Proton scalar polarizabilities from

real Compton scattering data

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Stefano Sconfietti in collaboration with

Barbara Pasquini & Paolo PedroniUniversity of Pavia & INFN (Pavia)

Carnegie Mellon University, Pittsburgh, June 4th 2019

Outline

WHAT?Extraction of dipole scalar polarizabilities (electric and magnetic) from proton RCS data

WHEN & WHERE? During my PhD in Physics at Università degli Studi di Pavia (Italy) & INFN

HOW?Dispersion Relation approach + data analysis (bootstrap)

WHY?“...sure, it may give some practical results, but that's not why we do it” - R. P. Feynman

Outline

1. PROTONwith DISPERSION

RELATIONS

Real Compton scattering (RCS)

Proton target + REAL photon in & REAL photon out

γ(q)+N (p)→γ(q ')+N ( p'

)

Real Compton scattering (RCS)

Proton target + REAL photon in & REAL photon out

Check the response of the proton to the external quasi-static field (the photon)

See Measurement of the proton polarizabilities at MAMI – E. Mornacchi

γ(q)+N (p)→γ(q ')+N ( p'

)

Real Compton scattering (RCS)

Proton target + REAL photon in & REAL photon out

Check the response of the proton to the external quasi-static field (the photon)

See Measurement of the proton polarizabilities at MAMI – E. Mornacchi

What’s going on here?

Dispersion Relations

γ(q)+N (p)→γ(q ')+N ( p'

)

The proton structure in RCS

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 field

B. Pasquini , D. Drechsel M. Vanderhaeghen, Phys.Rev. C76 (2007) 015203

The proton structure in RCS

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 field

spin-independent dipole

spin-dependent dipole

spin-dependent dipole-quadrupole

B. Pasquini , D. Drechsel M. Vanderhaeghen, Phys.Rev. C76 (2007) 015203

Unitarity: the sum over all possible processes has probability = 1

S=I+iT

2 ℑT f i=∑n

T f n* T n i

Sum over all the intermediate states!

Unitarity & analyticity: the basis of DRs

Sum over all the intermediate states!

Unitarity: the sum over all possible processes has probability = 1

S=I+iT

2 ℑT f i=∑n

T f n* T n i

Sum over all the intermediate states!

Unitarity & analyticity: the basis of DRs

Sum over all the intermediate states!

f (z )=1

2π i∮C

f (s)s−z

ds

Unitarity: the sum over all possible processes has probability = 1

S=I+iT

2 ℑT f i=∑n

T f n* T n i

Sum over all the intermediate states!

Unitarity & analyticity: the basis of DRs

BRANCH CUTS due to inelasticity

Sum over all the intermediate states!

f (z )=1

2π i∮C

f (s)s−z

ds

Ai(ν ,t )=A iB(ν , t)+ ∫

ν thr

νMAX

...+∫∩

...

ν energy → t transferred momentum→RCS differential cross section 6 amplitudes → Ai

B. Pasquini , D. Drechsel M. Vanderhaeghen, Phys.Rev. C76 (2007) 015203B. Pasquini, M. Vanderhaeghen, Ann.Rev.Nucl.Part.Sci. 68 (2018) 75-103

Inspecting the proton

Ai(ν ,t )=A iB(ν , t)+ ∫

ν thr

νMAX

...+∫∩

...

Ai(ν ,t )=A iB(ν , t)+∫

νthr

∞

...+0

Ai(ν ,t )=A iB(ν , t)+ ∫

ν thr

νMAX

...+A iAS

For i=3,…,6: “good” behavior

For i=1,2: “bad” behavior

ν energy → t transferred momentum→RCS differential cross section 6 amplitudes → Ai

B. Pasquini , D. Drechsel M. Vanderhaeghen, Phys.Rev. C76 (2007) 015203B. Pasquini, M. Vanderhaeghen, Ann.Rev.Nucl.Part.Sci. 68 (2018) 75-103

Inspecting the proton

Ai(ν ,t )=A iB(ν , t)+ ∫

ν thr

νMAX

...+∫∩

...

Contour behavior: that’s the problem! faster →convergence is needed...

Ai(ν ,t )=A iB(ν , t)+∫

νthr

∞

...+0

Ai(ν ,t )=A iB(ν , t)+ ∫

ν thr

νMAX

...+A iAS

For i=3,…,6: “good” behavior

For i=1,2: “bad” behavior

Asymptotic contribution meson exchange→

SUBTRACTED DISPERSION RELATIONS

ν energy → t transferred momentum→RCS differential cross section 6 amplitudes → Ai

B. Pasquini , D. Drechsel M. Vanderhaeghen, Phys.Rev. C76 (2007) 015203B. Pasquini, M. Vanderhaeghen, Ann.Rev.Nucl.Part.Sci. 68 (2018) 75-103

Inspecting the proton

Ai(0,0) = aiStatic polarizabilities

D. Drechsel, M. Gorchtein, B. Pasquini, M. Vanderhaeghen, Phys.Rev. C61 (1999) 015204B. Pasquini , D. Drechsel M. Vanderhaeghen, Phys.Rev. C76 (2007) 015203B. Pasquini, M. Vanderhaeghen, Ann.Rev.Nucl.Part.Sci. 68 (2018) 75-103

Dispersion Relations & polarizabilities

Subtracted Dispersion Relations (s-channel)

Ai(0,0) = aiStatic polarizabilities

D. Drechsel, M. Gorchtein, B. Pasquini, M. Vanderhaeghen, Phys.Rev. C61 (1999) 015204B. Pasquini , D. Drechsel M. Vanderhaeghen, Phys.Rev. C76 (2007) 015203B. Pasquini, M. Vanderhaeghen, Ann.Rev.Nucl.Part.Sci. 68 (2018) 75-103

Dispersion Relations & polarizabilities

s-CHANNEL

t-CHANNEL

Subtracted Dispersion Relations (s-channel)

Ai(0,0) = aiStatic polarizabilities

D. Drechsel, M. Gorchtein, B. Pasquini, M. Vanderhaeghen, Phys.Rev. C61 (1999) 015204B. Pasquini , D. Drechsel M. Vanderhaeghen, Phys.Rev. C76 (2007) 015203B. Pasquini, M. Vanderhaeghen, Ann.Rev.Nucl.Part.Sci. 68 (2018) 75-103

Dispersion Relations & polarizabilities

ELECTRIC FIELD

αE1

A naive picture of the polarizabilities

ELECTRIC FIELD

Electric dipole moment

αE1

A naive picture of the polarizabilities

ELECTRIC FIELD

Electric dipole moment

αE1

The proton is 1000 times “electrically” stiffer than the Hydrogen!

A naive picture of the polarizabilities

ELECTRIC FIELD

MAGNETIC FIELD

Electric dipole moment

αE1

βM1

The proton is 1000 times “electrically” stiffer than the Hydrogen!

A naive picture of the polarizabilities

PARAMAGNETIC (β >0) dipole moment

ELECTRIC FIELD

MAGNETIC FIELD

Electric dipole moment

αE1

βM1

The proton is 1000 times “electrically” stiffer than the Hydrogen!

A naive picture of the polarizabilities

PARAMAGNETIC (β >0) dipole moment

DIAMAGNETIC (β <0) dipole momentELECTRIC

FIELDMAGNETIC FIELD

Electric dipole moment

αE1

βM1

The proton is 1000 times “electrically” stiffer than the Hydrogen!

A naive picture of the polarizabilities

2. TRADITIONAL FITS

How fits are usually done

Experimental points assumed Gaussian distributed around the model predictions (in the best value of the parameters)

χ2=∑

i( E i−T̂ i

σ i )2

Ei∈G [T̂ i ,σi2 ]

How fits are usually done

Experimental points assumed Gaussian distributed around the model predictions (in the best value of the parameters)

χ2=∑

i( E i−T̂ i

σ i )2

Ei∈G [T̂ i ,σi2 ]

Inclusion of the systematic sources of uncertainties

Ei∉G [T̂ i ,σ i2 ] χ

2=∑

i , k( f k Ei−T̂ i

f k σ i)

2

+( 1−f k

Δk )2

Some pathological problems

Which is the probability distribution of the fitted parameters? A Gaussian?

How can we include the propagation of parameters that are not fitted? Squared sum?

Which probability distribution shall we use for the modified chi squared? A “traditional” one?

B. Pasquini, P. Pedroni, S. Sconfietti, Phys. Rev. C 98, 015204 (2018)B. Pasquini, P. Pedroni, S. Sconfietti, arXiv:1903.07952, to be submitted to J. Phys. G

Some pathological problems

Which is the probability distribution of the fitted parameters? A Gaussian?

How can we include the propagation of parameters that are not fitted? Squared sum?

Which probability distribution shall we use for the modified chi squared? A “traditional” one?

SOLUTION: change perspective and let the data define the rules!

B. Pasquini, P. Pedroni, S. Sconfietti, Phys. Rev. C 98, 015204 (2018)B. Pasquini, P. Pedroni, S. Sconfietti, arXiv:1903.07952, to be submitted to J. Phys. G

3. OUR CHOICE : bootstrap

Bootstrap in a nutshell

Whatever is the probability distribution of the experimental point, sample from there!

χ j2=∑

i( Bij−T̂ ij

σij )2

Bij∈P [ Ei ,σi2,Δk ]

Bootstrap in a nutshell

Whatever is the probability distribution of the experimental point, sample from there!

χ j2=∑

i( Bij−T̂ ij

σij )2

Bij∈P [ Ei ,σi2,Δk ]

Minimize the function and find the best value of the parameters. Store it.

DO j = 1, N

ENDDO

Bootstrap in a nutshell

Whatever is the probability distribution of the experimental point, sample from there!

χ j2=∑

i( Bij−T̂ ij

σij )2

Bij∈P [ Ei ,σi2,Δk ]

Minimize the function and find the best value of the parameters. Store it.

DO j = 1, N

ENDDO

We can reconstruct the probability distribution of the fitted parameters!

Bij=E i+γij σ i

Our bootstrap sampling

Bootstrapped points assumed Gaussian distributed around the measured value

Bij∈G [ Ei ,σi2 ]

Bij=E i+γij σ i

Our bootstrap sampling

Bootstrapped points assumed Gaussian distributed around the measured value

Additional shift due to systematic effect

Bij=( 1+δij ) (E i+γij σ i )

Bij∈G [ Ei ,σi2 ]

Bij∈G [ Ei ,σi2 ]⊗U [−Δk ,Δk ]

Bij=E i+γij σ i

Our bootstrap sampling

Bootstrapped points assumed Gaussian distributed around the measured value

Additional shift due to systematic effect

Bij=( 1+δij ) (E i+γij σ i )

Bij∈G [ Ei ,σi2 ]

Bij∈G [ Ei ,σi2 ]⊗U [−Δk ,Δk ]

The only assumption: which probability distribution for the systematic errors?

Bij=E i+γij σ i

Our bootstrap sampling

Bootstrapped points assumed Gaussian distributed around the measured value

Additional shift due to systematic effect

Bij=( 1+δij ) (E i+γij σ i )

Bij∈G [ Ei ,σi2 ]

Bij∈G [ Ei ,σi2 ]⊗U [−Δk ,Δk ]

The only assumption: which probability distribution for the systematic errors?

The sampling when more than one systematic source of error is included can be obtained from a convolution of all the sources

Bootstrap: pros & contra

Straightforward inclusion of systematic errors

No assumptions on the parameters probability distributions

Reconstruction of a realistic limit probability distribution for the chi squared (see paper)

Easy propagation of errors (for non-fitted parameters)

P. Pedroni, S.Sconfietti, in preparation

Bootstrap: pros & contra

Straightforward inclusion of systematic errors

No assumptions on the parameters probability distributions

Reconstruction of a realistic limit probability distribution for the chi squared (see paper)

Easy propagation of errors (for non-fitted parameters)

P. Pedroni, S.Sconfietti, in preparation

The number of iteration has to be big (from here on, N=10000 for me)

4. RESULTS& DISCUSSION

Our fitting framework

Fitting parameter: αE1 - β

M1

αE1 + β

M1 constrained

(Baldin’s sum rule)

Spin polarizabilities (γs) fixed from

experimental values (with errors)

Low-energy (below 150 MeV) RCS data

B. Pasquini, P. Pedroni, S. Sconfietti, arXiv:1903.07952, to be submitted to J. Phys. G

Our fitting framework

Fitting parameter: αE1 - β

M1

αE1 + β

M1 constrained

(Baldin’s sum rule)

Spin polarizabilities (γs) fixed from

experimental values (with errors)

Low-energy (below 150 MeV) RCS data

There is some discussion on the “definition” of the Compton data set →let’s discuss at coffee breaks if you want!

B. Pasquini, P. Pedroni, S. Sconfietti, arXiv:1903.07952, to be submitted to J. Phys. G

Probability distributions

Non-Gaussian shape (not that far from that, indeed)

Systematic errors → enlarging the shape (here, not always)

B. Pasquini, P. Pedroni, S. Sconfietti, arXiv:1903.07952, to be submitted to J. Phys. G

The χ2 cumulative distribution

P. Pedroni, S. Sconfietti, in preparationB. Pasquini, P. Pedroni, S. Sconfietti, arXiv:1903.07952, to be submitted to J. Phys. G

Differential cross section (with errors)

Error band at 68% CL, from the parameters probability distribution!

B. Pasquini, P. Pedroni, S. Sconfietti, arXiv:1903.07952, to be submitted to J. Phys. G

Our results – overview

B. Pasquini, P. Pedroni, S. Sconfietti, arXiv:1903.07952, to be submitted to J. Phys. G

Take-home messages

1. Extraction of electric and magnetic polarizabilities of the

proton from RCS data, with DRs and a detailed error propagation

Take-home messages

1. Extraction of electric and magnetic polarizabilities of the

proton from RCS data, with DRs and a detailed error propagation

2. Use bootstrap for your fits!

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