Alfred Švarc Ruđer Bošković Institute Croatia

The importance of The importance of inelastic channelsinelastic channels in eliminating in eliminating continuum ambiguitiescontinuum ambiguities in pion-nucleon partial wave analysesin pion-nucleon partial wave analyses

Alfred ŠvarcAlfred ŠvarcRuđer Bošković Institute

Croatia

Pg. 5Pg. 5

Pg. 6Pg. 6

constraints from fixed t-analyticity … constraints from fixed t-analyticity … resolveresolve the ambiguities the ambiguities

The effects of The effects of discreet and continuumdiscreet and continuum ambiguities ambiguities were separatedwere separated ….….

Dispersion relations were used to solve ambiguities Dispersion relations were used to solve ambiguities and to derive constraints…and to derive constraints…

1973

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2005

1985.

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1976

What does it mean What does it mean “continuum ambiguity”?“continuum ambiguity”?

Differential cross section is not Differential cross section is not sufficient to determine the sufficient to determine the scattering amplitude:scattering amplitude:

ifif

thenthenThe new function givesThe new function gives EXACTLY THE SAME CROSS EXACTLY THE SAME CROSS

SECTIONSECTION

S – matrix unitarity …………….. conservation of fluxS – matrix unitarity …………….. conservation of flux RESTRICTS THE PHASERESTRICTS THE PHASE

elastic region ……. unitarity relates real and imaginary part elastic region ……. unitarity relates real and imaginary part of each partial wave – of each partial wave – equalityequality constraint constraint

each partial wave must lie each partial wave must lie uponupon its unitary circle its unitary circle

inelastic region ……. unitarity provides only an inelastic region ……. unitarity provides only an inequalityinequality constraint between real and constraint between real and imaginary partimaginary parteach partial wave must lie each partial wave must lie uponupon or insideor inside its unitary its unitary circlecirclethere exists a whole there exists a whole family family of functions of functions ,, of limited of limited magnitude but of magnitude but of infinite infinite variety of functional form, which variety of functional form, which will behave exactly like thatwill behave exactly like that

These family of functions, though containing a continuum These family of functions, though containing a continuum infinity of points, infinity of points,

are limited in extend.are limited in extend.

TheThe ISLANDS OF AMBIGUITY ISLANDS OF AMBIGUITY are createdare created..

there exists a whole there exists a whole family family of functions of functions ,, of limited of limited magnitude but of magnitude but of infinite infinite variety of functional form, which variety of functional form, which will behave exactly like thatwill behave exactly like that

I M P O R T A N TI M P O R T A N T

Once the three body channels open up, this way of Once the three body channels open up, this way of eliminating continuum ambiguities (elastic channel eliminating continuum ambiguities (elastic channel

arguments) become arguments) become

in principle impossiblein principle impossible

I M P O R T A N TI M P O R T A N T

DISTINCTION DISTINCTION

theoretical islands of ambiguity / experimental uncertaintiestheoretical islands of ambiguity / experimental uncertainties

The treatment of continuum ambiguity The treatment of continuum ambiguity problemsproblems

a.a. Constraining the functional form Constraining the functional form mathematicalproblemb.b. Implementing the partial wave T – matrix continuityImplementing the partial wave T – matrix continuity

(energy smoothing and search for uniqueness)(energy smoothing and search for uniqueness)

a.

Finding the true boundaries of the phase function Finding the true boundaries of the phase function (z) is a (z) is a very difficult problem on which the very little progress has very difficult problem on which the very little progress has been made.been made.

b.

Let us formulate what the continuum ambiguityLet us formulate what the continuum ambiguity problem is in the language ofproblem is in the language of coupled channel formalismcoupled channel formalism

Continuum ambiguityContinuum ambiguity // T-matrix polesT-matrix poles

Each analytic function is uniquely defined with its poles and Each analytic function is uniquely defined with its poles and cuts.cuts.

If an analytic function contains a continuum ambiguity it is If an analytic function contains a continuum ambiguity it is not uniquely defined.not uniquely defined.

T matrix is an T matrix is an analytic function analytic function in s,tin s,t..

If an analytic function is not uniquely defined, we do not If an analytic function is not uniquely defined, we do not have a complete knowledge about its have a complete knowledge about its poles and cutspoles and cuts..

ConsequentlyConsequently fully constraining poles and cuts means eliminating fully constraining poles and cuts means eliminating continuum ambiguitycontinuum ambiguity

Basic idea: Basic idea: we wantwe want to demonstrate to demonstrate the rolethe role of inelastic channels of inelastic channels inin fully fully constrainingconstraining the poles of the partial wave T-matrix, the poles of the partial wave T-matrix,

or, alternatively said,or, alternatively said, for eliminating continuum ambiguity which arises if only for eliminating continuum ambiguity which arises if only

elastic channels a considered.elastic channels a considered.

We want We want as wellas well show that: show that: supplying only scarce information for supplying only scarce information for EACH EACH channel ischannel is

MUCH MORE CONSTRAINING MUCH MORE CONSTRAINING then supplying the perfect information in then supplying the perfect information in ONE ONE channel.channel.

Coupled channel T matrix formalismCoupled channel T matrix formalism

1.1. unitaryunitary2.2. fully analyticfully analytic

T-matrix polesT-matrix poles are connected to the are connected to the bare bare propagatorpropagator poles, but poles, but shifted shifted with the self energy term !with the self energy term !

Important:Important: real and imaginary parts of the self energy term real and imaginary parts of the self energy term are linked because of analyticityare linked because of analyticity

Constraining data:Constraining data: Elastic channel:Elastic channel: Pion elastic VPI SES solution FA02 Pion elastic VPI SES solution FA02

Karlsruhe - Helsinki KH 80 Karlsruhe - Helsinki KH 80

Inelastic channel:Inelastic channel:

recentrecent

CC PWA Pittsburgh/CC PWA Pittsburgh/ANLANL 20002000

however however NO PNO P11 11 is offeredis offered !!

olderolder

CC PWA Zagreb/CC PWA Zagreb/ANLANL 95/98 95/98 CC PWA CC PWA Zagreb/ Zagreb/ANLANL 95/9895/98

gives Pgives P1111

• Both: Both: three channel version of CMB model three channel version of CMB model

• N elastic T matrices + N elastic T matrices + N N N data + dummy channel N data + dummy channel

• Pittsburgh: Pittsburgh: VPI VPI + data + dummy channel + data + dummy channel• Zagreb: Zagreb: KH80KH80 + data + dummy channel + data + dummy channel

• fitted all partal waves up to L = 4fitted all partal waves up to L = 4

• Pittsburgh: Pittsburgh: offers Soffers S11 11 onlyonly • Zagreb: Zagreb: offers Poffers P11 11 as wellas well (nucl-th/9703023) (nucl-th/9703023)

SS1111 Let us compare Pittsburgh¸/ Zagreb SLet us compare Pittsburgh¸/ Zagreb S1111

Pittsburgh SPittsburgh S11 11 is taken as an “experimentally constrained” partial waveis taken as an “experimentally constrained” partial wave

PP1111

So wSo we offer Zagreb Pe offer Zagreb P11 11 as the “experimentally constrained” partial waveas the “experimentally constrained” partial wave as well as well. .

from from nucl-th/9703023nucl-th/9703023

TwoTwo--channelchannel-model-model

STEP 1 STEP 1 ::

Number of channels: Number of channels: 22 (pion elastic + effective)(pion elastic + effective)Number of Number of GF GF propagator poles: propagator poles: 33

(2 (2 background poles + 1 background poles + 1 physical pole)physical pole)

ONLYONLY ELASTIC ELASTIC CHANNEL IS FITTEDCHANNEL IS FITTED

• elastic channels is elastic channels is reproducedreproduced perfectly perfectly• inelastic channel is reproduced poorlyinelastic channel is reproduced poorly• we identify we identify one one polepole in the physical region in the physical region

STEP 2 STEP 2 ::

Number of channels: Number of channels: 22 (pion elastic + effective)(pion elastic + effective)Number of Number of GF GF propagator poles: propagator poles: 33

(2 (2 background poles + 1 background poles + 1 physical pole)physical pole)

ONLY ONLY INELASTIC INELASTIC CHANNEL IS FITTEDCHANNEL IS FITTED

• inelastic channel is reproduced perfectlyinelastic channel is reproduced perfectly• elastic channel is reproduced poorlyelastic channel is reproduced poorly• we identify we identify two polestwo poles in the physical region, in the physical region,

““RoperRoper”” and and 1700 MeV1700 MeV

STEP 3STEP 3::

Number of channels: Number of channels: 22 (pion elastic + effective)(pion elastic + effective)Number of Number of GF GF propagator poles: propagator poles: 33

(2 (2 background poles + 1 background poles + 1 physical pole)physical pole)

ELASTIC + INELASTIC ELASTIC + INELASTIC CHANNEL ARE FITTEDCHANNEL ARE FITTED

• elastic channel is reproduced OKelastic channel is reproduced OK• inelastic channel is reproduced tolerablyinelastic channel is reproduced tolerably• we identify we identify two two polespoles in the physical region, in the physical region,

but both are in the Roper –but both are in the Roper – resonance regionresonance region

We can not find a “single pole” solution which We can not find a “single pole” solution which would simultaneously reproduce would simultaneously reproduce

ELASTICELASTIC AND AND INELASTIC INELASTIC CHANNELCHANNELSS

STEP 4STEP 4::Number of channels: Number of channels: 22 (pion elastic + effective)(pion elastic + effective)

Number of Number of GF GF propagator poles: propagator poles: 44

(2 (2 background poles + background poles + 22 physical poles)physical poles)

ONLYONLY ELASTIC ELASTIC CHANNEL IS FITTEDCHANNEL IS FITTED

• elastic channels is reproduced perfectlyelastic channels is reproduced perfectly• inelastic channel is reproduced poorlyinelastic channel is reproduced poorly• we identify we identify two polestwo poles in the physical region, in the physical region,

Roper + one above 2200 MeVRoper + one above 2200 MeV

STEP 5STEP 5::Number of channels: Number of channels: 22 (pion elastic + effective)(pion elastic + effective)

Number of Number of GF GF propagator poles: propagator poles: 44

(2 (2 background poles + background poles + 22 physical poles)physical poles)

ONLYONLY INELASTIC INELASTIC CHANNEL IS FITTEDCHANNEL IS FITTED

We have found two possible solutionsWe have found two possible solutions which differ significantly in channels which are not which differ significantly in channels which are not

fittedfitted

Solution 1Solution 1

Solution Solution 22

For both solutionsFor both solutions::• elastic channel is poorly reproduced, elastic channel is poorly reproduced, AND AND differs differs

for both solutionsfor both solutions• inelastic channel is OKinelastic channel is OK• FoFor both solutions we identify r both solutions we identify two polestwo poles in the in the

physical region, physical region, Roper + Roper + 1700 MeV1700 MeV

STEP 6STEP 6::Number of channels: Number of channels: 22 (pion elastic + effective)(pion elastic + effective)

Number of Number of GF GF propagator poles: propagator poles: 44

(2 (2 background poles + background poles + 22 physical poles)physical poles)

ELASTIC +ELASTIC + INELASTIC INELASTIC CHANNELS ARE FITTEDCHANNELS ARE FITTED

We We offeroffer two possible solutions two possible solutions which differ significantly in channels which are not which differ significantly in channels which are not

fittedfitted

Solution 1Solution 1

Solution Solution 22

For both solutionsFor both solutions::• elastic channel is elastic channel is acceptablyacceptably reproduced, reproduced, AND AND

differs for both solutions differs for both solutions BUT ADDITIONAL STRUCTURE IN THE ELASTIC CHANNEL BUT ADDITIONAL STRUCTURE IN THE ELASTIC CHANNEL APPEARES IN THE ENEAPPEARES IN THE ENERGRGY RANGE OF1700 MeVY RANGE OF1700 MeV

• inelastic channel is OKinelastic channel is OK• For both solutions we identify For both solutions we identify two polestwo poles in the in the

physical region, physical region, Roper +Roper + 1700 MeV1700 MeV

An observationAn observation::

The structure in elastic channel, required by theThe structure in elastic channel, required by the presence of inelastic channels, appears:presence of inelastic channels, appears:

eexactly where error bars of the Fa02 solution are bigxactly where error bars of the Fa02 solution are big eexactly in the place where KH80 shows a structure xactly in the place where KH80 shows a structure

not observed in the FA02not observed in the FA02

ThreeThree--channelchannel-model-model

THE SAME STORRY AS FOR TWO THE SAME STORRY AS FOR TWO CHANNELSCHANNELS

BUT MUCH MORE FINE TUNING IS NEEDED BUT MUCH MORE FINE TUNING IS NEEDED (better input us required)(better input us required)

STEP 7STEP 7::

Number of channels: Number of channels: 33 (pion elastic + effective)(pion elastic + effective)

Number of Number of GF GF propagator poles: propagator poles: 44

(2 (2 background poles + background poles + 22 physical poles)physical poles)

ONLYONLY ELASTIC ELASTIC CHANNELS IS FITTEDCHANNELS IS FITTED

STEP 8STEP 8::

Number of channels: Number of channels: 33 (pion elastic + effective)(pion elastic + effective)

Number of Number of GF GF propagator poles: propagator poles: 44

(2 (2 background poles + background poles + 22 physical poles)physical poles)

ELASTIC + INELASTICELASTIC + INELASTIC CHANNELS ARE FITTEDCHANNELS ARE FITTED

We offer three solutionsWe offer three solutions

Solution Solution 11

Solution Solution 22

Solution Solution 33

ConclusionsConclusions

1.1. T matrix poles, invisible when only elastic channel is T matrix poles, invisible when only elastic channel is analyzed, spontaneously appear in the coupled channel analyzed, spontaneously appear in the coupled channel formalism when inelastic channels are added.formalism when inelastic channels are added.

2.2. It is demonstrated thatIt is demonstrated that:: the N(1710) Pthe N(1710) P11 11 state state existsexists thethe pole is hidden in the continuum ambiguity of pole is hidden in the continuum ambiguity of VPI/VPI/GWU FA02GWU FA02 it it spontaneously appears when inelastic channels are spontaneously appears when inelastic channels are

introduced in addition to the elastic ones. introduced in addition to the elastic ones.

How do we proceed?How do we proceed?

1.1. Instead of using raw data we have decided to Instead of using raw data we have decided to represent them in a form of partial represent them in a form of partial wave T-matrices (single channel PWA, something wave T-matrices (single channel PWA, something else…else…

2.2. We use them as a further constraint in a CC_PWA We use them as a further constraint in a CC_PWA

Partial wave T- matrices

Experiment

I II III

Different data sets

BRAG ?BRAG ?

A call for helpA call for helpAnyone who has some kind of partial wave T-matrices,Anyone who has some kind of partial wave T-matrices,

regardless of the way how they were createdregardless of the way how they were created please sent it to us, please sent it to us,

so that we could, so that we could, within the framework of our formalism within the framework of our formalism,,

establish which poles are responsible for their shape.establish which poles are responsible for their shape.

Topics to be resolvedTopics to be resolved

The background should be introduced The background should be introduced in a different wayin a different way, , because the recipe of simulating the background contribution because the recipe of simulating the background contribution with two distant poles raises severe technical problems in with two distant poles raises severe technical problems in fitting procedure. fitting procedure.

The formalism should be re-organized in such a way that the T The formalism should be re-organized in such a way that the T matrix poles, and not a bare propagator poles become a fitting matrix poles, and not a bare propagator poles become a fitting parameter.parameter.

The existence of other, low star PDG resonances, shouldThe existence of other, low star PDG resonances, should be be checked.checked.