Proton scalar polarizabilities from real Compton ...

Proton scalar polarizabilities from

real Compton scattering data

1 / 24

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'

)



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'

)



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



S=I+iT

2 ℑT f i=∑n

T f n* T n i




f (z )=1

2π i∮C

f (s)s−z

ds


S=I+iT

2 ℑT f i=∑n

T f n* T n i



BRANCH CUTS due to inelasticity


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




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




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)




s-CHANNEL

t-CHANNEL

Subtracted Dispersion Relations (s-channel)




ELECTRIC FIELD

αE1

A naive picture of the polarizabilities

ELECTRIC FIELD

Electric dipole moment

αE1


ELECTRIC FIELD


αE1

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


ELECTRIC FIELD

MAGNETIC FIELD


αE1

βM1



PARAMAGNETIC (β >0) dipole moment

ELECTRIC FIELD

MAGNETIC FIELD


αE1

βM1



PARAMAGNETIC (β >0) dipole moment

DIAMAGNETIC (β <0) dipole momentELECTRIC

FIELDMAGNETIC FIELD


αE1

βM1



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 ]



χ j2=∑

i( Bij−T̂ ij

σij )2


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

DO j = 1, N

ENDDO



χ j2=∑

i( Bij−T̂ ij

σij )2


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



Additional shift due to systematic effect

Bij=( 1+δij ) (E i+γij σ i )


Bij∈G [ Ei ,σi2 ]⊗U [−Δk ,Δk ]

Bij=E i+γij σ i







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

Bij=E i+γij σ i







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!


Probability distributions

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

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


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!


Our results – overview


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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