model-independent resonance parameters saša ceci, alfred Švarc, branimir zauner ruđer bošković...

Model-independent Model-independent RResonance esonance PParametersarameters

Saša CeciSaša Ceci, Alfred Švarc, Branimir Zauner, Alfred Švarc, Branimir ZaunerRuđer Bošković InstituteRuđer Bošković Institute

with special thanks towith special thanks toMike SadlerMike Sadler, , Shon WatsonShon Watson, , Jugoslav StahovJugoslav Stahov, , Pekko Pekko

MetsaMetsa, Mark Manley and Simon Capstick, Mark Manley and Simon Capstick

Helsinki 2007Helsinki 2007The 4th International PWA WorkshopThe 4th International PWA Workshop

Helsinki, June 27 (2007)Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA WorkshopSaša Ceci, The 4th International PWA Workshop

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AbstractAbstract

model-independent extractionmodel-independent extractionof the of the

resonance parametersresonance parametersis enabled by imposing of theis enabled by imposing of the

physical constraints physical constraints to theto the

scattering matrix.scattering matrix.


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Resonant Resonant SScatteringcattering• strong peaks of the cross section of the meson-nucleon scattering: strong peaks of the cross section of the meson-nucleon scattering:

a manifestation of the a manifestation of the resonance phenomena,resonance phenomena,

• resonances – excited baryons,resonances – excited baryons,

• baryon spectroscopy baryon spectroscopy – obtaining of the resonance parameters – obtaining of the resonance parameters (masses and widths) from the scattering data,(masses and widths) from the scattering data,

• at energies close to the baryon-resonance masses, the at energies close to the baryon-resonance masses, the microscopic model (QCD) is microscopic model (QCD) is still insolvablestill insolvable..


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HowHow

wouldwould

oneone

obtainobtain

resonanceresonance

parametersparameters

Breit - Wigner parameterizationFit of the experimental data

Ex

cit

ed

ba

ryo

n s

tate

s

??

Re

so

na

nt

pa

ram

ete

rsD

ata

se

t

Ex

pe

rim

en

tal

Da

tas

et

Dataset

ExpansionPartial wave

BW parameterizationFit to the partial waves

Theoreticalconstraints

+ Parameterization

Extraction

P wa ar vt e

i s al

set Functions

Parameterization of the partial waves

Procedure

Relevant analysisLegend:

Topicscovered here

Datasets Procedures

P wa ar vt e i s

al


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The Relevant AnalysesThe Relevant Analyses

What are relevant analysesWhat are relevant analyses

• Arndt FA02Arndt FA02

• Manley MS92Manley MS92

• Hoehler KH80Hoehler KH80

• Cutkosky CMB (‘79)Cutkosky CMB (‘79)

OriginOrigin:: Review of Particle Physics Review of Particle Physics (PDG).(PDG).

Getting spectra can be difficultGetting spectra can be difficult

• partial-wave functions are obtained partial-wave functions are obtained from the from the numerical calculationsnumerical calculations (DR, (DR, expansions on nontrivial functions),expansions on nontrivial functions),

• there are there are too many modelstoo many models, and it is , and it is not clear which one (and in which not clear which one (and in which cases) should be used, cases) should be used,

• in order for many extraction methods in order for many extraction methods to to workwork properlyproperly, some , some “additional”“additional” requirementsrequirements must be met must be met..


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It was (allegedly) easier It was (allegedly) easier in the old daysin the old days ... ...

• Transition matrixTransition matrix - T - T• T = TT = TRR + T + TBB

• like adding of the Feynman diagrams like adding of the Feynman diagrams oror

• CaleyCaley’s’s transform of unitary scattering matrixtransform of unitary scattering matrix - K - K• K = KK = KRR + K + KBB

• the scattering matrix unitarity is conservedthe scattering matrix unitarity is conservedoror

• Phase shiftPhase shift - - • = = RR + + BB

• addition of potentialsaddition of potentials


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...... when people were using these when people were using these simple simple resonant contributionsresonant contributions ......

• Resonant matrixResonant matrix T TRR

• TTRR = ( = (/2) / (M – W – i /2) / (M – W – i /2) /2)

oror• Resonant matrixResonant matrix K KRR

• KKRR = ( = (/2) / (M – W) /2) / (M – W)

oror• Resonant phase shiftResonant phase shift - - RR

• RR= arct= arctanan[([(/2) / (M – W)] /2) / (M – W)]

0.0

1.0

-0.5

0.5

M

W/a.u.

Im TR

Re TR


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... ... but, was it really?but, was it really?

• T-matrix additionT-matrix addition • generally violates the S-matrix unitarity,generally violates the S-matrix unitarity,

• K-matrix additionK-matrix addition• conserves unitarity, but the approach is not unique,conserves unitarity, but the approach is not unique,

• Phase shift additionPhase shift addition• comes down to comes down to S-matrix multiplicationS-matrix multiplication (yet another model), (yet another model),• in multichannel cases, in multichannel cases, order order of matrix multiplications of matrix multiplications is not definedis not defined

(instead of the scalar (instead of the scalar we have matrix we have matrix , coming from , coming from S = eS = e2i2i).).


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What are, then, resonant What are, then, resonant parameters?parameters?

Natural attemptsNatural attempts

• Matrices T and K have poles:Matrices T and K have poles:• T = K (I – i K)T = K (I – i K)-1-1, , • K = T (I + i T)K = T (I + i T)-1-1. .

• S has common poles with T:S has common poles with T:• S = I + 2 i T.S = I + 2 i T.

• What could be done with What could be done with ??

In baryon spectroscopyIn baryon spectroscopy

• Pole parameters:Pole parameters:• pole of T matrixpole of T matrix

• Breit-Wigner parameters:Breit-Wigner parameters:• fit of the BW parameterization fit of the BW parameterization

to the T (or K?) matrixto the T (or K?) matrix


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The Choices We Must Make: The Choices We Must Make: picking the resonance contributionpicking the resonance contribution

• there are many possible choices – there are many possible choices – how to pick the right onehow to pick the right one??• it is much simpler if the scattering is:it is much simpler if the scattering is:

• single channel (all inelastic channels closed), single channel (all inelastic channels closed), • single resonant (just one resonance contributes),single resonant (just one resonance contributes),• with constant resonant “parameters”.with constant resonant “parameters”.

• generalization generalization to to multichannelmultichannel and and multiresonancemultiresonance situations, with situations, with nontrivial energy dependencenontrivial energy dependence of of background background contributions and contributions and resonant resonant parametersparameters calls for quite calls for quite elaborate approacheselaborate approaches,,

• The AssumptionThe Assumption:: energy dependent partial waves has been established energy dependent partial waves has been established properlyproperly,,

• Now we must Now we must determinedetermine proper resonance parameters, and proper resonance parameters, and extractextract them.them.


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• To find To find physical physical criteriacriteria,,

• To determineTo determine resonant resonant parameters parameters based on based on those criteriathose criteria,,

• To testTo test ((i.e. i.e. applyapply) ) method on method on well known well known example.example.

CONCEPT

Non-uniqueness of the extraction results

Model dependence Scattered valuesof the resonant parameters

Framework for the method(s)Extraction methodEstablishing the method(s)

ApplicationTesting of the method(s)

Critical problem of baryon spectroscopy:Methods’ model dependence and non-uniqueness of the result

Resonant poleAmplitude analyticity

Time inversion symmetry

Probability conservation

Physical analysis (criteria)Extraction of the resonance

parametersComparison with initial

results

RESULTS

Uniqueness, analyticityunitarity and symmetric

Generalization:All matrices carry

the same information

Breit-WignerHybridsSpeed plot

Time delay

DisagreementsPoint to analysis problems

Physical approach

Legitimate methods:K matrix pole (trace method)T matrix pole (regularization)

Strong correlation ofT-matrix trace and poles of K

Physical criteria

PROBLEM(S)

OBJECTIVES

APPROACH

List of physical criteria: Non working methods: Choice: CMB approach

Breit-WignerK matrix

Phase shiftHybrids

T matrixSpeed plot Time delay

Main idea

of the extraction methods


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Physical criteriaPhysical criteria

• probabilitiesprobabilities add up to one, add up to one,

• time-inversion invariance,time-inversion invariance,

• parityparity is conserved in QCD, is conserved in QCD,

• total spin total spin is also conserved,is also conserved,

• quantum and relativistic quantum and relativistic mechanics – occurrence mechanics – occurrence of the of the propagators propagators

• scattering matrix scattering matrix unitarityunitarity,,

• symmetricalsymmetrical scattering matrix, scattering matrix,

• parityparity is a good quantum number, is a good quantum number,

• spinspin is a good quantum number, is a good quantum number,

• polespoles emerges. emerges.

Physical conditions Scattering matrix


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Infinite cross sectionInfinite cross section?!?!

• Propagator GPropagator G

• GGo o = = i (i (pp2 2 - m- m2 2 + + ii))-1-1

• GG(1)(1) = G = Goo

• masses of products greater then the masses of products greater then the propagator masspropagator mass – – infinite infinite contributioncontribution to cross section at some to cross section at some physical energy physical energy ((simple polesimple pole))

• next ordernext order – – loop contributionloop contribution i i• GG(2)(2) = G= Go o + G+ Goo iiGGo o

• even worseeven worse – – pole is now second pole is now second orderorder


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GGo o = = i (i (pp2 2 – m– m2 2 + + ii))-1-1

mm22 m m2 2 – – ii

i = i pfRe, , p = p = sqrt(psqrt(p22))

G =G = i ( i (pp2 2 – m– m2 2 + ip+ ipf + f + Re)Re)-1-1

Standard Breit-Wigner:Standard Breit-Wigner:Re = 0, f = 1, p = m.Re = 0, f = 1, p = m.

Flatte approximation:Flatte approximation:Re = 0, f = 1. Re = 0, f = 1.

( )( )

+

+=

+

+

+ . . .

=

G Go

Go Go

Go

Go

Go

Go

Go

Go Go

i

i

i

i

i i

G iGo G+

Go

=G i

-Go -1-1

G

i=

( )( )

+

+=

+

+

+ . . .

=

G Go

Go Go

Go

Go

Go

Go

Go

Go Go

i

i

i

i

i i

G iGo G+

Go

=G i

-Go -1-1

G

i=


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Scattering matrix Scattering matrix unitarityunitarity

• scattering matrix is unitary so it can scattering matrix is unitary so it can be diagonalized by some unitary be diagonalized by some unitary matrix Umatrix U

• S = US = U++ S SD D UU

• SSDD is diagonal matrixis diagonal matrix• SSD D = = s sa a EEaa

• matrixmatrix E Eaa is a vector of orthonormal is a vector of orthonormal basis for diagonal matrix basis for diagonal matrix decompositiondecomposition

• We define matricesWe define matrices aa

• aa = U = U++ E Eaa UU

• S matrix expansionS matrix expansion • S =S = a a ssaa= = a a eeii aa

Matrix Matrix propertiesproperties

• hermiticity hermiticity ( (aa))++ = = aa

• orthoidempotence orthoidempotence a a b b = = a a abab

• completeness completeness a a = I= I• trace trace Tr Tr aa = 1 = 1


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The rest of the physical demandsThe rest of the physical demands

• Other physical conditions on matrix Other physical conditions on matrix • II

JJIIJJparity, spin parity, spin andand isospin isospin conservationconservation

• T T = = time inversion time inversion invarianceinvariance• aa==OOT T EEa a OOmatrix has no matrix has no polespoles on the real axis on the real axis

(orthogonal O instead of unitary U).(orthogonal O instead of unitary U).

• Poles Poles come from come from aa and other functions of and other functions of aa::• sin sin aa e eii

aatan tan aaee

ii aa

• A new questionA new question::• What is going on with What is going on with outside of the real axes? outside of the real axes?


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GeneralizationGeneralization of S-matrix of S-matrix “offshoots” by“offshoots” by matricesmatrices

S-matrix “offshoots”S-matrix “offshoots”

• matrices T, K i matrices T, K i may be derived may be derived from S matrix,from S matrix,

• matrices S, T, K i matrices S, T, K i carry carry the the same informationsame information – what differs – what differs them is them is our capabilityour capability to extract to extract it,it,

• what is interesting is the fact thatwhat is interesting is the fact that matrices K, Im T i Re S are matrices K, Im T i Re S are diagonalized bydiagonalized by the same the same (real)(real) orthogonalorthogonal matrices matrices (O).(O).


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TraceTrace – – channel-dependence channel-dependence eliminationelimination

• T = T = a a ttaaTr Tr Tr Tr aa = 1 = 1

Tr T Tr T = = t taa= = sin sin aa e eii aa == ttr r + t + tbb

Tr K Tr K = = k kaa= = tg tg aa = = kkrr + k + kbb

Tr Tr = = aa= = aa = = rr + + bb

Tr S Tr S = = s saa= = e eii aa = = ssr r + s+ sbb

Trace of the matrixTrace of the matrix isis 1 1 on the real axison the real axis::

• is itis it ( (i.e. i.e. Tr Tr ) ) analytic functionanalytic function??

• if it isif it is, , it will have value 1 in the vicinity of the real axis as wellit will have value 1 in the vicinity of the real axis as well ( (in the resonant areain the resonant area););

• then we can say thisthen we can say this: : Pole positions of the T matrixPole positions of the T matrix, , as well as those of theas well as those of the K K matrix matrix, , will depend will depend on the energy dependence ofon the energy dependence of phase shift phase shift , , and will not care about the energy dependence ofand will not care about the energy dependence of . .


1919 / 33 / 33

Resonant parameter Resonant parameter extractionextraction: : types of parameterstypes of parameters

S or T have pole (pole T)S or T have pole (pole T)

K has pole (pole K)K has pole (pole K)

has “something” has “something” /2 /2 (?) (?)

aa is is /2/2 (pole (pole K)K)

tan(tan(aa) has pole) has pole (pole K) (pole K)

speed plot shows peak (“pole T”)speed plot shows peak (“pole T”)

time delay shows peak (“pole T”)time delay shows peak (“pole T”)

imaginary part of T matrix peak (BW?)imaginary part of T matrix peak (BW?)

Basically, two different waysBasically, two different ways::

• T-matrix poleT-matrix pole

• K-matrix poleK-matrix pole

But there could be some other But there could be some other possibilities ...possibilities ...


2020 / 33 / 33

Up-to-dateUp-to-date resonant parameters resonant parameters definitionsdefinitions

T-matrix poleT-matrix poleTTRR = r / ( = r / (– W)– W)

Speed plotSpeed plot|dT|dTRR/dW| = |r| / [(Re /dW| = |r| / [(Re – W)– W)2 2 + + Im Im 22]]

Time delayTime delayddRR/dW = (/dW = (/2) / [(M – W)/2) / [(M – W)2 2 + + 22/4]/4]

HybridsHybrids??

K-matrix poleK-matrix poleKKRR = ( = (/2) / (M – W) /2) / (M – W)

Breit-Wigner fitBreit-Wigner fitTTRR = ( = (/2) / (M – W – i/2) / (M – W – i/2) /2)

Phase shift is Phase shift is /2/2RR = arctan[( = arctan[(/2) / (M – W)] /2) / (M – W)]


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Poles ofPoles of K K andand T T matrices matrices

These are not (necessarily) parameterizations!


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K-matrix PolesK-matrix Poles: : The Recipe The Recipe I I

S. Ceci, A. Švarc, B. Zauner, D. M. Manley and S. Capstick, hep-ph/0611094v1


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T-matrix PolesT-matrix Poles: : Regularization MethodRegularization Method

channel opening

resonance pole

Almost ideal case: single resonance case

parabolic behavior

unstable

region

convergent region

convergent segment


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T-matrix Poles: The Recipe IIT-matrix Poles: The Recipe II

S. Ceci, J. Stahov, A. Švarc, S. Watson and B. Zauner, hep-ph/060923v1

elastic channel opening continuum opening eta channel opening

resonance pole N(1535) resonance pole N(1650)

Nontrivial case of the N(1535)

N(1535) convergent segment

Small N(1535) convergence region

Unstable regions


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What isWhat is in fact in fact speed plotspeed plot??

N N == 11

Speed plotSpeed plot

|dT|dTRR/dW| = |r| / [(Re /dW| = |r| / [(Re – W) – W)22 + (Im + (Im ))22]]


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OK, but who does really care what the speed plot OK, but who does really care what the speed plot in fact isin fact is??

• KH80 pole parameters are obtained by using of the speed plot (the relevant KH80 pole parameters are obtained by using of the speed plot (the relevant values),values),

• it is generally considered as a nifty way of finding pole position (NSTAR 2004),it is generally considered as a nifty way of finding pole position (NSTAR 2004),

• many PWA groups use it (NSTAR 2007),many PWA groups use it (NSTAR 2007),

• Excited Baryon Analysis Center (EBAC) at JLab in fact wants to study the validity of Excited Baryon Analysis Center (EBAC) at JLab in fact wants to study the validity of the speed plot and time delay (this years whitepaper),the speed plot and time delay (this years whitepaper),

• it is considered as a model-independent extraction method,it is considered as a model-independent extraction method,

• many papers utilizing speed plot or time delay are already published, and some many papers utilizing speed plot or time delay are already published, and some might be published as we speak.might be published as we speak.


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Up-to-dateUp-to-date resonant parameters resonant parameters definitionsdefinitions

T-matrix poleT-matrix poleTTRR = r / ( = r / (– W)– W)

Speed plotSpeed plot|dT|dTRR/dW| = |r| / [(Re /dW| = |r| / [(Re – W)– W)2 2 + + Im Im 22]]

Time delayTime delayddRR/dW = (/dW = (/2) / [(M – W)/2) / [(M – W)2 2 + + 22/4]/4]

HybridsHybrids??

K-matrix poleK-matrix poleKKRR = ( = (/2) / (M – W) /2) / (M – W)

Breit-Wigner fitBreit-Wigner fitTTRR = ( = (/2) / (M – W – i/2) / (M – W – i/2) /2)

Phase shift is Phase shift is /2/2RR = arct = arctanan[([(/2) / (M – W)] /2) / (M – W)]

T-matrix pole: first

order approximation

T-matrix pole: first

order approximation

K-matrix pole

Model dependent

Model dependent


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Which came firstWhich came first ? ?

K-matrix and its poles T-matrix and its poles


2929 / 33 / 33

Choice Choice of the analysis of the analysis

CMBCMB• unitary (S)unitary (S)

• analyticanalytic

• multiresonancemultiresonance

• coupled channelcoupled channel

experimentally obtained

theory-produced functions

parameterization by elementary functions

functions from other sources


3030 / 33 / 33

K-matrix pole K-matrix pole parametersparameters – – valuesvalues

It seems like there is some relation between BW and K-

matrix pole parameters?


3131 / 33 / 33

CorrelationCorrelation – – K-poles and T-trace K-poles and T-trace

Peak

Peak

Zero

Zero


3232 / 33 / 33

T-matrix polesT-matrix poles: : comparison ofcomparison of regularization regularization andand analytic analytic continuation.continuation.

Interesting is that

strong discrepancies

occur at the same

places as in the case

of the K-matrix poles.


3333 / 33 / 33

Some results of Some results of this researchthis research

Physics:Physics:• two extraction methods are developedtwo extraction methods are developed

(regularization method i trace method),(regularization method i trace method),

• correlationcorrelation between pole positions of K between pole positions of K matrix and trace of matrix T is matrix and trace of matrix T is quitequite strongstrong (but also (but also unexpectedunexpected),),

• stronger departuresstronger departures from this correlation from this correlation indicated (practically always) to known indicated (practically always) to known issues of the issues of the original analysisoriginal analysis,,

• we expect that this could be improved by we expect that this could be improved by addition of a few additional important addition of a few additional important inelastic channels,inelastic channels,

• projector matrices projector matrices most likely most likely do not do not have have resonant poles.resonant poles.

Mathematics:Mathematics:• a method is developeda method is developed that determines that determines

simple poles or zeros of complex functions simple poles or zeros of complex functions (verified on various independent cases),(verified on various independent cases),

• projector matricesprojector matrices are introduced as a are introduced as a basis for expansion of the normal matrices.basis for expansion of the normal matrices.

Influence to standard approaches:Influence to standard approaches:• origin origin andand misinterpretations misinterpretations are given are given

of the of the speed plotspeed plot method (similarly for the method (similarly for the time delaytime delay),),

• critics is drawn on the critics is drawn on the model-dependentmodel-dependent methodsmethods (like various (like various BW BW parameterizationsparameterizations and and hybridhybrid approaches)approaches)

• most methods are developed for most methods are developed for narrow narrow resonancesresonances – – in baryon physics they are in baryon physics they are simply simply not narrow enoughnot narrow enough..


3535 / 33 / 33

OK, but who really cares what the speed plot OK, but who really cares what the speed plot in fact isin fact is??

...

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model-independent resonance parameters saša ceci, alfred Švarc, branimir zauner ruđer bošković...

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