The importance of inelastic channels in eliminating continuum ambiguities in pion-nucleon partial wave analyses The importance of inelastic channels in eliminating continuum ambiguities in pion-nucleon partial wave analyses Alfred Švarc Ruđer Bošković Institute Croatia

Pg. 5 Pg. 6 constraints from fixed t-analyticity … resolve the ambiguities

The effects of discreet and continuum ambiguities were separated …. Dispersion relations were used to solve ambiguities and to derive constraints…

What does it mean “continuum ambiguity”?

Differential cross section is not sufficient to determine the scattering amplitude: if then The new function gives EXACTLY THE SAME CROSS SECTION

S – matrix unitarity …………….. conservation of flux RESTRICTS THE PHASE elastic region ……. unitarity relates real and imaginary part of each partial wave – equality constraint each partial wave must lie upon its unitary circle inelastic region ……. unitarity provides only an inequality constraint between real and imaginary part each partial wave must lie upon or inside its unitary circle there exists a whole family of functions , of limited magnitude but of infinite variety of functional form, which will behave exactly like that

These family of functions, though containing a continuum infinity of points, are limited in extend. The ISLANDS OF AMBIGUITY are created. there exists a whole family of functions , of limited magnitude but of infinite variety of functional form, which will behave exactly like that

I M P O R T A N T Once the three body channels open up, this way of eliminating continuum ambiguities (elastic channel arguments) become in principle impossible

I M P O R T A N T DISTINCTION theoretical islands of ambiguity / experimental uncertainties

The treatment of continuum ambiguity problems a.Constraining the functional form  a.Constraining the functional form  mathematical problem b.Implementing the partial wave T – matrix continuity (energy smoothing and search for uniqueness)

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

b.

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

Continuum ambiguity / T-matrix poles Each analytic function is uniquely defined with its poles and cuts. If an analytic function contains a continuum ambiguity it is not uniquely defined. T matrix is an analytic function in s,t. If an analytic function is not uniquely defined, we do not have a complete knowledge about its poles and cuts. Consequently fully constraining poles and cuts means eliminating continuum ambiguity

Basic idea: we want to demonstrate the role of inelastic channels in fully constraining the poles of the partial wave T-matrix, or, alternatively said, for eliminating continuum ambiguity which arises if only elastic channels a considered.

We want as well show that: supplying only scarce information for EACH channel is supplying only scarce information for EACH channel is MUCH MORE CONSTRAINING MUCH MORE CONSTRAINING then supplying the perfect information in ONE channel. then supplying the perfect information in ONE channel.

Coupled channel T matrix formalism 1.unitary 2.fully analytic

T-matrix poles are connected to the bare propagator poles, but shifted with the self energy term ! Important: real and imaginary parts of the self energy term are linked because of analyticity

Constraining data: Elastic channel: Pion elastic VPI SES solution FA02 Karlsruhe - Helsinki KH 80 Pion elastic VPI SES solution FA02 Karlsruhe - Helsinki KH 80

Inelastic channel: Inelastic channel:recent      CC PWA Pittsburgh/ANL 2000 however NO P 11 is offered ! older    CC PWA Zagreb/ANL 95/98      CC PWA Zagreb/ANL 95/98 gives P 11

Both: three channel version of CMB modelBoth: three channel version of CMB model  N elastic T matrices +  N   N data + dummy channel  N elastic T matrices +  N   N data + dummy channel Pittsburgh: VPI + data + dummy channel Pittsburgh: VPI + data + dummy channel Zagreb: KH80 + data + dummy channelZagreb: KH80 + data + dummy channel fitted all partal waves up to L = 4fitted all partal waves up to L = 4 Pittsburgh: offers S 11 only Pittsburgh: offers S 11 only Zagreb: offers P 11 as well (nucl-th/ )Zagreb: offers P 11 as well (nucl-th/ )

S 11 Let us compare Pittsburgh¸/ Zagreb S 11

Pittsburgh S 11 is taken as an “experimentally constrained” partial wave Pittsburgh S 11 is taken as an “experimentally constrained” partial wave

P 11 So we offer Zagreb P 11 as the “experimentally constrained” partial wave as well. from nucl-th/

Two-channel-model

STEP 1 : Number of channels: 2 (pion elastic + effective) (pion elastic + effective) Number of GF propagator poles: 3 (2 background poles + 1 physical pole) (2 background poles + 1 physical pole) ONLY ELASTIC CHANNEL IS FITTED

elastic channels is reproduced perfectlyelastic channels is reproduced perfectly inelastic channel is reproduced poorlyinelastic channel is reproduced poorly we identify one pole in the physical regionwe identify one pole in the physical region

STEP 2 : Number of channels: 2 (pion elastic + effective) (pion elastic + effective) Number of GF propagator poles: 3 (2 background poles + 1 physical pole) (2 background poles + 1 physical pole) ONLY INELASTIC CHANNEL IS FITTED

inelastic channel is reproduced perfectlyinelastic channel is reproduced perfectly elastic channel is reproduced poorlyelastic channel is reproduced poorly we identify two poles in the physical region, “Roper” and 1700 MeVwe identify two poles in the physical region, “Roper” and 1700 MeV

STEP 3: Number of channels: 2 (pion elastic + effective) (pion elastic + effective) Number of GF propagator poles: 3 (2 background poles + 1 physical pole) (2 background poles + 1 physical pole) ELASTIC + INELASTIC CHANNEL ARE FITTED

elastic channel is reproduced OKelastic channel is reproduced OK inelastic channel is reproduced tolerablyinelastic channel is reproduced tolerably we identify two poles in the physical region, but both are in the Roper – resonance regionwe identify two poles in the physical region, but both are in the Roper – resonance region

We can not find a “single pole” solution which would simultaneously reproduce ELASTIC AND INELASTIC CHANNELS

STEP 4: Number of channels: 2 (pion elastic + effective) (pion elastic + effective) Number of GF propagator poles: 4 (2 background poles + 2 physical poles) (2 background poles + 2 physical poles) ONLY ELASTIC CHANNEL IS FITTED

elastic channels is reproduced perfectlyelastic channels is reproduced perfectly inelastic channel is reproduced poorlyinelastic channel is reproduced poorly we identify two poles in the physical region, Roper + one above 2200 MeVwe identify two poles in the physical region, Roper + one above 2200 MeV

STEP 5: Number of channels: 2 (pion elastic + effective) (pion elastic + effective) Number of GF propagator poles: 4 (2 background poles + 2 physical poles) (2 background poles + 2 physical poles) ONLY INELASTIC CHANNEL IS FITTED

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

Solution 1

Solution 2

For both solutions: elastic channel is poorly reproduced, AND differs for both solutionselastic channel is poorly reproduced, AND differs for both solutions inelastic channel is OKinelastic channel is OK For both solutions we identify two poles in the physical region, Roper MeVFor both solutions we identify two poles in the physical region, Roper MeV

STEP 6: Number of channels: 2 (pion elastic + effective) (pion elastic + effective) Number of GF propagator poles: 4 (2 background poles + 2 physical poles) (2 background poles + 2 physical poles) ELASTIC + INELASTIC CHANNELS ARE FITTED

We offer two possible solutions which differ significantly in channels which are not fitted

Solution 1

Solution 2

For both solutions: elastic channel is acceptably reproduced, AND differs for both solutions BUT ADDITIONAL STRUCTURE IN THE ELASTIC CHANNEL APPEARES IN THE ENERGY RANGE OF1700 MeVelastic channel is acceptably reproduced, AND differs for both solutions BUT ADDITIONAL STRUCTURE IN THE ELASTIC CHANNEL APPEARES IN THE ENERGY RANGE OF1700 MeV inelastic channel is OKinelastic channel is OK For both solutions we identify two poles in the physical region, Roper MeVFor both solutions we identify two poles in the physical region, Roper MeV

An observation : The structure in elastic channel, required by the presence of inelastic channels, appears:  exactly where error bars of the Fa02 solution are big  exactly in the place where KH80 shows a structure not observed in the FA02

Three-channel-model

THE SAME STORRY AS FOR TWO CHANNELS BUT MUCH MORE FINE TUNING IS NEEDED (better input us required)

STEP 7: Number of channels: 3 (pion elastic + effective) (pion elastic + effective) Number of GF propagator poles: 4 (2 background poles + 2 physical poles) (2 background poles + 2 physical poles) ONLY ELASTIC CHANNELS IS FITTED

STEP 8: Number of channels: 3 (pion elastic + effective) (pion elastic + effective) Number of GF propagator poles: 4 (2 background poles + 2 physical poles) (2 background poles + 2 physical poles) ELASTIC + INELASTIC CHANNELS ARE FITTED ELASTIC + INELASTIC CHANNELS ARE FITTED

We offer three solutions

Solution 1

Solution 2

Solution 3

Conclusions 1.T matrix poles, invisible when only elastic channel is analyzed, spontaneously appear in the coupled channel formalism when inelastic channels are added. 2.It is demonstrated that:  the N(1710) P 11 state exists  the pole is hidden in the continuum ambiguity of VPI/GWU FA02  it spontaneously appears when inelastic channels are introduced in addition to the elastic ones.

How do we proceed? 1.Instead of using raw data we have decided to represent them in a form of partial wave T-matrices (single channel PWA, something else… 2.We use them as a further constraint in a CC_PWA

Partial wave T- matrices Experiment III III Different data sets BRAG ? A call for help Anyone who has some kind of partial wave T-matrices, regardless of the way how they were created please sent it to us, so that we could, within the framework of our formalism, establish which poles are responsible for their shape.

Topics to be resolved  The background should be introduced in a different way, because the recipe of simulating the background contribution with two distant poles raises severe technical problems in fitting procedure.  The formalism should be re-organized in such a way that the T matrix poles, and not a bare propagator poles become a fitting parameter.  The existence of other, low star PDG resonances, should be checked.