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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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**Overview Overview Introduction and Concepts Spin Formalisms**

Dynamical Functions Technical Issues

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**Overview – Introduction and Concepts**

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

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**Overview – Spin Formalisms**

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

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**Overview - Dynamical Functions**

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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**Overview – Technical Issues / Fitting**

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

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**Header – Introduction and Concepts**

Spin Formalisms Dynamical Functions Technical issues

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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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**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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**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 (b1, a1 decays)

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**n-Particle Phase space, n=3**

2 Observables From four vectors 12 Conservation laws -4 Meson masses -3 Free rotation -3 Σ 2 Usual choice Invariant mass m12 Invariant mass m13 Dalitz plot π1 pp π2 π3

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**J/ψ ® π+π-π0 Angular distributions are easily seen in the Dalitz plot**

cosθ -1 +1

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**Phase Space Plot - Dalitz Plot**

Q small Q large dN ~ (E1dE1) (E2dE2) (E3dE3)/(E1E2E3) Energy conservation E3 = Etot-E1-E2 Phase space density ρ = dN/dEtot ~ dE1 dE2 Kinetic energies Q=T1+T2+T3 Plot x=(T2-T1)/√3 y=T3-Q/3 Flat, if no dynamics is involved

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**The first plots τ/θ-Puzzle**

Dalitz applied it first to KL-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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**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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**It’s All a Question of Statistics ...**

pp ® 3p0 with 100 events Crystal Barrel 3pi0 Plots

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**It’s All a Question of Statistics ... ...**

pp ® 3p0 with 100 events 1000 events

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**It’s All a Question of Statistics ... ... ...**

pp ® 3p0 with 100 events 1000 events 10000 events

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**It’s All a Question of Statistics ... ... ... ...**

pp ® 3p0 with 100 events 1000 events 10000 events events

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**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 Ds decays FNAL, Babar, Belle

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**Introducing Partial Waves**

Schrödinger‘s Equation Angular Amplitude Dynamic Amplitude

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Argand Plot

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**Standard Breit-Wigner**

Full circle in the Argand Plot Phase motion from 0 to π Argand Plot Intensity I=ΨΨ* Phase δ Speed dφ/dm

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**Breit-Wigner in the Real World**

e+e- ® ππ mππ ρ-ω

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**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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**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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**The Full Amplitude For each node an amplitude f(I,I3,s,Ω) is obtained.**

The full amplitude is the sum of all nodes. Summed over all unobservables

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**Example: Isospin Dependence**

pp initial states differ in isospin Calculate isospin Clebsch-Gordan 1S0 destructive interferences 3S1 ρ0π0 forbidden

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**Header – Spin Formalisms**

Introduction and Concepts Spin Formalisms Dynamical Functions Technical Issues

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**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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**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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**Zemach Formalism For particle with spin S traceless tensor of rank S**

with indices Similar for orbital angular momentum L

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**Example: Zemach – pp (0-+)®f2π0**

Construct total spin 0 amplitude Angular distribution (Intensity) A=Af2π x Aππ

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**The Original Zemach Paper**

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**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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**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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**possibility/simplicity**

Properties Helicity Transversity Canonical property possibility/simplicity partial wave expansion simple complicated parity conservation no yes crossing relation good bad specification of kinematical constraints

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Rotation of States Canonical System Helicity System

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**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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**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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**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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**Two-Particle State Helicity similar procedure no recoupling needed**

normalization

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**Completeness and Normalization**

Canonical completeness normalization Helicity completeness normalization

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**Canonical Decay Amplitudes**

From two-particle state LS-Coefficients

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**Helicity Decay Amplitudes**

From two-particle state Helicity amplitude

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**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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**Relations Canonical ⇔ Helicity**

Recoupling coefficients Start with Canonical to Helicity Helicity to Canonical

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**Clebsch-Gordan Tables**

Clebsch-Gordan Coefficients are usually tabled in a graphical form (like in the PDG) Two cases coupling two initial particles with |j1m1> and |j2m2> to final system <JM| decay of an initial system |JM> to <j1m1| and <j2m2| j1 and j2 do not explicitly appear in the tables all values implicitly contain a square root Minus signs are meant to be used in front of the square root j1 x j2 J M m1 m2 <j1m1j2m2|JM>

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**Using Clebsch-Gordan Tables, Case 1**

1 x 1 2 +2 1 +1 1/2 -1/2 -1 1/6 1/3 2/3 -1/3 -2

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**Using Clebsch-Gordan Tables, Case 2**

1 x 1 2 +2 1 +1 1/2 -1/2 -1 1/6 1/3 2/3 -1/3 -2

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**Parity Transformation and Conservation**

single particle two particles helicity amplitude relations (for P conservation)

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**f2 ® ππ (Ansatz) Initial: f2(1270) IG(JPC) = 0+(2++)**

Final: π0 IG(JPC) = 1-(0-+) Only even angular momenta, since ηf=ηπ2(-1)l Total spin s=2sπ=0 Ansatz

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f2 ® ππ (Rates) Amplitude has to be symmetrized because of the final state particles

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**ω ® π0γ (Ansatz) Initial: ω IG(JPC) = 0-(1--)**

Final: π0 IG(JPC) = 1-(0-+) γ IG(JPC) = 0(1--) Only odd angular momenta, since ηω=ηπηγ(-1)l Only photon contributes to total spin s=sπ+sγ Ansatz

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ω ® π0γ (Rates) λγ=±1 do not interfere, λγ=0 does not exist for real photons Rate depends on density matrix Choose uniform density matrix as an example

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**f0,2 ® γγ (Ansatz) Initial: f0,2 IG(JPC) = 0+(0,2++)**

Final: γ IG(JPC) = 0(1--) Only even angular momenta, since ηf=ηγ2(-1)l Total spin s=2sγ=2, l=0,2 (f0), l=0,2,4 (f2) Ansatz

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**f0,2 ® γγ (cont‘d) Ratio between a00 and a22 is not measurable**

Problem even worse for J=2

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f0,2 ® γγ (cont‘d) Usual assumption J=λ=2

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**pp (2++) ® ππ Proton antiproton in flight into two pseudo scalars**

Initial: pp J,M=0,±1 Final: π IG(JPC) = 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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**pp ® π0ω Two step process First step pp®π0ω - Second step ω®π0γ**

Combine the amplitudes helicity constant aω,11 factorizes and is unimportant for angular distributions

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**pp (0-+) ® f2π0 Initial: pp IG(JPC) = 1-(0-+)**

Final: f2(1270) IG(JPC) = 0+(2++) π0 IG(JPC) = 1-(0-+) is only possible from L=2 Ansatz

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**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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**Moments Analysis Consider reaction Total differential cross section**

expand H leading to

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**Moments Analysis cont‘d**

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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**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 D0KSK+K-

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**Proton-Antiproton Annihilation @ 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 JPC 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 S P D F n=4 n=3 n=2 n=1 Stark-Effect S-Wave P-Wave (99% of 2P) Annihilation ext. Auger-Effect b a L K (1%) rad. Transition

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**Initial States @ Rest JPC IG L S 1S0 0-+ pseudo scalar 1-;0+ 3S1 1--**

3S1 1-- vector 1+;0- 1 1P1 1+- axial vector 3P0 0++ scalar 3P1 1++ 3P2 2++ tensor Quantumnumbers G=(-1)I+L+S P=(-1)L+1 C=(-1)L+S CP=(-1)2L+S+1 I=0 I=1

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**Proton-Antiproton 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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**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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**Scattering Amplitudes in Flight (II)**

Singlett even L JPC L S H++ H+- 1S0 0-+ Yes No 1D2 2-+ 2 1G4 4-+ 4 Singlett odd L JPC L S H++ H+- 1P1 1+- 1 Yes No 1F3 3+- 3 1G5 5+- 5 Triplett odd L JPC L S H++ H+- 3P0 0++ 1 Yes No 3P1 1++ 3P2 2++ 3F2 3 3F3 3++ 3F4 4++ Triplett even L JPC L S H++ H+- 3S1 1-- 1 Yes 3D1 2 3D2 2-- 3D3 3--

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**Header – Dynamical Functions**

Introduction and concepts Spin Formalisms Dynamical Functions Technical issues

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**S-Matrix Differential cross section Scattering amplitude**

Total scattering cross section S-Matrix with and

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**Harmonic Oscillator Free oscillator Damped oscillator Solution**

External periodic force Oscillation strength and phase shift Lorentz function

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**T-Matrix from Scattering**

Back to Schrödinger‘s equation Incoming wave Solves the equation solution without interaction solution with interaction outgoing wave incoming wave inelasticity and phase shift

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**T-Matrix from Scattering (cont’d)**

wave function Scattering amplitude and T-Matrix Example: ππ-Scattering below 1 GeV/c2

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**(In-)Elastic cross sections and T-Matrix**

Total cross section Identify elastic and inelastic part using the optical theorem

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**Breit-Wigner Function**

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

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**Dressed Resonances – T-Matrix & Field Theory**

Suppose we have a resonance with mass m0 We can describe this with a propagator But we may have a self-energy term leading to

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**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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**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 E0 has now changed into E0-<{b}

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**Relativistic Breit-Wigner**

Argand Plot Phase δ Intensity I=ΨΨ* By migrating from Schrödinger‘s equation (non-relativistic) to Klein-Gordon‘s equation (relativistic) the energy term changes different energy-momentum relation E=p2/m vs. E2=m2c4+p2c2 The propagators change to sR-s from mR-m

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**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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**Blatt-Weisskopf Barrier Factors**

The energy dependence is usually parameterized in terms of spherical Hankel-Functions we define Fl(q) with the following features Main problem is the choice of the scale parameter qR=qscale

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

Usage

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**Resonant daughters 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 <Fl(q)>=∫Fl(q)dBW Fl(q)~ql complex even above threshold meaning of mass and width are mixed up Im(q) threshold Re(q)

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**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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**T-Matrix Dispersion Relations**

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

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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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**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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**Example: ππ-Scattering**

1 channel 2 channel

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**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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**Relativistic Treatment – 2 channel**

S-Matrix 2 channel T-Matrix to be compared with the non-relativistic case

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**K-Matrix Poles Now we introduce resonances as poles (propagators)**

One may add cij a real polynomial of m2 to account for slowly varying background (not experimental background!!!) Width/Lifetime For a single channel and one pole we get

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**Example: 1x2 K-Matrix Strange effects in subdominant channels**

Argand Plot Phase δ Intensity I=ΨΨ* Strange effects in subdominant channels Scalar resonance at 1500 MeV/c2, Γ=100 MeV/c2 All plots show ππ channel Blue: ππ dominated resonance (Γππ=80 MeV and ΓKK=20 MeV) Red: KK dominated resonance (ΓKK=80 MeV and Γππ=20 MeV) Look at the tiny phase motion in the subdominant channel

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**Example: 2x1 K-Matrix Overlapping Poles**

two resonances overlapping with different (100/50 MeV/c2) widths are not so dramatic (except the strength) The width is basically added 2 BW K-Matrix FWHM FWHM

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**Example: 1x2 K-Matrix Nearby Poles**

2 BW K-Matrix Two nearby poles (1.27 and 1.5 GeV/c2) show nicely the effect of unitarization

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**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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**Flatté Example a0(980) decaying into πη and KK Real Part Argand Plot**

BW πη Flatte πη Flatte KK Example a0(980) decaying into πη and KK Intensity I=ΨΨ* Phase δ

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**Example: K-Matrix Parametrizations**

Au, Morgan and Pennington (1987) Amsler et al. (1995) Anisovich and Sarantsev (2003)

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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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**P-Vector Poles The resonance poles are constructed as in the K-Matrix**

and one may add a polynomial di 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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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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**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 m1+m2>m therefore the real part of the denominator (mass term) changes and imaginary part (width term) vanishes completely

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**Complex Analysis Revisited (cont’d)**

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 2n complex planes, called Riemann sheets they are continuously connected. The borderlines are called CUTS.

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**Riemann Sheets in a 2 Channel Problem**

Usual definition sheet sgn(q1) sgn(q2) I II III IV This implies for the T-Matrix Complex Energy Plane Complex Momentum Plane

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**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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**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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**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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**N/D Method To get the proper behavior for the left-hand cuts**

Use Nl(s) and Dl(s) which are correlated by dispersion relations An example for this is the work of Bugg and Zhou (1993)

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**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 f0(980) or a0(980) have not

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**Pole and Shadows near Threshold (2 Channel)**

For a real resonance one always obtains poles on sheet II and III due to symmetries in Tl 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 ρ2 smoothly from q2 to –q2

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**Summary Acknowledgements I‘d 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 I’d like to thank R.S. Longacre, from whom I have stolen some paragraphs from his Lecture in Maryland 1991

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**Legend Master Intensity I=ΨΨ* Polarangle φ Phase δ Speed dφ/dm**

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* Polarangle φ Phase δ Speed dφ/dm Inelasticity η Real Part Imaginary Part Argand Plot

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Copyright © 2011, Elsevier Inc. All rights reserved. Chapter 6 Author: Julia Richards and R. Scott Hawley.

Copyright © 2011, Elsevier Inc. All rights reserved. Chapter 6 Author: Julia Richards and R. Scott Hawley.

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