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Partial Wave Analysis Lectures at the School on Hadron Physics Klaus Peters Ruhr Universität Bochum Varenna, June 2004 E. Fermi CLVII Course

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2 Overview Introduction and Concepts Spin Formalisms Dynamical Functions Technical Issues

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3 Overview – Introduction and Concepts Overview Introduction and Concepts Spin Formalisms Dynamical Functions Technical Issues Goals Wave Approach Isobar-Model Level of Detail

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4 Overview – Spin Formalisms Overview Introduction and Concepts Spin Formalisms Dynamical Functions Technical Issues Overview Zemach Formalism Canonical Formalism Helicity Formalism Moments Analysis

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5 Overview - Dynamical Functions Overview Introduction and Concepts Spin Formalisms Dynamical Functions Technical Issues Breit-Wigner S-/T-Matrix K-Matrix P/Q-Vector N/D-Method Barrier Factors Interpretation

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6 Overview – Technical Issues / Fitting Overview Introduction and Concepts Spin Formalisms Dynamical Functions Technical Issues Coding Amplitudes Speed is an Issue Fitting Methods Caveats FAQ

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7 Header – Introduction and Concepts Introduction & Concepts Introduction and Concepts Spin Formalisms Dynamical Functions Technical issues

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8 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 provide toolkit to analyze heavy quark processes thus an important tool 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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9 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 dont need just a good description, you need the right one Many solutions may look alike but only one is right

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10 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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11 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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12 J/ψ π + π - π 0 Angular distributions are easily seen in the Dalitz plot cosθ 0+1

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13 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 densityρ = 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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14 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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15 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 but only if spins are properly assigned bias due to hypothetical spin-parity assumption Optics Dalitz plot

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16 Its All a Question of Statistics... pp 3 0 with 100 events

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17 Its All a Question of Statistics pp 3 0 with 100 events 1000 events

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18 Its All a Question of Statistics pp 3 0 with 100 events 1000 events events

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19 Its All a Question of Statistics pp 3 0 with 100 events 1000 events events events

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20 Experimental Techniques Scattering Experiments πN - N* measurement πN - meson spectroscopy E818, AGS, GAMS pp meson threshold production Celsius, COSY pp or πp in the central region WA76, WA91, WA102 γN – photo production Cebaf, Mami, Elsa, Graal At-rest Experiments rest at LEAR Asterix, Obelix, Crystal Barrel J/ψ decays MarkIII,DM2,BES,CLEO-c ф(1020) decays Dafne, VEPP D and D s decays FNAL, Babar, Belle

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21 Introducing Partial Waves Schrödingers Equation Angular Amplitude Dynamic Amplitude

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22 Argand Plot

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23 Standard Breit-Wigner Full circle in the Argand Plot Phase motion from 0 to π Intensity I=ΨΨ * Phase δSpeed dφ/dm Argand Plot

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24 Breit-Wigner in the Real World e + e -ππ m ππ ρ-ωρ-ω

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25 Dynamical Functions are Complicated 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 distort line shapes due to available phase space A more general approach is needed for a detailed understanding (see last lecture!)

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26 Isobar Model 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 We need need two-body spin-algebra various formalisms need two-body scattering formalism final state interaction, e.g. Breit-Wigner Isobar

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27 The Full Amplitude For each node an amplitude f(I,I 3,s,Ω) is obtained. The full amplitude is the sum of all nodes. Summed over all unobservables

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28 Example: Isospin Dependence pp initial states differ in isospin Calculate isospin Clebsch-Gordan 1 S 0 destructive interferences 3 S 1 ρ 0 π 0 forbidden

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29 Header – Spin Formalisms Spin Formalisms Introduction and Concepts Spin Formalisms Dynamical Functions Technical Issues

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30 Formalisms – on overview Tensor formalisms in non-relativistic (Zemach) or covariant form Fast computation, simple for small L and S Spin-projection formalisms where a quantization axis is chosen and proper rotations are used to define a two-body decay Efficient formalisms, even large L and S easy to handle Formalisms based on Lorentz invariants (Rarita-Schwinger) where each operator is constructed from Mandelstam variables only Elegant, but extremely difficult for large L and S

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31 How To Construct a Formalism Key steps are Definition of single particle states of given momentum and spin component (momentum-states), Definition of two-particle momentum-states in the s-channel center-of-mass system and of amplitudes between them, Transformation to states and amplitudes of given total angular momentum (J-states), Symmetry restrictions on the amplitudes, Derive Formulae for observable quantities.

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32 Zemach Formalism For particle with spin S traceless tensor of rank S with indices Similar for orbital angular momentum L

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33 Example: Zemach – pp (0 -+ )f 2 π 0 Construct total spin 0 amplitude Angular distribution (Intensity) A=A f 2 π x A ππ

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34 The Original Zemach Paper

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35 Spin-Projection Formalisms Differ in choice of quantization axis Helicity Formalism parallel to its own direction of motion Transversity Formalism the component normal to the scattering plane is used Canonical (Orbital) Formalism the component m in the incident z-direction is diagonal

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36 Generalized Single Particle State In general all single particle states are derived from a lorentz transformation and the rotation of the basic state with the Wigner rotation

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37 Properties HelicityTransversityCanonical propertypossibility/simplicity partial wave expansionsimplecomplicated parity conservationnoyes crossing relationnogoodbad specification of kinematical constraints noyes

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38 Rotation of States Canonical SystemHelicity System

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39 Single Particle State Canonical 1) momentum vector is rotated via z-direction. Secondly 2) absolute value of the momentum is Lorentz boosted along z 3) z-axis is rotated to the momentum direction

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40 Single Particle State Helicity 1) z-axis is rotated to the momentum direction 2) Lorentz Boost Therefore the new z-axis, z, is parallel to the momentum

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41 Two-Particle State Canonical constructed from two single-particle states (back-to-back) Couple s and t to S Couple L and S to J Spherical Harmonics

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42 Two-Particle State Helicity similar procedure no recoupling needed normalization

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43 Completeness and Normalization Canonical completeness normalization Helicity completeness normalization

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44 Canonical From two-particle state LS-Coefficients Canonical Decay Amplitudes

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45 Helicity Decay Amplitudes Helicity From two-particle state Helicity amplitude

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46 Spin Density and Observed Number of Events To finally calculate the intensity i.e. the number of events observed Spin density of the initial state Sum over all unobservables taking into account

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47 Relations Canonical Helicity Recoupling coefficients Start with Canonical to Helicity Helicity to Canonical

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48 Clebsch-Gordan Tables Clebsch-Gordan Coefficients are usually tabled in a graphical form (like in the PDG) Two cases coupling two initial particles with |j 1 m 1 > and |j 2 m 2 > to final system

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49 Using Clebsch-Gordan Tables, Case 1 1 x / /2-1/ /61/21/3 002/30-1/ /6-1/21/3 01/2 2 01/2-1/2-2 1

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50 1 x / /2-1/ /61/21/3 002/30-1/ /6-1/21/3 01/2 2 01/2-1/2-2 1 Using Clebsch-Gordan Tables, Case 2

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51 Parity Transformation and Conservation Parity transformation single particle two particles helicity amplitude relations (for P conservation)

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52 f 2ππ (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=2s π =0 Ansatz

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53 f 2ππ (Rates) Amplitude has to be symmetrized because of the final state particles

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54 ω π 0 γ (Ansatz) Initial:ωI G (J PC ) = 0 - (1 -- ) Final:π 0 I G (J PC ) = 1 - (0 -+ ) γI G (J PC ) = 0(1 -- ) Only odd angular momenta, since η ω =η π η γ (-1) l Only photon contributes to total spin s=s π +s γ Ansatz

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55 ω π 0 γ (Rates) λ γ =±1 do not interfere, λ γ =0 does not exist for real photons Rate depends on density matrix Choose uniform density matrix as an example

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56 f 0,2γγ (Ansatz) Initial:f 0,2 I G (J PC ) = 0 + (0,2 ++ ) Final:γI G (J PC ) = 0(1 -- ) Only even angular momenta, since η f =η γ 2 (-1) l Total spin s=2s γ =2, l=0,2 (f 0 ), l=0,2,4 (f 2 ) Ansatz

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57 f 0,2 γγ (contd) Ratio between a 00 and a 22 is not measurable Problem even worse for J=2

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58 f 0,2γγ (contd) Usual assumption J=λ=2

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59 pp (2 ++ ) ππ Proton antiproton in flight into two pseudo scalars Initial:ppJ,M=0,±1 Final:πI G (J PC ) = 1 - (0 -+ ) Ansatz Problem: d-functions are not orthogonal, if φ is not observed ambiguities remain in the amplitude – polarization is needed

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60 pp π 0 ω Two step process First step ppπ 0 ω - Second step ωπ 0 γ Combine the amplitudes helicity constant a ω,11 factorizes and is unimportant for angular distributions

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61 pp (0 -+ ) f 2 π 0 Initial:ppI G (J PC ) = 1 - (0 -+ ) Final:f 2 (1270) I G (J PC ) = 0 + (2 ++ ) π 0 I G (J PC ) = 1 - (0 -+ ) is only possible from L=2 Ansatz

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62 General Statements Flat angular distributions General rules for spin 0 initial state has spin 0 0 any both final state particles have spin 0 J 0+0 Special rules for isotropic density matrix and unobserved azimuth angle one final state particle has spin 0 and the second carries the same spin as the initial state J J+0

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63 Moments Analysis Consider reaction Total differential cross section expand H leading to

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64 Moments Analysis contd Define now a density tensor the d-function products can be expanded in spherical harmonics and the density matrix gets absorbed in a spherical moment

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65 Example: Where to start in Dalitz plot anlysis Sometimes a moment-analysis can help to find important contributions best suited if no crossing bands occur D 0 K S K + K -

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66 Proton-Antiproton Rest Atomic initial system formation at high n, l (n~30) slow radiative transitions de-excitation through collisions (Auger effect) Stark mixing of l-levels (Day, Snow, Sucher 1960) Advantages J PC varies with target density isospin varies with n (d) or p target incoherent initial states unambiguous PWA possible Disadvantages phase space very limited small kaon yield L L K (1%) rad. Transition Stark-Effect ext. Auger-Effect S-Wave P-Wave (99% of 2P) Annihilation S PD F n=4 n=3 n=2 n=1

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67 Initial Rest Quantumnumbers G=(-1) I+L+S P=(-1) L+1 C=(-1) L+S CP=(-1) 2L+S+1 I=0 I=1 J PC IGIG LS 1S01S pseudo scalar1 - ; S13S vector1 + ; P11P axial vector1 + ; P03P scalar1 - ; P13P axial vector1 - ; P23P tensor1 - ;0 + 11

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68 Proton-Antiproton Annihilation in Flight Annihilation in flight scattering process: no well defined initial state maximum angular momentum rises with energy Advantages larger phase space formation experiments Disadvantages many waves interfere with each other many waves due to large phase space

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69 Scattering Amplitudes in Flight (I) pp helicity amplitude only H ++ and H +- exist C-Invariance H ++ =0 if L+S-J odd CP-Invariance H +- =0 if S=0 and/or J=0 CP transform CP=(-1) 2L+S+1 S and CP directly correlated CP conserved in strong int. singlet and triplet decoupled C transform L and P directly correlated C conserved in strong int. (if total charge is q=0) odd and even L decouples 4 incoherent sets of coherent amplitudes

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70 Scattering Amplitudes in Flight (II) Singlett even L J PC LSH ++ H +- 1S01S YesNo 1D21D YesNo 1G41G YesNo Triplett even L J PC LSH ++ H +- 3S13S Yes 3D13D Yes 3D23D Yes 3D33D Yes Singlett odd L J PC LSH ++ H +- 1P11P YesNo 1F31F YesNo 1G51G YesNo Triplett odd L J PC LSH ++ H +- 3P03P YesNo 3P13P NoYes 3P23P Yes 3F23F YesNo 3F33F NoYes 3F43F Yes

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71 Header – Dynamical Functions Dynamical Functions Introduction and concepts Spin Formalisms Dynamical Functions Technical issues

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72 S-Matrix Differential cross section Scattering amplitude Total scattering cross section S-Matrix with and

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73 Harmonic Oscillator Free oscillator Damped oscillator Solution External periodic force Oscillation strength and phase shift Lorentz function

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74 T-Matrix from Scattering Back to Schrödingers equation Incoming wave Solves the equation solution without interaction solution with interaction incoming wave outgoing wave inelasticity and phase shift

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75 T-Matrix from Scattering (contd) Scattering wave function Scattering amplitude and T-Matrix Example: ππ-Scattering below 1 GeV/c 2

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76 (In-)Elastic cross sections and T-Matrix Total cross section Identify elastic and inelastic part using the optical theorem

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77 Breit-Wigner Function Wave function for an unstable particle Fourier transformation for E dependence Finally our first Breit-Wigner

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78 Dressed Resonances – T-Matrix & Field Theory Suppose we have a resonance with mass m 0 We can describe this with a propagator But we may have a self-energy term leading to

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79 T-Matrix Perturbation We can have an infinite number of loops inside our propagator every loop involves a coupling b, so if b is small, this converges like a geometric series +... = + +

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80 T-Matrix Perturbation – Retaining Breit-Wigner So we get and the full amplitude with a dressed propagator leads to which is again a Breit-Wigner like function, but the bare energy E 0 has now changed into E 0 - < {b}

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81 Relativistic Breit-Wigner By migrating from Schrödingers equation (non-relativistic) to Klein-Gordons equation (relativistic) the energy term changes different energy-momentum relation E=p 2 /m vs. E 2 =m 2 c 4 +p 2 c 2 The propagators change to s R -s from m R -m Intensity I=ΨΨ * Phase δArgand Plot

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82 Barrier Factors - Introduction At low energies, near thresholds but is not valid far away from thresholds -- otherwise the width would explode and the integral of the Breit-Wigner diverges Need more realistic centrifugal barriers known as Blatt-Weisskopf damping factors We start with the semi-classical impact parameter and use the approximation for the stationary solution of the radial differential equation with we obtain

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83 Blatt-Weisskopf Barrier Factors The energy dependence is usually parameterized in terms of spherical Hankel-Functions we define F l (q) with the following features Main problem is the choice of the scale parameter q R =q scale

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84 Blatt-Weisskopf Barrier Factors (l=0 to 3) Usage

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85 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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86 T-Matrix Unitarity Relations Unitarity is a basic feature since probability has to be conserved T is unitary if S is unitary since we get in addition

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87 T-Matrix Dispersion Relations Cauchy Integral on a closed contour By choosing proper contours and some limits one obtains the dispersion relation for T l (s) Satisfying this relation with an arbitrary parameterization is extremely difficult and is dropped in many approaches

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88 K-Matrix Definition T is n x n matrix representing n incoming and n outgoing channel If the matrix K is a real and symmetric also n x n then the T is unitary by construction

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89 K-Matrix Properties T is then easily computed from K T and K commute K is the Caley transform of S Some more properties

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90 Example: ππ-Scattering 1 channel2 channel

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91 Relativistic Treatment So far we did not care about relativistic kinematics covariant description or and with therefore and K is changed as well

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92 Relativistic Treatment – 2 channel S-Matrix 2 channel T-Matrix to be compared with the non-relativistic case

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93 K-Matrix Poles Now we introduce resonances as poles (propagators) One may add c ij a real polynomial of m 2 to account for slowly varying background (not experimental background!!!) Width/Lifetime For a single channel and one pole we get

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94 Example: 1x2 K-Matrix Strange effects in subdominant channels Scalar resonance at 1500 MeV/c 2, Γ=100 MeV/c 2 All plots show ππ channel Blue: ππ dominated resonance (Γ ππ =80 MeV and Γ KK =20 MeV) Red: KK dominated resonance (Γ KK =80 MeV and Γ ππ =20 MeV) Look at the tiny phase motion in the subdominant channel Intensity I=ΨΨ * Phase δArgand Plot

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95 Example: 2x1 K-Matrix Overlapping Poles two resonances overlapping with different (100/50 MeV/c 2 ) widths are not so dramatic (except the strength) The width is basically added FWHM 2 BW K-Matrix

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96 Example: 1x2 K-Matrix Nearby Poles Two nearby poles (1.27 and 1.5 GeV/c 2 ) show nicely the effect of unitarization 2 BW K-Matrix

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97 Example: Flatté 1x2 K-Matrix 2 channels for a single resonance at the threshold of one of the channels with Leading to the T-Matrix and with we get

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98 Flatté Example a 0 (980) decaying into πη and KK BW πη Flatte πη Flatte KK Intensity I=ΨΨ * Phase δ Real Part Argand Plot

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99 Example: K-Matrix Parametrizations Au, Morgan and Pennington (1987) Amsler et al. (1995) Anisovich and Sarantsev (2003)

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100 P-Vector Definition But in many reactions there is no scattering process but a production process, a resonance is produced with a certain strength and then decays Aitchison (1972) with

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101 P-Vector Poles The resonance poles are constructed as in the K-Matrix and one may add a polynomial d i again For a single channel and a single pole If the K-Matrix contains fake poles... for non s-channel processes modeled in an s-channel model...the corresponding poles in P are different

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102 Q-Vector A different Ansatz with a different picture: channel n is produced and undergoes final state interaction For channel 1 in 2 channels

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103 Complex Analysis Revisited The Breit-Wigner example shows, that Γ(m) implies ρ(m) but below threshold ρ(m) gets complex because q (breakup-momentum) gets complex, since m 1 +m 2 >m therefore the real part of the denominator (mass term) changes and imaginary part (width term) vanishes completely

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104 Complex Analysis Revisited (contd) But furthermore for each ρ(m) which is a squareroot, one has two solutions for p>0 or p<0 resp. But the two values (w=2q/m) have some phase in between and are not identical So you define a new complex plane for each solution, which are 2 n complex planes, called Riemann sheets they are continuously connected. The borderlines are called CUTS.

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105 Riemann Sheets in a 2 Channel Problem Usual definition sheetsgn(q 1 )sgn(q 2 ) I + + II - + III - - IV + + This implies for the T-Matrix Complex Energy Plane Complex Momentum Plane

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106 States on Energy Sheets Singularities appear naturally where Singularities might be 1 – bound states 2 – anti-bound states 3 – resonances or artifacts due to wrong treatment of the model

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107 States on Momentum Sheets Or in the complex momentum plane Singularities might be 1 – bound states 2 – anti-bound states 3 – resonances

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108 Left-hand and Right-hand Cuts The right hand CUTS (RHC) come from the open channels in an n channel problem But also exchange processes and other effects introduce CUTS on the left-hand side (LHC)

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109 N/D Method To get the proper behavior for the left-hand cuts Use N l (s) and D l (s) which are correlated by dispersion relations An example for this is the work of Bugg and Zhou (1993)

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110 Nearest Pole Determines Real Axis The pole nearest to the real axis or more clearly to a point with mass m on the real axis determines your physics results Far away from thresholds this works nicely At thresholds, the world is more complicated While ρ(770) in between two thresholds has a beautiful shape the f 0 (980) or a 0 (980) have not

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111 Pole and Shadows near Threshold (2 Channel) For a real resonance one always obtains poles on sheet II and III due to symmetries in T l Usually To make sure that pole an shadow match and form an s-channel resonance, it is mandatory to check if the pole on sheets II and III match This is done by artificially changing ρ 2 smoothly from q 2 to –q 2

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112 Summary Id like to thank the organizers U. Wiedner and T. Bressani for their kind invitation to Varenna and for the pressure to prepare the lecture and to write it down for later use I also would like to thank S.U. Chung and M.R. Pennington for teaching me, what I hopped to have taught you! and finally Id like to thank R.S. Longacre, from whom I have stolen some paragraphs from his Lecture in Maryland 1991 Acknowledgements

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113 Style 1 Style 2 Style 3 Style 4 Style 5 Style 6 Style 1 Style 2 Style 1 Style 2 Style 3 Style 1 Style 2 Style 3 Style 1 Style 2 Style 3 Intensity I=ΨΨ * Inelasticity η Phase δ Polarangle φ Real Part Imaginary Part Speed dφ/dm Argand Plot Legend Master

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