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Measuring the Proton Spin Polarizabilities in Real Compton
Scattering
Philippe Martel – UMass AmherstAdvisor: Rory Miskimen
TUNL (Triangle Universities Nuclear Lab)Bosen 2009
April 18, 2023 P. Martel - Bosen 2009 2
Table of Contents
• Concerning spin-polarizabilities
• What are they? Where do they
come from?
• What is currently known?
• Concerning the sensitivities to
them
• Observe changes in asymmetries
after perturbing one
• Smearing of effects from multiple
perturbations
• Concerning the fitting method
• Constructing asymmetries and
partials
• Minimization checks
• Results
April 18, 2023 P. Martel - Bosen 2009 3
Nuclear Compton Scattering
Compton scattering refers to scattering a photon
off of a bound electron (atomic) or off of a
nucleon (nuclear). Below about 20 MeV, this
process is described by the Hamiltonian:*
Above 20 MeV, the photon begins to probe the
nucleon structure. To second order, an effective
Hamiltonian can be written:
Here, E1 represents the electric, and M1 the
magnetic, dipole (scalar) polarizabilities.*
*B. Holstein, GDH Convenor’s Report: Spin polarizabilities (2000)
April 18, 2023 P. Martel - Bosen 2009 4
Spin Polarizabilities
These scalar polarizabilities have been measured
for the proton through real Compton scattering
experiments.*
Advancing to third order, four new terms arise in
the eff. Hamiltonian:*
These terms are the spin (vector)
polarizabilities. The subscript notation denotes
their relation to a multipole expansion.*R.P. Hildebrandt, Elastic Compton Scattering from the Nucleon and Deuteron (2005) - Dissertation thesis
April 18, 2023 P. Martel - Bosen 2009 5
S.P. Measurements
The GDH experiments at Mainz and ELSA used
the Gell-Mann, Goldberger, and Thirring sum rule
to evaluate the forward S.P.:
The Backward S.P. was determined from
dispersive analysis of backward angle Compton
scattering:
*B. Pasquini et al., Proton Spin Polarizabilities from Polarized Compton Scattering (2007)
April 18, 2023 P. Martel - Bosen 2009 6
S.P. Theoretical Values
O(p3) O(p4) O(p4) LC3 LC4 SSE BGLMN HDPV KS DPV Experiment
E1E1 -5.7 -1.4 -1.8 -3.2 -2.8 -5.7 -3.4 -4.3 -5.0 -4.3 No data
M1M1 -1.1 3.3 2.9 -1.4 -3.1 3.1 2.7 2.9 3.4 2.9 No data
E1M2 1.1 0.2 .7 .7 .8 .98 0.3 -0.01 -1.8 0 No data
M1E2 1.1 1.8 1.8 .7 .3 .98 1.9 2.1 1.1 2.1 No data
0 4.6 -3.9 -3.6 3.1 4.8 .64 -1.5 -.7 2.3 -.7 -1.01 ±0.08 ±0.10
4.6 6.3 5.8 1.8 -.8 8.8 7.7 9.3 11.3 9.3 8.0± 1.8
The pion-pole contribution has been subtracted from the experimental value for
Calculations labeled O(pn) are ChPT
LC3 and LC4 are O(p3) and O(p4) Lorentz invariant ChPT calculations
SSE is small scale expansion
Other calculations are dispersion theory
Lattice QCD calculation by Detmold is in progress
April 18, 2023 P. Martel - Bosen 2009 7
Dispersion Analysis Program
The theoretical cross sections used here are
produced in a fixed-t dispersion analysis code,
provided to us by Barbara Pasquini. For further
information, see B. Pasquini, D. Drechsel, M.
Vanderhaeghen, Phys. Rev. C 76 015203 (2007).The program was run for three different
experimental runs:
• Transversely polarized target with a circularly
polarized beam
• Longitudinally polarized target with a circularly
polarized beam
• Unpolarized target with a linearly polarized
beam
The former two return cross sections for
unpolarized, left helicity, and right helicity
beams. The latter returns the beam asymmetry.
April 18, 2023 P. Martel - Bosen 2009 8
Asymmetries
After producing tables of cross sections with
various values for the polarizabilities (their
HDPV values, and those + 1 unit we construct
the asymmetries:
Using the counts, the statistical errors can be
propagated through:
April 18, 2023 P. Martel - Bosen 2009 18
Fitting Program
• Cross sections → Counts →
Asymmetries
• Solid Angle of Detector
• Real/Effective Polarization
• Energy Bin Width
• Partials with Respect to
Polarizabilities
• Pseudodata
• 300 hours 2x, 300 hours 2z, 100
hours 3
• Fitting
• 2 Construction
• Minimization
April 18, 2023 P. Martel - Bosen 2009 19
Solid Angle
The solid angle for the chosen polar angle bin is
given by:
In order to tag the event (as likely described
before), the reaction requires a proton recoil
energy of at least 40 MeV, limiting our minimum
forward angle of acceptance to:
This, however, neglects the different events that
different parts of the detector observe for a
given run configuration, which will be corrected
for in the next section.
April 18, 2023 P. Martel - Bosen 2009 20
Real/Effective Polarization
The cross sections produced in the code assume
100% beam and target polarization (if
applicable). The cross sections with a real
polarization:
Transversely Longitudinally
Unpolarized
Where red is target polarization, and gray is
beam polarization direction
For 2x and
3
April 18, 2023 P. Martel - Bosen 2009 21
Real/Effective Polarization
The effective polarizations can then be written
as:
With the expected experimental polarizations,
the resulting effective polarizations are:
April 18, 2023 P. Martel - Bosen 2009 22
Minimization Partials
The energy bin averaged counts can be written
in a linear expansion:
Where kmax, kmin, and k0 are the energy bin max,
min, and centroid respectively, i is the S.P.
perturbation, and F is the flux factor:
The Ci term represents the partials of the counts
with respect to the S.P.s
April 18, 2023 P. Martel - Bosen 2009 23
Construction
The fitting program uses a minimization routine
on 2, defined as:
The algorithm is actually the summation of
various 2 components, including the selected
constraints for the particular run.
Is it reasonable, however, to assume that the
theoretical component can be approximated by a
linear expansion? Run a 2 check.
April 18, 2023 P. Martel - Bosen 2009 34
Results
The tabulated results for running with both the
0 and constraints:
The tabulated results for running with no
constraints:
April 18, 2023 P. Martel - Bosen 2009 35
Conclusions
• By simply plotting the changes in the
asymmetries, the sensitivities to the
polarizabilities is seen to be appreciable.
• Checks of the fitting method being used
demonstrates reasonable behavior (near
linearity and containment)
• Fitting results are very promising:
• Running the full program with 2x, 2z, and 3
appears to be sufficient to extract the S.P.s
without invoking the 0 or constraints.
• This would provide the first set of
experimental values for these important
quantities!