proton scalar polarizabilities from real compton
TRANSCRIPT
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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
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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
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1. PROTONwith DISPERSION
RELATIONS
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Real Compton scattering (RCS)
Proton target + REAL photon in & REAL photon out
γ(q)+N (p)→γ(q ')+N ( p'
)
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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'
)
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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'
)
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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
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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
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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!
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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
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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
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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
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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
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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
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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
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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
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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
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ELECTRIC FIELD
αE1
A naive picture of the polarizabilities
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ELECTRIC FIELD
Electric dipole moment
αE1
A naive picture of the polarizabilities
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ELECTRIC FIELD
Electric dipole moment
αE1
The proton is 1000 times “electrically” stiffer than the Hydrogen!
A naive picture of the polarizabilities
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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
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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
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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
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2. TRADITIONAL FITS
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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 ]
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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
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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
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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
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3. OUR CHOICE : bootstrap
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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 ]
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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
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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!
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Bij=E i+γij σ i
Our bootstrap sampling
Bootstrapped points assumed Gaussian distributed around the measured value
Bij∈G [ Ei ,σi2 ]
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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 ]
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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?
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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
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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
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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)
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4. RESULTS& DISCUSSION
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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
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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
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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
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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
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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
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Our results – overview
B. Pasquini, P. Pedroni, S. Sconfietti, arXiv:1903.07952, to be submitted to J. Phys. G
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Take-home messages
1. Extraction of electric and magnetic polarizabilities of the
proton from RCS data, with DRs and a detailed error propagation
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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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Backup Slides BACKUP
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