Measuring the Proton Spin Polarizabilities in Real Compton Scattering

Measuring the Proton Spin Polarizabilities in Real Compton

Scattering

Philippe Martel – UMass AmherstAdvisor: Rory Miskimen

TUNL (Triangle Universities Nuclear Lab)Bosen 2009

April 21, 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


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)


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


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)


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


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.


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:


2x – 0 and Constrained


2z – 0 and Constrained


3 – 0 and Constrained


2x – All Four Multipole S.P.s


2z – All Four Multipole S.P.s


3 – All Four Multipole S.P.s





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


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.


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


Real/Effective Polarization

The effective polarizations can then be written

as:

With the expected experimental polarizations,

the resulting effective polarizations are:


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


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.


Check – 240 MeV


Check – 280 MeV


S.P. Fitting – 240 MeV (0, con.)


Con. Values – 240 MeV (0, con.)


S.P. Fitting – 240 MeV (no con.)


Con. Values – 240 MeV (no con.)


S.P. Fitting – 280 MeV (0, con.)


Con. Values – 280 MeV (0, con.)


S.P. Fitting – 280 MeV (no con.)


Con. Values – 280 MeV (no con.)


Results

The tabulated results for running with both the

0 and constraints:

The tabulated results for running with no

constraints:


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!

Documents

Measuring the Proton Spin Polarizabilities in Real Compton Scattering