From Experimental Data to Pole Parameters in a Model Independent Way (Angle Dependent Continuum Ambiguity and Laurent + Pietarinen Expansion)   A. Švarc Rudjer Bošković Institute, Zagreb, Croatia BLED 2017

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Poles as resonance signal Main task Poles as resonance signal Experiment BLED 2017

Poles as resonance signal Main task Poles as resonance signal PWA Experiment SE PWA MODEL MODEL BLED 2017

Poles as resonance signal Main task Poles as resonance signal PWA Experiment SE PWA MODEL MODEL Phase ambiguity Laurent + Pietarinen From Experimental Data to Pole Parameters in a Model Independent Way BLED 2017

Continuum ambiguity 𝑶 ≡ 𝒇( 𝑨 𝒊 ∙𝑩 𝒋 ∗ ) All observables in single-channel processes are described as bilinears of invariant amplitudes! 𝑶 ≡ 𝒇( 𝑨 𝒊 ∙𝑩 𝒋 ∗ ) BLED 2017

Example: η photoproduction BLED 2017

Therefore, the simultaneous transformation 1-fold rotation does not change anything! A consequence: ALL SC OBSERVABLES ARE INVARIANT WITH RESPECT TO ENERGY AND ANGLE DEPENDENT PHASE ROTATION where Φ(W,θ) is an arbitrary energy and ANGLE dependent real function! This is called CONTINUUM AMBIGUITY! BLED 2017

It is usually done on the level of partial waves: Energy dependent phase rotation has been discussed very often when two different models are compared! It is usually done on the level of partial waves: We choose one multipole Match the phase of that multipole Multiply all other multipoles with that phase Angle dependent phase rotation has hardly been ever discussed! BLED 2017

For η photoproduction partial wave decomposition is defined as Let us see what is the influence of angle dependent continuum ambiguity upon a partial wave decomposition! Example For η photoproduction partial wave decomposition is defined as Let us simplify things somewhat! BLED 2017

Legendre polynomials satisfy recursive relations All derivatives of Legendre polynomials can be reduced to lower index Legendre polynomials. So, in practice we deal with the general problem BLED 2017

So, our problem is: BLED 2017

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Angle dependent phase rotations mix multipoles is a long time ago given in However, Yannick Wunderlich has in his thesis Derived a very nice closed formula: Angle dependent phase rotations mix multipoles BLED 2017

Explicitly: BLED 2017

Importance of angle dependent phase rotations in PWA Influence upon unconstrained SE PWA BLED 2017

Fitted invariant amplitudes General testing scheme of a procedure by using numeric or pseudo data Theoretical multipoles Invariant amplitudes Observables (complete set) SE PWA I D E N T I C A L Fitted invariant amplitudes Fitted multipoles Fitted observables BLED 2017

truncated PWA of η photoproduction pseudo-data We perform SINGLE ENERGY UNCONSTRAINED LMAX = 8 truncated PWA of η photoproduction pseudo-data BLED 2017

G dσ/dΩ Data F P Oz’ T Σ Cx’ BLED 2017

Two different initial values two different solutions with the same χ2 Sol 1: MAID16a Sol 2: Bon-Gatchina BLED 2017

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PROBLEM ! Complex analysis Sol 1 and Sol 2 must be identical to the generating model as we fitted a complete set …. Continuum ambiguity Sol 1 and Sol 2 must be identical up to a phase Conclusion There must exist a phase which connects Sol 1 and Sol 2. And the phase MUST be defined at the level of AMPLITUDES and NOT partial waves BLED 2017

Let me illustrate the problem BLED 2017

Let me illustrate the problem BLED 2017

Let us plot THE PHASE of helicity amplitude H2 BLED 2017

generating model BLED 2017

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Let me solve the problem BLED 2017

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PWA9/ATHOS4 Bad Honnef 2017 31 BLED 2017

This means that all three SE solutions: Generating solution MAID15a can be transformed into each other by a phase rotation at arbitrary energy and arbitrary ANGLE! SO THEY ARE EQUIVALENT! MAID15a Sol 1 Sol 2 Question: Can we use it to stabilize unconstrained SE PWA? BLED 2017

𝑯 𝒌 𝑺𝑬𝟏𝟔𝒂 (W, 𝒙)= 𝑯 𝒌 𝑺𝑬𝟏𝟔𝒂 𝒆 𝒊 𝚽 𝑯 𝟐 𝟏𝟓𝒂 𝑾,𝒙 −𝒊 𝚽 𝑯 𝟐 𝑺𝑬𝟏𝟔𝒂 𝑾,𝒙 Let us introduce the following angle dependent phase rotation Sol 1: 𝑯 𝒌 𝑺𝑬𝟏𝟔𝒂 (W, 𝒙)= 𝑯 𝒌 𝑺𝑬𝟏𝟔𝒂 𝒆 𝒊 𝚽 𝑯 𝟐 𝟏𝟓𝒂 𝑾,𝒙 −𝒊 𝚽 𝑯 𝟐 𝑺𝑬𝟏𝟔𝒂 𝑾,𝒙 Sol 2: 𝑯 𝒌 𝑺𝑬𝑩𝒐𝑮𝒂 (W, 𝒙)= 𝑯 𝒌 𝑺𝑬𝑩𝒐𝑮𝒂 𝒆 𝒊 𝚽 𝑯 𝟐 𝟏𝟓𝒂 𝑾,𝒙 −𝒊 𝚽 𝑯 𝟐 𝑺𝑬𝑩𝒐𝑮𝒂 𝑾,𝒙 Both rotated SE solutions now have the same H2 phase; phase of generating 15a model BLED 2017

Problem solved BLED 2017

we obtain the „anchor” point VERY IMPORTANT! we obtain the „anchor” point BLED 2017

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References: CLAS BLED 2017

II How to extract T-matrix poles from partial waves BLED 2017

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 BLED 2017

Direct problems for global solutions: Many models Complicated and different analytic structure Elaborated method for solving the problem SINGLE USER RESULTS BLED 2017

Local approach BLED 2017

Eliminate or reduce the dependence upon background contribution Speed plot Idea behind it? Eliminate or reduce the dependence upon background contribution BLED 2017

Padé expansion BLED 2017

Regularization method BLED 2017

PROBLEMS for local solutions ! In both cases we have n-TH DERIVATIVE of the function PROBLEMS for local solutions ! BLED 2017

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 BLED 2017

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. BLED 2017

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. BLED 2017

Expansion of the T-matrix in terms of constant coefficients cannot be valid everywhere in C 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. BLED 2017

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 BLED 2017

1. Analyticity Analyticity is introduced via generalized Laurent’s decomposition (Mittag-Leffler theorem) BLED 2017

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 BLED 2017

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 EXPLICT analytic form! BLED 2017

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 ! BLED 2017

By L. Tiator 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...” By L. Tiator BLED 2017

In practice this means: BLED 2017

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. BLED 2017

Illustration: Powes series for Z(ω) = BLED 2017

Z(ω) BLED 2017

Z(ω)2 BLED 2017

Z(ω)3 BLED 2017

A resonance CANNOT be well described by Pietarinen series. Important! A resonance CANNOT be well described by Pietarinen series. BLED 2017

Finally, the area of convergence for Laurent expansion of P11 partial wave BLED 2017

The model BLED 2017

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 BLED 2017

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 BLED 2017

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! BLED 2017

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 BLED 2017

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 BLED 2017

Conclusion for L+P The L+P method defined as: WORKS BLED 2017

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L+P method is tested and well documented Conformal-mapping generated fast converging expansion of regular part of Laurent decomposition around physical branchpoints in the limited part of 2-nd Riemann sheet of complex energy plane near real axes BLED 2017

SUM MARY BLED 2017

Future prospects We need hard work… We have the tool BLED 2017

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