A. Švarc Rudjer Bošković Institute, Zagreb, Croatia Partial wave Analysis and Baryon Resonances A. Švarc Rudjer Bošković Institute, Zagreb, Croatia I will mostly talk about single energy (energy independent) partial wave analysis! Boppard 2016

Some introductory remarks: Start with the full set of formula for η - photoproduction Discuss essential parts Focus on knowledge which is not to be found in text-books Problem of precise definitions of terms in use Process or elimination errors: What is true and what is wrong when text-books do not give you the answer Boppard 2016

Partial wave analysis in η photoproduction Boppard 2016

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How does one obtain partial waves? One has to measure the COMPLETE set of observables Extracting partial waves Using theoretical models (STANDARD) Performing single energy (energy independent) PWA NO MODEL Boppard 2016

Extracting partial waves using theoretical models (STANDARD) Boppard 2016

In SE PWA with NO THEORETICAL MODEL these assumptions simply pop-up ! Problem: In such a procedure many assumptions are hidden! In SE PWA with NO THEORETICAL MODEL these assumptions simply pop-up ! I will focus my talk on SE PWA and stress what exactly which are additional assumptions which ARE AUTHOMATICALLY DEFINED if one uses theoretical models. Boppard 2016

Title contains two notions: Partial waves Baryon resonances We defined partial waves. Let us define the notion baryon resonances. What are we looking for in partial waves? Boppard 2016

What is a resonance Graz 2015

Various definitions ! Graz 2015

For me, the most transparent discussion is given in: Graz 2015

"Adventures in Mathematical Physics" (Proceedings, Cergy-Pontoise 2006), Contemporary Mathematics, 447 (2007) 73-81 Graz 2015

Interrelation of scattering and resolvent resonances: Concept of resolvent resonances is at length discussed in Graz 2015

This is a reason for numerous disputes and controversies. Both definitions of resonances are being used in the literature sometimes without full awareness that they are different, and that both are in principle allowed. This is a reason for numerous disputes and controversies. Knowing that we are dealing with the two equivalent quantifications of the same phenomenon solves the issue. Let us remember: our task is not only describe resonances, but describe them in a way which is identical to QCD. The final answer to which of the two resonance definitions should be used comes from analyzing what is precisely calculated in QCD! Is it a time delay or a resolvent pole? Graz 2015

What is a resonance in QCD? Talk presented at the Workshop "Light-cone Physics: Particles and Strings" at ECT* in Trento, Sep 3-11, 2001 ...solving the bound-state problem in gauge filed theory, particularly QCD... Graz 2015

Light-cone approach How is it done? resonances Hamiltonian proper values Remember Dalitz-Moorhouse Hamiltonian proper values poles Graz 2015

QCD is analyzing resolvent resonances ! So, in principle: QCD is analyzing resolvent resonances ! Graz 2015

So we recently agreed that poles are the true quantities which quantify resonances! Illustration: PDG PDG 2015 PDG 2015 – encoder tools Boppard 2016

I. Continuum ambiguity Boppard 2016

PHASE AMBIGUITY - CONTINUUM AMBIGUITY We start looking for poles. Here comes the first ambiguity non-existing in theoretical models: PHASE AMBIGUITY - CONTINUUM AMBIGUITY What is continuum ambiguity? In theoretical models : automatically defined In SE PWA: FREE ENERGY AND ANGLE DEPENDENT PHASE Boppard 2016

Discrete and continuum ambiguities Trento 2014

Continuum ambiguities Trento 2014

Question: What does it do to our problem? Direct consequence: Unconstrained SE PWA is discontinuous in energy We need stabilization procedures ! Boppard 2016

One way: fixed-t analiticitity Boppard 2016

t – analyticity in fixed-t minimization Boppard 2016

s –channel penalty function Boppard 2016

II. Cut direction Boppard 2016

parameterize partial amplitudes directly in term of poles How to extract poles? First idea: parameterize partial amplitudes directly in term of poles   B(W) …………. background Unfortunately, this is IMPOSSIBLE! In scattering theory resonances are poles on the II - Riemann sheet, and the only poles on the I-st Riemann are bound states. Unfortunately, constant poles appear on all Riemann sheets, hence on the I-st one as well. FORBIDDEN! Boppard 2016

For more details see: Boppard 2016

Therefore, we have to create poles with more complicated functions! The simplest choice is to have functions with energy dependent parameters and cuts in the complex energy plane. Lothar Tiator constructed a very simple, educative model with only square root cuts! See at: http://www-f1.ijs.si/BledPub/bled2015.pdf Boppard 2016

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Where do you rotate cuts? Question: Where do you rotate cuts? Mathematics says that you can rotate cuts ANY WAY YOU LIKE! Does physics agree with that? Let us show what is happening with poles if square root cut is rotated in different directions! Simple rules how to define the direction of the cut!   Boppard 2016

Example: Boppard 2016

We see that we have 4 possible branch-points and a lot of combinations of directions! (D, D, D, D) (L, D, D, D) (D, D, D, R) Boppard 2016

And 3D plot for (D, D, D, D) Boppard 2016

Model is simple, and produces unphysical poles below threshold! Let us focus on: Physical range Only on the physical cut m + mπ Three possible directions of rotations (L,D,R) Boppard 2016

(D, D, D, L) (D, D, D, D) (D, D, D, R) Boppard 2016

HAS TO BE ROTATED TO THE RIGHT! Important! If we want to agree with scattering theory demand that only bounds states can have poles on the I-st Riemann sheet, and that resonances are poles on the II-nd Riemann sheet, physical cut HAS TO BE ROTATED TO THE RIGHT! Discussion! When can we rotate cuts? Boppard 2016

New example: elastic cut + inelastic cut on the real axes Boppard 2016

Elastic cut to the right Elastic cut down Elastic cut to the right Boppard 2016

? bound states resonances bound states resonances How do I see what is our main task? Experiment Matching point Theory QCD bound states resonances ? bound states resonances What is a resonance in QCD? What is a resonance in experiment? Boppard 2016

So the answer to our question is:  POLES Boppard 2016

II How to extract T-matrix poles from partial waves Boppard 2016

Speed plot, expansions in power series, etc The usual answer was: Do it globally One first has to make a model which fits the data, SOLVE IT, and obtain an explicit analytic function in the full complex energy plane. Second, one has to look for the complex poles of the obtained analytic functions. Do it locally Speed plot, expansions in power series, etc Boppard 2016

Direct problems for global solutions: Many models Complicated and different analytic structure Elaborated method for solving the problem SINGLE USER RESULTS Boppard 2016

Local approach Boppard 2016

Eliminate or reduce the dependence upon background contribution Speed plot Idea behind it? Eliminate or reduce the dependence upon background contribution Graz 2015

Padé expansion Graz 2015

Regularization method Boppard 2016

PROBLEMS for local solutions ! In both cases we have n-TH DERIVATIVE of the function PROBLEMS for local solutions ! Boppard 2016

I have tryed to do it starting from very general principles: Is it possible to create universal approach, usable for everyone, and above all REPRODUCIBLE? I have tryed to do it starting from very general principles: Analyticity Unitarity Idea: TRADING ADVANTAGES GLOBALITY FOR SIMPLICITY Boppard 2016

THEORETICAL MODELS If you create a model, the advantage is that your solution is absolutely global, valid in the full complex energy plane (all Riemann sheets). The drawback is that the solution is complicated, pole positions are usually energy dependent otherwise you cannot ensure simple physical requirements like absence of the poles on the first, physical Riemann sheet, Schwartz reflection principle, etc. It is complicated and demanding to solve it. WE PROPOSE Construct an analytic function NOT in the full complex energy plane, but CLOSE to the real axes in the area of dominant nucleon resonances, which is fitting the data by using LAURENT EXPANSION. Boppard 2016

Why Laurent’s decomposition? It is a unique representation of the complex analytic function on a dense set in terms of pole parts and regular background It explicitly seperates pole terms from regular part It has constant pole parameters It is not a representation in the full complex energy plane, but has its well defined area of convergence IMPORTANT TO UNDERSTAND: It is not an expansion in pole positions with constant coefficients (as some referees reproached), because it is defined only in a part of the complex energy plane. Boppard 2016

Expansion of the T-matrix in terms of constant coefficients cannot be valid in principle. Namely, poles with constant coefficients have poles on ALL physical sheets, and that violates common sense because only bound states are allowed to be located on the physical sheet. Boppard 2016

However, even this function has its Laurent decomposition The only way how to accomodate both, requirements of absence of poles on the physical sheet, and Schwartz principle requires that pole positions become energy dependent: However, even this function has its Laurent decomposition But it is valid only in the part of the complex energy plane Boppard 2016

1. Analyticity Analyticity is introduced via generalized Laurent’s decomposition (Mittag-Leffler theorem) Boppard 2016

Now, we have two parts¸of Laurent’s decomposition: Poles Regular part Assumption: We are working with first order poles so all negative powers in Laurent’s expansion lower than n -1 are suppressed Now, we have two parts¸of Laurent’s decomposition: Poles Regular part Boppard 2016

The problem is how to determine regular function B(w).  The problem is how to determine regular function B(w). What do we know about it? We know it’s analytic structure for each partial wave! We do not know its EXPLICIT analytic form! Boppard 2016

So, instead of „guessing” its exact form by using model assumptions we EXPAND IT IN FASTLY CONVERGENT POWER SERIES OF PIETARINEN („Z”) FUNCTIONS WITH WELL KNOWN BRANCH-POINTS! S. Ciulli and J. Fischer in Nucl. Phys. 24, 465 (1961) I. Ciulli, S. Ciulli, and J. Fisher, Nuovo Cimento 23, 1129 (1962). Original idea: S. Ciulli and J. Fischer in Nucl. Phys. 24, 465 (1961) Detailed proof in I. Caprini and J. Fischer: "Expansion functions in perturbative QCD and the determination of αs", Phys.Rev. D84 (2011) 054019, Convergence proven in: E. Pietarinen, Nuovo Cimento Soc. Ital. Fis. 12A, 522 (1972). Hoehler – Landolt Boernstein „BIBLE” (1983) Applied in πN scattering on the level of invariant amplitudes PENALTY FUNCTION INTRODUCED NAMING ! Boppard 2016

What is Pitarinen’s expansion? In principle, in mathematical language, it is ” ...a conformal mapping which maps the physical sheet of the ω-plane onto the interior of the unit circle in the Z-plane...” In practice this means: Boppard 2016

Or in another words, Pietarinen functions Z(ω) are a complet set of functions for an arbitrary function F(ω) which HAS A BRANCH POINT AT xP ! Observe: Pietarinen functions do not form a complete set of functions for any function, but only for the function having a well defined branch point. Boppard 2016

Illustration: Powes series for Z(ω) = Boppard 2016

Z(ω) Boppard 2016

Z(ω)2 Boppard 2016

Z(ω)3 Boppard 2016

A resonance CANNOT be well described by Pietarinen series. Important! A resonance CANNOT be well described by Pietarinen series. Boppard 2016

Finally, the area of convergence for Laurent expansion of P11 partial wave Boppard 2016

The model Boppard 2016

Pietarinen power series We use Mittag-Leffleur decomposition of „analyzed” function: regular background k - simple poles We know analytic properties (number and position of cuts) of analyzed function ONE Pietarinen power series per cut Boppard 2016

Method has problems, and the one of them definitely is: There is a lot of cuts, so it is difficult to imagine that we shall be able to represent each cut with one Pietarinen series (too many possibly interfering terms). Answer: We shall use „effective” cuts to represent dominant effects. We use three Pietarinen series: One to represent subthreshold, unphysical contributions Two in physical region to represent all inelastic channel openings Strategy of choosing branchpoint positions is extremely important and will be discussed later Boppard 2016

The method is „self-checking” ! It might not work. Advantage: The method is „self-checking” ! It might not work. But, if it works, and if we obtain a good χ2, then we have obtained AN ANALYTIC FUNCTION WITH WELL KNOWN POLES AND CUTS WHICH DEFINITELY DESCRIBES THE INPUT! So, if we have disagreements with other methods, then we are looking at two different analytic functions which are almost identical on a discrete set, so we may discuss the general stability of the problem. However, our solution definitely IS A SOLUTION! Boppard 2016

What can we do with this model? We may analyze various kinds of inputs Theoretical curves coming from ANY model but also Information coming directly from experiment (partial wave data) Observe: Partial wave data are much more convenient to analyze! To fit „theoretical input” we have to „guess” both: pole position AND analyticity structure of the background imposed by the analyzed model exact To fit „experimental input” we have to „guess” only: pole position AND analyticity structure of the background as no information about functional type is imposed the simplest Boppard 2016

Testing is a very simple procedure. It comes to: Does it work? Testing is a very simple procedure. It comes to: Doesn’t work Works TESTING Testing on a toy model: Testing and application on realistic amplitudes πN elastic scattering ED PW amplitudes (some solutions from GWU/SAID) ED PW amplitudes (some solutions from Dubna-Mainz-Taipei) Photo – and electroproduction on nucleon ED multipoles (all solutions from MAID and SAID) SES multipoles (all solutions from MAID and SAID) arXiv nucl-th 1212.1295 Boppard 2016

a. Toy model Boppard 2016

We have constructed a toy model using two poles and two cuts, used it to construct the input data set, attributed error bars of 5%, and tried to use L+P method to extract pole parameters under different conditions. C1, C2, B1, B2 = -1, 0, 1 Boppard 2016

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b. Testing on realistic amplitude πN elastic GWU/SAID FA02 GWU/SAID SP06 GWU/SAID WI08 DMT Photoproduction GWU/SAID ZN11 ED Boppard 2016

Quality of the fit Boppard 2016

πN elastic scattering SAID FA02 ED Boppard 2016

Conclusion The L+P method defined as: WORKS Boppard 2016

World recognition Graz 2015

Graz 2015

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