 K S ππ

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6Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Intermediate State Mixing Many states may contribute to a final state not only ones with well defined (already measured) properties not only expected ones Many mixing parameters are poorly known K-phases SU(3) phases In addition also D/S mixing (b 1, a 1 decays)

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7Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Introduction & Concepts Introduction and concepts Dynamical aspects Limitations of the models Technical issues

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8Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques n-Particle Phase space, n=3 2 Observables From four vectors12 Conservation laws-4 Meson masses-3 Free rotation-3 Σ2 Usual choice Invariant mass m 12 Invariant mass m 13 π3π3 π2π2 p π1π1 Dalitz plot

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9Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Phase Space Plot - Dalitz Plot dN ~ (E 1 dE 1 ) (E 2 dE 2 ) (E 3 dE 3 )/(E 1 E 2 E 3 ) Energy conservation E 3 = E tot -E 1 -E 2 Phase space density ρ = dN/dE tot ~ dE 1 dE 2 Kinetic energies Q=T 1 +T 2 +T 3 Plotx=(T 2 -T 1 )/ √3 y=T 3 -Q/3 Flat, if no dynamics is involved Q small Q large

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10Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques The first plots τ/θ-Puzzle Dalitz applied it first to K L -decays The former τ/θ puzzle with only a few events goal was to determine spin and parity And he never called them Dalitz plots

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11Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Zemach Formalism Refs Phys Rev 133, B1201 (1964), Phys Rev 140, B97 (1965), Phys Rev 140, B109 (1965) Amplitude M = Σ i M I,i M F,i M JP,i M I,i = isospin dependence M F,i = form factors M JP,i = spin-parity factors Tensors (M JP spin-parity factors) 0 T = 1 1 T i = t i 2 T ij = (3/2)-1/2 [t i t j - (1/3) t 2 δ ij ] Formalism Multiply tensors for each angular momentum involved and contract over unobservable indices

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12Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Interference problem PWA The phase space diagram in hadron physics shows a pattern due to interference and spin effects This is the unbiased measurement What has to be determined ? Analogy Optics PWA # lamps # level # slits # resonances positions of slits masses sizes of slits widths bias due to hypothetical spin-parity assumption Optics I(x)=|A 1 (x)+A 2 (x)e -iφ | 2 Dalitz plot I(m)=|A 1 (m)+A 2 (m)e -iφ | 2 but only if spins are properly assigned

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13Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques It’s All a Question of Statistics... p p 3 0 with 100 events

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14Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques It’s All a Question of Statistics p p 3 0 with 100 events 1000 events

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15Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques It’s All a Question of Statistics p p 3 0 with 100 events 1000 events events

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16Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques It’s All a Question of Statistics p p 3 0 with 100 events 1000 events events events

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17Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Isobar model Go back to Zemach‘s approach M = Σ i M I,i M F,i M JP,i M I,i = isospin dependence M F,i = form factors M JP,i = spin-parity factors Generalization construct any many-body system as a tree of subsequent two-body decays the overall process is dominated by two-body processes the two-body systems behave identical in each reaction different initial states may interfere need two-body „spin“-algebra various formalisms need two-body scattering formalism final state interaction, e.g. Breit-Wigner s

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18Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Particle Decays - Revisited Fourier-Transform of a short lived state wave-function + decay transformed from time to energy spectrum m ππ ρ-ωρ-ω

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19Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques J/ψ π + π - π 0 cosθ 0+1

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20Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Rotations Single particle states Rotation R Unitary operator U D function represents the rotation in angular momentum space Valid in an inertial system Relativistic state

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21Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques From decays to the helicity amplitude J1+2 Basic idea Two state system constructed from two single particle states Amplitude Transition from | JM > to the observed two body system Helicity amplitude is F and gives the strength of the initial system to split up into the helicities λ 1 and λ 2

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22Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Properties of the helicity amplitude LS Scheme Parity

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23Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Example: f 2 ππ (Ansatz) Initial:f 2 (1270) I G (J PC ) = 0 + (2 ++ ) Final: π 0 I G (J PC ) = 1 - (0 -+ ) Only even angular momenta since η f =η π 2 (-1) l Total spin s=2 s π =0 Ansatz

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24Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Example: f 2 ππ (Rates) Amplitude has to be symmetrized because of the final state particles

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25Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Dynamical Aspects Introduction and concepts Dynamical aspects Limitations of the models Technical issues

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26Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Introduction Search for resonance enhancements is a major tool in meson spectroscopy The Breit-Wigner Formula was derived for a single resonance appearing in a single channel But: Nature is more complicated Resonances decay into several channels Several resonances appear within the same channel Thresholds distorts line-shapes due to available phase space A more general approach is needed for a detailed understanding

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27Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Outline of the Unitarity Approach The only granted feature of an amplitude is UNITARITY Everything which comes in has to get out again no source and no drain of probability Idea: Model a unitary amplitude Realization: n-Rank Matrix of analytic functions, T ij one row (column) for each decay channel What is a resonance? A pole in the complex energy plane T ij (m) with m being complex Parameterizations: e.g. sum of poles

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28Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Unitarity, cont‘d Goal: Find a reasonable parameterization The parameters are used to model the analytic function to follow the data Only a tool to identify the resonances in the complex energy plane Poles and couplings have not always a direct physical meaning Problem: Freedom and unitarity Find an approach where unitarity is preserved by construction Leave a lot of freedom for further extension

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29Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Some Basics Considering two-body processes Scattering amplitude f fi Cross section for a partial wave by integration over Ω Note that T J has no unit, the unit is carried by q i 2 J (spin), M (z-component of J) Conservation of angular momentum preserves J and M s a b c d

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30Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques S-Matrix and Unitarity S fi is the amplitude for an initial state | i > found in the final state | f > An operator K can be defined Caley Transform which is hermitian by construction from time reversal invariance it follows that K is symmetric and commutes with T

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31Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Lorentz Invariance But Lorentz invariance has to be considered introduces phase space factor ρ

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32Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Resonances in K-Matrix Formalism Resonances are introduced as sum of poles, one pole per resonance expected It is possible to parameterize non- resonant backgrounds by additional unit-less real constants or functions c ij Unitarity is still preserved Partial widths are energy dependent

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33Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Blatt-Weisskopf Barrier Factors The energy dependence is usually parameterized in terms of Hankel-Functions Normalization is done that F l (q) = 1 at the pole position Main problem is the choice of the scale parameter q R

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34Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Single Resonance In the simple case of only one resonance in a single channel the classical Breit-Wigner is retained

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35Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Overlapping resonances This simple example of resonances at 1270 MeV/c 2 and 1565 MeV/c 2 illustrate the effect of nearby resonances Sum of Breit-Wigner Sum of K-matrix poles

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36Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Resonances near threshold Line-shapes are strongly distorted by thresholds if strong coupling to the opening channel exists unitarity implies cusp in one channel to give room for the other channel

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37Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Production of Resonances: P-Vector So far only s-channel resonances Generalization for production processes Aitchison approach T is used to propagate the production vector P to the observed amplitude F P contains the same poles as K An arbitrary real function may be added to accommodate for background amplitudes The production vector P has complex strengths β α for each resonance s c d ?

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38Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Coupled channels K-Matrix/P-Vector approach imply coupled channel functionality same intermediate state but different final states Isospin relations (pure hadronic) combine different channels of the same gender, like π + π - and π 0 π 0 (as intermediate states) or combining pp, pn and nn or X 0 KKπ, Example K* in K + K L π - J C =0 + I=0 J C =0 + I=1 J C =1 - I=0 J C =1 - I=1

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39Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Limitations of the models Introduction and concepts Dynamical aspects Limitations of the models Technical issues

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40Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Limits of the Isobar approach The isobar model implies s-channel reactions all two-body combinations undergo FSI FSI dominates all amplitudes can be added coherently Failures of the model t-channel exchange Rescattering s

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41Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques t-channel Effects (also u-channel) They may appear resonant and non-resonant Formally they cannot be used with Isobars But the interaction is among two particles To save the Isobar Ansatz (workaround) they may appear as unphysical poles in K-Matrices or as polynomial of s in K-Matrices background terms in unitary form t

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42Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Rescattering Most severe Problem Example JLab‘s θ + of neutrons No general solution Specific models needed d ? ? + n K + K - p K - p n

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43Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Barrier factors Scales and Formulae formula was derived from a cylindrical potential the scale (197.3 MeV/c) may be different for different processes valid in the vicinity of the pole definition of the breakup- momentum Breakup-momentum may become complex (sub-threshold) set to zero below threshold need =∫F l (q)dBW F l (q)~q l complex even above threshold meaning of mass and width are mixed up Resonant daughters Re(q) Im(q) threshold

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44Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Technical Issues Introduction and concepts Dynamical aspects Limitations of the models Technical issues

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45Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Boundary problem I Most Dalitz plots are symmetric Problem: sharing of events Solution: transform DP

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46Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Boundary Problem II Efficiencies often factorize in mass and angular distribution 2 n d Approach Use mass and cosθ Not always applicable

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47Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Fit methods - χ 2 vs. Likelihood χ 2 small # of independent phase space observables usually not more than 2 High statistics >10k if there are only a few well known resonances >50k for complicated final states with discovery potential e.g. CB found events of the type p p ® 3π 0 -log L more than 2 independent phase space observables low statistics (compared to size of phase space) narrow structures [like Φ (1020) ]

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48Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Adaptive binning cut-off Finite size effects in a bin of the Dalitz plot limited line shape sensitivity for narrow resonances Entry cut-off for bins of a Dalitz plots χ 2 makes no sense for small #entries cut-off usually 10 entries Problems the cut-off method may deplete important regions of the plot to much circumvent this by using a bin-by-bin Poisson-test for these areas alternatively: adaptive Dalitz plots, but one may miss narrow depleted regions, like the f 0 (980) dip systematic choice-of-binning-errors

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49Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Background subtraction and/or fitting Experiments at LEAR did a great job, but backgrounds were low and statistics were extremely high Background was usually not an issue In D (s) -Decays we know this is a severe problem Backgrounds can exceed 50% Approaches Likelihood compensation add log L i of all background events (from sidebands) Background parameterization (added incoherently) combined fit fit to sidebands and fix for Dalitz plot fit Try all to get a feeling on the systematic error

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50Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Finite Resolution Due to resolution or wrong matching: True phase space coordinates of MC events are different from the reconstructed coordinates In principle amplitudes of MC-events have to be calculated at the generated coordinate, not the reconstructed location But they are plotted at the reconstructed location Applies to: Experiments with “bad” resolution (like Asterix) For narrow resonances [like Φ or f 1 (1285) or f 0 (980) ] Wrongly matched tracks Cures phase-smearing and non-isotropic resolution effects

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51Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Strategy Where to start the fit Is one more resonance significant Indications for a bad solution Where to stop the sophistication/fit ? ? ? ?

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52Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Where to start Problem dependent start with obvious signatures Sometimes a moment-analysis can help to find important contributions best suited if no crossing bands occur D0KSK+K-D0KSK+K-

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53Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Is one more resonance significant ? Base your decision on objective bin-by-bin χ 2 and χ 2 /N dof visual quality is the trend right? is there an imbalance between different regions compatibility with expected L structure Produce Toy MC for Likelihood Evaluation many sets with full efficiency and Dalitz plot fit each set of events with various amplitude hypotheses calc L expectation L expectation is usually not just ½/dof sometimes adding a wrong (not necessary) resonance can lead to values over 100! compare this with data Result: a probability for your hypothesis!

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54Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Where to stop Apart from what was said before Additional hypothetical trees (resonances, mechanisms) do not improve the description considerably Don‘t try to parameterize your data with inconsistent techniques If the model don‘t match, the model might be the problem reiterate with a better model

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55Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Indications for a bad solution Apart from what was said before one indication can be a large branching fraction of interference terms Definition of BF of channel j BF j = ∫ |A j | 2 dΩ /∫ | Σ i A i | 2 But due to interferences, something is missing IncoherentI=|A| 2 +|B| 2 CoherentI=|A+e i φ B| 2 = |A| 2 +|B| 2 +2[Re(AB * )sinφ+Im(AB * )cosφ If Σ j BF j is much different from 100% there might be a problem The sum of interference terms must vanish if integrated from -∞ to +∞ But phase space limits this region If the resonances are almost covered by phase space then the argument holds......and large residual interference intensities signal overfitting

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56Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Other important topics Amplitude calculation Symbolic amplitude manipulations (Mathematica) On-the-fly amplitude construction (Tara) CPU demand Minimization strategies and derivatives Coupled channel implementation Variants, Pros and Cons Numerical instabilities Unitarity constraints Constraining ambiguous solutions with external information Constraining resonance parameters systematic impact if wrong masses are used

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57Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques Summary & Outlook Dalitz plot analysis is an important tool for Light and Heavy Hadron spectroscopy CP-Violation studies Multi-body phase space parameterization Stable solutions need High statistics Good angular coverage Good efficiency knowledge High Statistics need Precise modeling Huge amount of CPU and Memory Joint Spin Analysis Group

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