Scalar Dipole Dynamical Polarizabilities from proton Real Compton - PowerPoint PPT Presentation

15th International Workshop on Meson Physics Krakóv, Poland, June 7 th - 12 th 2018 Scalar Dipole Dynamical Polarizabilities from proton Real Compton Scattering data University of Pavia & INFN, Pavia (Italy) Stefano Sconfjetti 1 / 24

Outline ✔ RCS: theoretical framework ✔ Static polarizabilities ✔ Low Energy expansion ✔ Dynamical polarizabilities MESON 2018, KRAKÓW 2 / 24

Outline ✔ RCS: theoretical framework ✔ Static polarizabilities ✔ Low Energy expansion ✔ Dynamical polarizabilities ✔ Data set inconsistency ✔ Very high correlations ✔ T oo much parameters ✔ New approach: simplex + bootstrap MESON 2018, KRAKÓW 2 / 24

Outline ✔ RCS: theoretical framework ✔ Static polarizabilities ✔ Low Energy expansion ✔ Dynamical polarizabilities ✔ Data set inconsistency ✔ Very high correlations ✔ T oo much parameters ✔ New approach: simplex + bootstrap ✔ Static polarizabilities: cross check ✔ Systematical errors ✔ Dynamical polarizabilities from data ✔ Conclusions and future perspectives MESON 2018, KRAKÓW 2 / 24

RCS amplitudes and Dispersion Relations D. Drechsel, M. Gorchtein, B. Pasquini, M. Vanderhaeghen, Phys.Rev. C61 (1999) 015204 3 / 24 B. Pasquini , D. Drechsel M. Vanderhaeghen, Phys.Rev. C76 (2007) 015203

RCS amplitudes and Dispersion Relations Static polarizabilities A i (0,0) = a i Subtracted Dispersion Relations ( s -channel) D. Drechsel, M. Gorchtein, B. Pasquini, M. Vanderhaeghen, Phys.Rev. C61 (1999) 015204 3 / 24 B. Pasquini , D. Drechsel M. Vanderhaeghen, Phys.Rev. C76 (2007) 015203

RCS amplitudes and Dispersion Relations Static polarizabilities A i (0,0) = a i Subtracted Dispersion Relations ( s -channel) s -CHANNEL t -CHANNEL D. Drechsel, M. Gorchtein, B. Pasquini, M. Vanderhaeghen, Phys.Rev. C61 (1999) 015204 3 / 24 B. Pasquini , D. Drechsel M. Vanderhaeghen, Phys.Rev. C76 (2007) 015203

Static polarizabilities Powell cross section: point-like nucleon with anomalous magnetic moment Static polarizabilities : response of the internal nucleon degrees of freedom to a static electric and magnetic fjeld A.I. L'vov, V.A. Petrun'kin, Martin Schumacher. Phys.Rev. C55 (1997) 359-377 4 / 24 D. Drechsel, B. Pasquini, M. Vanderhaeghen. Phys.Rept. 378 (2003) 99-205

Static polarizabilities Powell cross section: point-like nucleon with anomalous magnetic moment Static polarizabilities : response of the internal nucleon degrees of freedom to a static electric and magnetic fjeld spin-independent dipole A.I. L'vov, V.A. Petrun'kin, Martin Schumacher. Phys.Rev. C55 (1997) 359-377 4 / 24 D. Drechsel, B. Pasquini, M. Vanderhaeghen. Phys.Rept. 378 (2003) 99-205

Static polarizabilities Powell cross section: point-like nucleon with anomalous magnetic moment Static polarizabilities : response of the internal nucleon degrees of freedom to a static electric and magnetic fjeld spin-independent dipole spin-dependent dipole spin-dependent dipole- quadrupole A.I. L'vov, V.A. Petrun'kin, Martin Schumacher. Phys.Rev. C55 (1997) 359-377 4 / 24 D. Drechsel, B. Pasquini, M. Vanderhaeghen. Phys.Rept. 378 (2003) 99-205

Static polarizabilities Powell cross section: point-like nucleon with anomalous magnetic moment Static polarizabilities : response of the internal nucleon degrees of freedom to a static electric and magnetic fjeld spin-independent dipole spin-dependent dipole spin-dependent dipole- quadrupole A.I. L'vov, V.A. Petrun'kin, Martin Schumacher. Phys.Rev. C55 (1997) 359-377 4 / 24 D. Drechsel, B. Pasquini, M. Vanderhaeghen. Phys.Rept. 378 (2003) 99-205

Low Energy Expansion (LEX) (I) D. Babusci, G. Giordano, A.I. L'vov, G. Matone, A.M. Nathan. Phys.Rev.C58 (1998) 1013-1041 5 / 24

Low Energy Expansion (LEX) (I) ν and t as independent variables Lorentz invariant amplitudes (ready for R i (A i ) multipole expansion) Need to choose a ref-frame: CM D. Babusci, G. Giordano, A.I. L'vov, G. Matone, A.M. Nathan. Phys.Rev.C58 (1998) 1013-1041 5 / 24

Multipole expansion and DYNAMICAL polarizabilities D. Babusci, G. Giordano, A.I. L'vov, G. Matone, A.M. Nathan.Phys.Rev. C58 (1998) 1013-1041 6 / 24 V. Lensky, J. McGovern, and V. Pascalutsa, Eur. Phys. J. C75, 604 (2015)

Multipole expansion and DYNAMICAL polarizabilities DYNAMICAL POLARIZABILITIES D. Babusci, G. Giordano, A.I. L'vov, G. Matone, A.M. Nathan.Phys.Rev. C58 (1998) 1013-1041 6 / 24 V. Lensky, J. McGovern, and V. Pascalutsa, Eur. Phys. J. C75, 604 (2015)

Multipole expansion and DYNAMICAL polarizabilities DYNAMICAL POLARIZABILITIES DIPOLE DYNAMICAL POLARIZABILITIES (DDPs) β M1 (ω) α E1 (ω) D. Babusci, G. Giordano, A.I. L'vov, G. Matone, A.M. Nathan.Phys.Rev. C58 (1998) 1013-1041 6 / 24 V. Lensky, J. McGovern, and V. Pascalutsa, Eur. Phys. J. C75, 604 (2015)

DDPs: physical meaning F . Hagelstein, R. Miskimen, V. Pascalutsa. Prog.Part.Nucl.Phys. 88 (2016) 29-97 7 / 24

DDPs: physical meaning DDPs : response of the internal nucleon degrees of freedom to Pion cusp Delta resonance an electric and magnetic fjeld with and explicit dependence on energy F . Hagelstein, R. Miskimen, V. Pascalutsa. Prog.Part.Nucl.Phys. 88 (2016) 29-97 7 / 24

Low Energy Expansion (LEX) (II) α E1 (ω) = α E10 + α E11 ω + α E12 ω 2 + α E13 ω 3 + α E14 ω 4 + α E15 ω 5 β M1 (ω) = β M10 + β M11 ω + β M12 ω 2 + β M13 ω 3 + β M14 ω 4 + β M15 ω 5 8 / 24

Low Energy Expansion (LEX) (II) α E1 (ω) = α E10 + α E11 ω + α E12 ω 2 + α E13 ω 3 + α E14 ω 4 + α E15 ω 5 β M1 (ω) = β M10 + β M11 ω + β M12 ω 2 + β M13 ω 3 + β M14 ω 4 + β M15 ω 5 β M1 (10 -4 fm 3 ) α E1 (10 -4 fm 3 ) 0 ω cm (MeV) 130 0 ω cm (MeV) 130 Full DRs calculation LEX + fjt to DRs 8 / 24

DRs vs ( LEX + multipoles) θ = 1 1 2 ° LEX + multipoles DRS calculation LEX: up to ω 5 multipoles: up to l=3 9 / 24

The GOAL Extract scalar Dipole Dynamical Polarizabilites ( DDPs ) from RCS data B. Pasquini, P . Pedroni, S. S., arXiv:1711.07401v1 10 / 24

Complications Gradient method to fjnd the χ 2 minimum VERY high correlations between parameters! 11 / 24

Complications Gradient method to fjnd the χ 2 minimum VERY high correlations between parameters! MINUIT WARNING IN HESSE ============== MATRIX FORCED POS-DEF BY ADDING 0.13727E-01 TO DIAGONAL. 11 / 24

Complications Gradient method to fjnd the χ 2 minimum VERY high correlations between parameters! MINUIT WARNING IN HESSE ============== MATRIX FORCED POS-DEF BY ADDING 0.13727E-01 TO DIAGONAL. VERY low sensitivity of the data to dynamical polarizabilities NO WAY to fjnd the “right” minimum and to defjne “right” errors on fjt parameters 11 / 24

Complications Gradient method to fjnd the χ 2 minimum VERY high correlations between parameters! MINUIT WARNING IN HESSE ============== MATRIX FORCED POS-DEF BY ADDING 0.13727E-01 TO DIAGONAL. VERY low sensitivity of the data to dynamical polarizabilities NO WAY to fjnd the “right” minimum and to defjne “right” errors on fjt parameters Combination of SIMPLEX method and BOOTSTRAP technique 11 / 24

The DATA set Half of the Spartans that King Leonidas led to the Battle of Thermopylae... 12 / 24

The DATA set Half of the Spartans θ θ = 4 5 ° = 1 1 2 ° that King Leonidas led to the Battle of Thermopylae... θ = 1 3 5 ° θ = 6 0 ° θ = 1 5 5 ° θ = 8 5 ° 12 / 24

DRs vs ( LEX + multipoles) θ = 1 1 2 ° FIRST cross check: comparison with LEX + multipoles and DRs B. Pasquini, P . Pedroni, S. S., arXiv:1711.07401v1 13 / 24

DRs vs ( LEX + multipoles) θ = 1 1 2 ° FIRST cross check: comparison with LEX + multipoles and DRs α E1 (10 -4 fm 3 ) β M1 (10 -4 fm 3 ) DRs 11.9 ± 0.2 1.9 ± 0.2 LEX + MUL TIPOLES 11.8 ± 0.2 2.0 ± 0.2 B. Pasquini, P . Pedroni, S. S., arXiv:1711.07401v1 13 / 24

Bootstrap sampling and systematics S i,exp boot = S i,exp ±γ σ i,exp Gaussian distributed 14 / 24

Bootstrap sampling and systematics S i,exp boot = S i,exp ±γ σ i,exp Gaussian distributed How can we include systematical errors? ...one normalization factor per data set is needed! 14 / 24

Bootstrap sampling and systematics S i,exp boot = S i,exp ±γ σ i,exp Gaussian distributed How can we include systematical errors? ...one normalization factor per data set is needed! S i,exp boot =ξ i [ S i,exp ±γ σ i,exp ] At every bootstrap cycle the systematical errors for each set can vary independently! 14 / 24

Bootstrap vs G radient : systematics ON α E1 β M1 11.8 ± 0.2 2.0 ± 0.2 BOOTSTRAP 11.8 ± 0.2 2.0 ± 0.2 LEX + MUL TIPOLES 11.8 ± 0.3 2.0 ± 0.3 BOOTSTRAP SYS ON Systematical errors enlarge the error band of polarizabilities! 15 / 24

“χ 2 ” probability distribution in bootstrap framework (static pol.) S y s t e ma t i c s O N S y s t e ma t i c s O F F Not even more a “true” χ 2 distribution (due to point correlations given by systematics) Work in collaboration with B.Pasquini, P . Pedroni, A. Rotondi (paper in preparation) 16 / 24

The efgect of systematics (static spin- independent polarizabilities) S y s t e ma t i c s O N S y s t e ma t i c s O F F Expected Gaussian shape + systematics enlarging B. Pasquini, P . Pedroni, S. S., in preparation 17 / 24