Data Interpretation of Baryon Resonances A. Švarc Rudjer Bošković Institute, Zagreb, Croatia CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017

Poles as resonance signal CLAS Collaboration Meeting 13-16 June 2017 Main task Poles as resonance signal Experiment CLAS Collaboration Meeting 13-16 June 2017

Poles as resonance signal CLAS Collaboration Meeting 13-16 June 2017 Main task Poles as resonance signal PWA Experiment SE PWA MODEL MODEL CLAS Collaboration Meeting 13-16 June 2017

Poles as resonance signal CLAS Collaboration Meeting 13-16 June 2017 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 CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 Continuum ambiguity All observables in single-channel processes are described as bilinears of invariant amplitudes! 𝑶 ≡ 𝒇( 𝑨 𝒊 ∙𝑩 𝒋 ∗ ) CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 Example: η photoproduction CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 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! CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 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! CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 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! CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 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 CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 So, our problem is: CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 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 CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 Explicitly: CLAS Collaboration Meeting 13-16 June 2017

Importance of angle dependent phase rotations in PWA Influence upon unconstrained SE PWA CLAS Collaboration Meeting 13-16 June 2017

Fitted invariant amplitudes CLAS Collaboration Meeting 13-16 June 2017 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 CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 We perform SINGLE ENERGY UNCONSTRAINED LMAX = 8 truncated PWA of η photoproduction pseudo-data CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 dσ/dΩ Data F P Oz’ T Σ Cx’ CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 Two different initial values two different solutions with the same χ2 Sol 1: MAID16a Sol 2: Bon-Gatchina CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 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 CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 Let me illustrate the problem CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 Let me illustrate the problem CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 generating model CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 Let me solve the problem CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 PWA9/ATHOS4 Bad Honnef 2017 30 CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 This means that all three SE solutions: Generating solution MAID15a Sol 1 Sol 2 can be transformed into each other by an angle dependent phase rotation! SO THEY ARE EQUIVALENT! MAID15a Sol 1 Sol 2 Question: Can we use it to stabilize unconstrained SE PWA? CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 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 CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 Problem solved CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 VERY IMPORTANT! we obtain the „anchor” point CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 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 CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 References: CLAS CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 CLAS seminar at JLab Friday, August 18th, at 11.00 CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 SUMMARY CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017 Future prospects We need hard work… We have the tool CLAS Collaboration Meeting 13-16 June 2017

CLAS Collaboration Meeting 13-16 June 2017