Partial Wave AnalysisLectures at the “School on Hadron Physics”

Klaus PetersRuhr Universität Bochum

Varenna, June 2004

E. Fermi CLVII Course

2

Overview

Overview

Introduction and Concepts

Spin Formalisms

Dynamical Functions

Technical Issues

3

Overview – Introduction and Concepts

Overview

Introduction and Concepts

Spin Formalisms

Dynamical Functions

Technical Issues

GoalsWave ApproachIsobar-ModelLevel of Detail

4

Overview – Spin Formalisms

Overview

Introduction and Concepts

Spin Formalisms

Dynamical Functions

Technical Issues

Overview

Zemach Formalism

Canonical Formalism

Helicity Formalism

Moments Analysis

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

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

7

Header – Introduction and Concepts

Introduction& Concepts

Introduction and Concepts

Spin Formalisms

Dynamical Functions

Technical issues

8

What is the mission ?

Particle physics at small distances is well understoodOne Boson Exchange, Heavy Quark Limits

This is not true at large distancesHadronization, Light mesonsare barely understood compared to their abundance

Understanding interaction/dynamics of light hadrons willimprove our knowledge about non-perturbative QCDparameterizations will give provide toolkit to analyze heavy quark processesthus an important tool also for precise standard model tests

We needAppropriate parameterizations for the multi-particle phase spaceA translation from the parameterizations to effective degrees of freedom for a deeper understanding of QCD

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

masswidthspin and parities

as well as properties of the initial stateand 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

10

Intermediate State Mixing

Many states may contribute to a final state

not only ones with well defined (already measured) propertiesnot only expected ones

Many mixing parameters are poorly known

K-phasesSU(3) phases

In additionalso D/S mixing(b1, a1 decays)

11

n-Particle Phase space, n=3

2 ObservablesFrom four vectors 12Conservation laws -4Meson masses -3Free rotation -3Σ 2

Usual choiceInvariant mass m12

Invariant mass m13

π3

π2pp

π1

Dalitz plot

12

J/ψ π+π-π0

Angular distributions are easily seen in the Dalitz plot

cosθ

-1 0 +1

13

Phase Space Plot - Dalitz Plot

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

Q smallQ large

14

The first plots τ/θ-Puzzle

Dalitz applied it first to KL-decays

The former τ/θ puzzle with only a few eventsgoal was to determine spin and parity

And he never called them Dalitz plots

15

Interference problem

PWAThe phase space diagram in hadron physics shows a patterndue to interference and spin effectsThis is the unbiased measurementWhat has to be determined ?

Analogy Optics ⇔ PWA# lamps ⇔ # level# slits ⇔ # resonancespositions of slits ⇔ massessizes of slits ⇔ widths

but only if spins are properly assigned

bias due to hypothetical spin-parity assumption

Optics

Dalitz plot

16

It’s All a Question of Statistics ...

pp 30 with

100 events

17

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

pp 30 with

100 events1000 events

18

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

pp 30 with

100 events1000 events10000 events

19

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

pp 30 with

100 events1000 events10000 events100000 events

20

Experimental Techniques

Scattering Experiments

πN - N* measurementπN - meson spectroscopy

E818, E852 @ AGS, GAMSpp meson threshold production

Wasa @ Celsius, COSYpp or πp in the central region

WA76, WA91, WA102γN – photo production

Cebaf, Mami, Elsa, Graal

“At-rest” Experiments

pN @ rest at LEARAsterix, Obelix, Crystal Barrel

J/ψ decaysMarkIII,DM2,BES,CLEO-c

ф(1020) decaysKloe @ Dafne, VEPP

D and Ds decays

FNAL, Babar, Belle

21

Introducing Partial Waves

Schrödinger‘s Equation

Angular Amplitude

Dynamic Amplitude

22

Argand Plot

23

Standard Breit-Wigner

Full circle in the Argand PlotPhase motion from 0 to π

Intensity I=ΨΨ*

Phase δ Speed dφ/dm

Argand Plot

24

Breit-Wigner in the Real World

e+e- ππ

mππ

ρ-ω

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 complicatedResonances decay into several channelsSeveral resonances appear within the same channelThresholds distort line shapes due to available phase space

A more general approach is needed for a detailed understanding (see last lecture!)

26

Isobar Model

Generalizationconstruct any many-body system as a tree of subsequent two-body decaysthe overall process is dominated by two-body processesthe two-body systems behave identical in each reactiondifferent initial states may interfere

We needneed two-body “spin”-algebra

various formalisms

need two-body scattering formalismfinal state interaction, e.g. Breit-Wigner

Isobar

27

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

28

Example: Isospin Dependence

pp initial states differ in isospin

Calculate isospin Clebsch-Gordan

1S0 destructive interferences3S1 ρ0π0 forbidden

29

Header – Spin Formalisms

Spin Formalisms

Introduction and Concepts

Spin Formalisms

Dynamical Functions

Technical Issues

30

Formalisms – on overview

Tensor formalismsin non-relativistic (Zemach) or covariant formFast computation, simple for small L and S

Spin-projection formalismswhere a quantization axis is chosen and proper rotations are used to define a two-body decayEfficient formalisms, even large L and S easy to handle

Formalisms based on Lorentz invariants (Rarita-Schwinger)where each operator is constructed from Mandelstam variables onlyElegant, but extremely difficult for large L and S

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.

32

Zemach Formalism

For particle with spin Straceless tensor of rank S

with indices

Similar for orbital angular momentum L

33Example: Zemach – pp (0-+)f2π0

Construct total spin 0 amplitude

Angulardistribution(Intensity)

A=Af2π x Aππ

34

The Original Zemach Paper

35

Spin-Projection Formalisms

Differ in choice of quantization axis

Helicity Formalismparallel to its own direction of motion

Transversity Formalismthe component normal to the scattering plane is used

Canonical (Orbital) Formalismthe component m in the incident z-direction is diagonal

37

Properties

Helicity Transversity Canonical

property possibility/simplicity

partial wave expansion simple complicated complicated

parity conservation no yes yes

crossing relation no good bad

specification of kinematical constraints

no yes yes

38

Rotation of States

Canonical System Helicity System

39

Single Particle State

Canonical

1) momentum vector is rotated via z-direction. Secondly2) absolute value of the momentum is Lorentz boosted along z3) z-axis is rotated to the momentum direction

40

Single Particle State

Helicity

1) z-axis is rotated to the momentum direction 2) Lorentz BoostTherefore the new z-axis, z’, is parallel to the momentum

41

Two-Particle State

Canonicalconstructed from two single-particle states(back-to-back)

Couple s and t to S

Couple L and S to J

Spherical Harmonics

42

Two-Particle State

Helicitysimilar procedure

no recoupling needed

normalization

43

Completeness and Normalization

Canonical

completeness

normalization

Helicity

completeness

normalization

44

CanonicalFrom two-particle state

LS-Coefficients

Canonical Decay Amplitudes

45

Helicity Decay Amplitudes

HelicityFrom two-particle state

Helicity amplitude

46

Spin Density and Observed Number of Events

To finally calculate the intensityi.e. the number of eventsobserved

Spin density of the initial state

Sum over all unobservables

taking into account

47

Relations Canonical ⇔ Helicity

Recoupling coefficients

Start with

Canonical to Helicity

Helicity to Canonical

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 |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 j2J J

M M

m1 m2

<j1m1j2m2|JM>m1 m2

49

Using Clebsch-Gordan Tables, Case 1

1 x 12

+2 2 1

+1 +1 1 +1 +1

+1 0 1/2 1/2 2 1 0

0 +1 1/2 -1/2 0 0 0

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

0 0 2/3 0 -1/3 2 1

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

-1 0 1/2 1/2 2

0 -1 1/2 -1/2 -2

-1 -1 1

( )

( )

1 1

2 2

1 1 2 2

JM 20

j m 11

j m 1 1

j m j m JM 11 1 1 20

16

=

=

= -

= -

=

50

1 x 12

+2 2 1

+1 +1 1 +1 +1

+1 0 1/2 1/2 2 1 0

0 +1 1/2 -1/2 0 0 0

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

0 0 2/3 0 -1/3 2 1

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

-1 0 1/2 1/2 2

0 -1 1/2 -1/2 -2

-1 -1 1

Using Clebsch-Gordan Tables, Case 2

1 1

2 2

1 1 2 2

JM 00

j m 10

j m 10

j m j m JM 10 10 00

13

=

=

=

=

= -

51

Parity Transformation and Conservation

Parity transformationsingle particle

two particles

helicity amplitude relations (for P conservation)

52f2 ππ (Ansatz)

Initial: f2(1270) IG(JPC) = 0+(2++)

Final: π0 IG(JPC) = 1-(0-+)Only even angular momenta, since ηf=ηπ

2(-1)l

Total spin s=2sπ=0

Ansatz( )

( )

( )( )

( )

1 2 1 2

J M J J*λ λ J λ λ Mλ

1 2

2M 2 2*00 2 00 M0

22 00 20 20

1 1

2M 2*00 20 M0

A N F D φ,θ

λ λ λ 0

J 2

A N F D φ,θ

N F 5 20 00 20 00 00 00 a 5a

A 5a D φ,θ

=

= - =

=

=

= =

=

E555555F E555555F

53f2 ππ (Rates)

Amplitude has to be symmetrized because of the final state particles

( ) ( )( ) ( )

( )( )( )

( )

( ) ( )

( ) ( )

( )

2 2iφ2 0

2 iφ1 0

1M 200 20 00

2 iφ102 2iφ20

1M 1M'*00 MM' 00

M,M'

2 22 0

21 0

2 200

d θ ed θ e

A 5a d θd θ ed θ e

I θ A ρ A

11ρ

2J 1 1

6d θ sin θ

4

3d θ sinθcosθ

23 1

d θ cos θ2 2

--

--

±

±

é ùê úê úê úê ú=ê úê úê úê úë û

=

æ ö÷ç ÷ç= ÷ç ÷ç+ ÷÷çè ø

=

= -

æ ö÷ç= - ÷ç ÷÷çè ø

å

O

( )

4 2 2 4 2

22 4 2 2 2

20

1 3 1 115 sin θ sin θcos θ cos θ cos θ

4 4 2 12

2

20

15 3 1I θ a sin θ 15sin θcos θ 5 cos θ

4 2 2

a const

æ ö÷ç + + - + ÷ç ÷÷çè ø

æ öæ ö ÷ç ÷ç ÷ç= + + - ÷÷çç ÷÷ç ÷è øç ÷çè ø

= =

E5555555555555555555555555555555F

54

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( )

( )

( )( )

( )

1 2 1 2

J M J J*λ λ J λ λ Mλ

1 2 γ 1

1M 1 1*λ0 1 λ0 Mλ

11 λ0 11 11

λ 1

2

1M 1*λ0 11 Mλ

A N F D φ,θ

λ λ λ λ λ

J 1

A N F D φ,θ

3N F 3 10 1λ J λ 1λ 00 1λ a λ a

2

3A λ a D φ,θ

2

-

=

= - = =

=

=

= = -

= -

E55555F E55555F

55

ω π0γ (Rates)

λγ=±1 do not interfere, λγ=0 does not exist for real photons

Rate depends on density matrixChoose uniform density matrix as an example

( )( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 iφ 1 iφ1 1 1 1

1M 1 1λ0 010 1

1 iφ 1 iφ111 1

1M 1M'*λ0 MM' λ '0 λλ '

M,M',λ,λ '

1 11 1 1 1

1 1 110 01 0 1

d θ e 0 d θ e3

A d θ 0 d θ2

d θ e 0 d θ e

I θ A ρ A δ

1 0 01ρ 0 1 0

3 0 0 1

1 cosθd θ d θ

2sinθ

d θ d θ d θ2

- -- - -

-

-

±

-

é ù-ê úê ú= - -ê úê ú-ê úë û

=

æ ö÷ç ÷ç= ÷ç ÷ç ÷÷çè ø

±= =

= - = =

å

m

2 21 1 cosθ 1 cosθ

I 2 2 22 2 2

æ ö æ ö- +÷ ÷ç ç= + +÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

2sin θ

2

2 211 cos θ sin θ 1

2const

é ùê úê úê úë û

é ù= + + =ê úë û

=

56f0,2 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 ( )

( ) ( )( )

( ) ( )( )

( ) ( )( )

1 2 1 2

1 2

00 0 0*λ λ 1 λ λ 0λ

00 λ λ 1 1 2 2 ls

ls

00 1 2

22 1 2

00 22

J 0

A N F D φ,θ

N F l0 sλ J λ s λ s λ sλ a

1a 00 00 00 1λ 1 λ 0λ

5a 20 20 00 1λ 1 λ 2λ

1 1a a

3 6

=

=

= -

= -

+ -

= +

å

( )1 2 1 2

JM J J*λ λ J λ λ Mλ

1 2

A N F D φ,θ

λ λ λ

=

= -

57f0,2 γγ (cont‘d)

Ratio between a00 and a22 is not measurable

Problem even worse for J=2

( )

( )

1 1 1 1

00 0 0*λ λ 1 λ λ 00

0*00 22 00

const

A N F D φ,θ

1 1a a D φ,θ

3 6

const

=

é ùê ú= +ê úê úë û

=

E5555F

( )

( ) ( )( )

( ) ( )( )

( ) ( )( )

( ) ( )( )

1 2 1 2

1 2

2M 2 2*λ λ 2 λ λ Mλ

22 λ λ 1 1 2 2 ls

ls

20 1 2

22 1 2

42 1 2

J 2

A N F D φ,θ

N F l0 sλ J λ s λ s λ sλ a

5a 20 00 2λ 1λ 1 λ 00

5a 20 2λ 2λ 1λ 1 λ 2λ

9a 40 2λ 2λ 1λ 1 λ 2λ

=

=

= -

= -

+ -

+ -

å

58f0,2 γγ (cont‘d)

Usual assumption J=λ=2

( ) ( )( )

( )( )

( )( )

( ) ( )( ) ( )

( )

22 1 2 ls1 1

ls

22

127

42

t.b.d. 1

2M 2 2 2*2 M21 1 1 1

2*2 M2

000

N F l0 s2 2 s 1 s 1 s2 a

5a 20 22 22 11 11 22

9a 40 22 22 11 11 22

Symmetrization

A N F F D φ,θ

N 'D φ,θ

Comparison

A N '

-

- -

=

= +

+

= +

=

=

å

E555555F E55555F

E555555F 14444244443

59

pp (2++) ππ

Proton antiproton in flight into two pseudo scalarsInitial: pp J,M=0,±1Final: π IG(JPC) = 1-(0-+)

Ansatz

Problem: d-functions are not orthogonal, if φ is not observedambiguities remain in the amplitude – polarization is needed

( )

( )

( )( )

( ) ( )

1 2 1 2

lJ

JM J J*λ λ J λ λ Mλ

1 2

JM J J*00 J 00 M0

JJ 00 l0 J0

l δ 1

JM J* J* iMφ00 J0 M0 J0 M0

A N F D φ,θ

λ λ λ 0

J l

A N F D φ,θ

N F 2J 1 l0 00 J0 00 00 00 a 2J 1a

A 2J 1a D φ,θ 2J 1a d θ e-

=

= - =

=

=

= + = +

= + = +

å E55555F E555555F

60

pp π0ω

Two step processFirst step ppπ0ω - Second step ωπ0γCombine the amplitudes

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

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

ω

ω γ γ ω

γ ω γ ω ω

ω γ ω

ω γ ω

1λJM JMλ λ ω γ λ 0 γ λ 0 ω

1 J* J J*ω,1 λ 0 λ λ γ pp,J λ 0 Mλ ω

1* J*ω,11 λ λ γ ω ω pp,l1 Mλ ω

l

1* J*ω,11 λ λ γ Mλ ω ω ω pp,l1

l

A Ω ,Ω A Ω A Ω

N F D Ω N F D Ω

3λ a D Ω 2l 1 l0 1λ J λ a D Ω

2

3λ a D Ω D Ω 2l 1 l0 1λ J λ a

2

=

=

= - +

= - +

å

å

61pp (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 ( )

( ) ( ) ( ) ( )

( )( )

( )( )

( ) ( )

1 2 1 2

2 2 2

2 2

2 2

J M J J*λ λ J λ λ Mλ

00 20 0 0* 2 2*00 f 00 π pp,0 00 00 ff ,2 00 00 π

0pp,0 00 pp,22

115

2f ,2 00 f ,20

1 1

00 20 2*00 f 00 π pp,22 f ,20 00

A N F D Ω

A Ω A Ω N F D Ω N F D Ω

N F 1 20 20 00 20 00 20 a

N F 5 20 00 20 00 00 00 a

A Ω A Ω 5a a D

=

=

=

=

=

E555555F E555555F

E555555F E555555F

( )

( )

2

2

2π pp,22 f ,20

22 2

pp,22 f ,20

3 1Ω 5a a cos θ

2 2

3 1I cosθ 5 a a cos θ

2 2

æ ö÷ç= - ÷ç ÷÷çè ø

æ ö÷ç= - ÷ç ÷÷çè ø

2

1 2

pp

f

λ λ λ 0

J 0

J 2

= - =

=

=

62

General Statements

Flat angular distributions

General rules for spin 0initial state has spin 0

0 any

both final state particles have spin 0J 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

63

Moments Analysis

Consider reaction

Total differential cross section

expand H

leading to

64

Moments Analysis cont‘d

Define now a density tensor

the d-function productscan be expanded in spherical harmonics

and the density matrix gets absorbed in a spherical moment

65

Example: Where to start in Dalitz plot anlysis

Sometimes a moment-analysis can help to find important contributionsbest suited if no crossing bands occur

( ) ( )

( ) ( )

LM0

LM0

t LM D φ,θ,0

I Ω D φ,θ,0 dΩ

=

= ò

D0KSK+K-

66

Proton-Antiproton Annihilation @ Rest

Atomic initial systemformation at high n, l (n~30)slow radiative transitionsde-excitation through collisions (Auger effect)Stark mixing of l-levels (Day, Snow, Sucher‚ 1960)

AdvantagesJPC varies with target densityisospin varies with n (d) or p targetincoherent initial statesunambiguous PWA possible

Disadvantagesphase space very limitedsmall kaon yield

LL

K(1%) rad. Transition

Stark-Effectext. Auger-Effect

S-Wave

P-Wave(99% of 2P)

Annihilation

S P D F

n=4

n=3

n=2

n=1

67

Initial States @ Rest

Quantumnumbers

G=(-1)I+L+S

P=(-1)L+1

C=(-1)L+S CP=(-1)2L+S+1

I=0

I=1

JPC IG L S

1S0 0-+ pseudo scalar 1-;0+ 0 0

3S1 1-- vector 1+;0- 0 1

1P1 1+- axial vector 1+;0- 1 0

3P0 0++ scalar 1-;0+ 1 1

3P1 1++ axial vector 1-;0+ 1 1

3P2 2++ tensor 1-;0+ 1 1

68

Proton-Antiproton Annihilation in Flight

0.5 1.0 1.5 2.0 2.5 3.0

10

1

P [GeV/c]lab

l=4

l=3

l=2l=1

ang. mom. ~ /0.2 GeV/

ll p ccms

ann= l l

l(p)=(2l+1) [1-exp(- (p))] / p l2

l(p)=N(p) exp(-3l(l+1)/4p R ) 2 2100

l [

mb]

Annihilation in flightscattering process:no well defined initial statemaximum angular momentum rises with energy

Advantageslarger phase spaceformation experiments

Disadvantagesmany waves interfere with each othermany waves due to large phase space

69

Scattering Amplitudes in Flight (I)

pp helicity amplitude

only H++ and H+- exist

C-InvarianceH++=0 if L+S-J odd

CP-InvarianceH+-=0 if S=0 and/or J=0

( )( )( )

( ) ( )

1 2

1 2 2 1

Jν ν 1 1 2 2

L,S

JJ Jν ν J ν ν

2L 1H L0Sν J ν s ν s ν Sν J MLS J M

2J 1

H η 1 H- -

+= -

+

= -

å M

CP transformCP=(-1)2L+S+1

S and CP directly correlatedCP conserved in strong int.singlet and triplet decoupled

C transformL 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

70

Scattering Amplitudes in Flight (II)

Singlett even L

JPC L S H++ H+-

1S0 0-+ 0 0 Yes No

1D2 2-+ 2 0 Yes No

1G4 4-+ 4 0 Yes No

Triplett even L

JPC L S H++ H+-

3S1 1-- 0 1 Yes Yes

3D1 1-- 2 1 Yes Yes

3D2 2-- 2 1 Yes Yes

3D3 3-- 2 1 Yes Yes

Singlett odd L

JPC L S H++ H+-

1P1 1+- 1 0 Yes No

1F3 3+- 3 0 Yes No

1G5 5+- 5 0 Yes No

Triplett odd L

JPC L S H++ H+-

3P0 0++ 1 1 Yes No

3P1 1++ 1 1 No Yes

3P2 2++ 1 1 Yes Yes

3F2 2++ 3 1 Yes No

3F3 3++ 3 1 No Yes

3F4 4++ 3 1 Yes Yes

71

Header – Dynamical Functions

Dynamical Functions

Introduction and concepts

Spin Formalisms

Dynamical Functions

Technical issues

72

S-Matrix

Differential cross section

Scattering amplitude

Total scattering cross section

S-Matrix

with

and

73

Harmonic Oscillator

Free oscillator

Damped oscillator

Solution

External periodic force

Oscillation strengthand phase shiftLorentz function

74

T-Matrix from Scattering

Back to Schrödinger‘s equation

Incoming wave

Solves the equation

solution without interaction

solution with interaction

incomingwave

outgoingwave

inelasticity and phase shift

75

T-Matrix from Scattering (cont’d)

Scatteringwave function

Scattering amplitudeand T-Matrix

Example: ππ-Scatteringbelow 1 GeV/c2

76

(In-)Elastic cross sections and T-Matrix

Total cross section

Identify elastic

and inelastic part

using the optical theorem

77

Breit-Wigner Function

Wave function for an unstable particle

Fourier transformation for E dependence

Finally our first Breit-Wigner

78

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

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

+ ... =

+ +

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 E0 has now changed into E0-<{b}

81

Relativistic Breit-Wigner

By migrating from Schrödinger‘s equation (non-relativistic)to Klein-Gordon‘s equation (relativistic) the energy term changesdifferent energy-momentum relation E=p2/m vs. E2=m2c4+p2c2

The propagators change to sR-s from mR-m

Intensity I=ΨΨ*Phase δArgand Plot

82

Barrier Factors - Introduction

At low energies, near thresholdsbut 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

withwe obtain

83

Blatt-Weisskopf Barrier Factors

The energy dependence is usually parameterized in terms of spherical Hankel-Functions

we define Fl(q) with thefollowing features

Main problem is the choice of the scale parameter qR=qscale

84

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

Usage

85

Barrier factors

Scales and Formulaeformula was derived from a cylindrical potentialthe scale (197.3 MeV/c) may be different for different processesvalid in the vicinity of the poledefinition of the breakup-momentum

Breakup-momentummay become complex (sub-threshold)set to zero below threshold

need <Fl(q)>=∫Fl(q)dBWFl(q)~ql

complex even above thresholdmeaning of mass and width are mixed up

Resonant daughters

2 2

2 a b a bii i

m m m m2q1 as m ; 1 1

m m mρ ρ

é ùé ùæ ö æ ö+ -ê úê ú÷ ÷ç ç® ® ¥ = = - -÷ ÷ç çê úê ú÷ ÷÷ ÷ç çè ø è øê úê úë ûë ûRe(q)

Im(q

)

threshold

86

T-Matrix Unitarity Relations

Unitarity is a basic featuresince probability has to be conserved

T is unitary if S is unitary

since

we get in addition

87

T-Matrix Dispersion Relations

Cauchy Integral on a closed contour

By choosing proper contoursand some limits one obtains the dispersion relation for Tl(s)

Satisfying this relation with an arbitraryparameterization is extremely difficultand is dropped in many approaches

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 symmetricalso n x n

then the T is unitary by construction

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

90

Example: ππ-Scattering

1 channel 2 channel

91

Relativistic Treatment

So far we did not care about relativistic kinematics

covariant description

or

and

with

therefore

and K is changed as well

92

Relativistic Treatment – 2 channel

S-Matrix

2 channel T-Matrix

to be compared with the non-relativistic case

93

K-Matrix Poles

Now we introduce resonancesas 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

94

Example: 1x2 K-Matrix

Strange effects in subdominant channels

Scalar resonance at 1500 MeV/c2, Γ=100 MeV/c2

All plots show ππ channelBlue: 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

Intensity I=ΨΨ*Phase δArgand Plot

95

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

FWHM

FWHM

2 BWK-Matrix

96

Example: 1x2 K-Matrix Nearby Poles

Two nearby poles (1.27 and 1.5 GeV/c2)show nicely the effect of unitarization

2 BWK-Matrix

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

98

Flatté

Examplea0(980) decaying into πη and KK

BW πηFlatte πηFlatte KK

Intensity I=ΨΨ* Phase δ

Real PartArgand Plot

99

Example: K-Matrix Parametrizations

Au, Morgan and Pennington (1987)

Amsler et al. (1995)

Anisovich and Sarantsev (2003)

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

101

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

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

103

Complex Analysis Revisited

The Breit-Wigner example

shows, that Γ(m) implies ρ(m)

but below threshold ρ(m) gets complexbecause 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

104

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 betweenand are not identical

So you define a new complex plane for each solution,which are 2n complex planes, called Riemann sheetsthey are continuously connected. The borderlines are called CUTS.

105

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

106

States on Energy Sheets

Singularities appear naturally where

Singularities might be

1 – bound states2 – anti-bound

states3 – resonances

or

artifacts due to wrong treatment of the model

107

States on Momentum Sheets

Or in the complex momentum plane

Singularities might be

1 – bound states2 – anti-bound states3 – resonances

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)

109

N/D Method

To get the proper behavior for the left-hand cutsUse Nl(s) and Dl(s) which are correlated by dispersion relations

An example for this is the work of Bugg and Zhou (1993)

110

Nearest Pole Determines Real Axis

The pole nearest to the real axisor 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 morecomplicated

While ρ(770) in between twothresholds has a beautiful shapethe f0(980) or a0(980) have not

111

Pole and Shadows near Threshold (2 Channel)

For a real resonance one always obtains poles on sheet II and IIIdue 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 changingρ2 smoothly from q2 to –q2

112

Summary

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

Acknowledgements