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Measuring the Proton Spin Polarizabilities in Real Compton Scattering Philippe Martel – UMass Amherst Advisor: Rory Miskimen TUNL (Triangle Universities Nuclear Lab) Bosen 2009

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May 2, 2015P. 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

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May 2, 2015P. 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)

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May 2, 2015P. 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

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May 2, 2015P. 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)

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May 2, 2015P. Martel - Bosen 2009 6 S.P. Theoretical Values O(p 3 )O(p 4 ) LC3LC4SSEBGLMNHDPVKSDPVExperiment E1E1 -5.7-1.4-1.8-3.2-2.8-5.7-3.4-4.3-5.0-4.3No data M1M1 -1.13.32.9-1.4-3.13.12.72.93.42.9No data E1M2 1.10.2.7.8.980.3-0.01-1.80No data M1E2 1.11.8.7.3.981.92.11.12.1No data 00 4.6-3.9-3.63.14.8.64-1.5-.72.3-.7-1.01 ±0.08 ±0.10 4.66.35.81.8-.88.87.79.311.39.38.0± 1.8 The pion-pole contribution has been subtracted from the experimental value for Calculations labeled O(p n ) are ChPT LC3 and LC4 are O(p 3 ) and O(p 4 ) Lorentz invariant ChPT calculations SSE is small scale expansion Other calculations are dispersion theory Lattice QCD calculation by Detmold is in progress

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May 2, 2015P. 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.

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May 2, 2015P. 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:

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May 2, 2015P. Martel - Bosen 2009 9 2x – 0 and Constrained

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May 2, 2015P. Martel - Bosen 2009 10 2z – 0 and Constrained

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May 2, 2015P. Martel - Bosen 2009 11 3 – 0 and Constrained

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May 2, 2015P. Martel - Bosen 2009 12 2x – All Four Multipole S.P.s

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May 2, 2015P. Martel - Bosen 2009 13 2z – All Four Multipole S.P.s

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May 2, 2015P. Martel - Bosen 2009 14 3 – All Four Multipole S.P.s

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May 2, 2015P. 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

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May 2, 2015P. 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.

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May 2, 2015P. 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 LongitudinallyUnpolarized Where red is target polarization, and gray is beam polarization direction For 2x and 3

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May 2, 2015P. 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:

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May 2, 2015P. Martel - Bosen 2009 22 Minimization Partials The energy bin averaged counts can be written in a linear expansion: Where k max, k min, and k 0 are the energy bin max, min, and centroid respectively, i is the S.P. perturbation, and F is the flux factor: The C i term represents the partials of the counts with respect to the S.P.s

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May 2, 2015P. 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.

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May 2, 2015P. Martel - Bosen 2009 24 Check – 240 MeV

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May 2, 2015P. Martel - Bosen 2009 25 Check – 280 MeV

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May 2, 2015P. Martel - Bosen 2009 26 S.P. Fitting – 240 MeV ( 0, con.)

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May 2, 2015P. Martel - Bosen 2009 27 Con. Values – 240 MeV ( 0, con.)

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May 2, 2015P. Martel - Bosen 2009 28 S.P. Fitting – 240 MeV (no con.)

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May 2, 2015P. Martel - Bosen 2009 29 Con. Values – 240 MeV (no con.)

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May 2, 2015P. Martel - Bosen 2009 30 S.P. Fitting – 280 MeV ( 0, con.)

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May 2, 2015P. Martel - Bosen 2009 31 Con. Values – 280 MeV ( 0, con.)

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May 2, 2015P. Martel - Bosen 2009 32 S.P. Fitting – 280 MeV (no con.)

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May 2, 2015P. Martel - Bosen 2009 33 Con. Values – 280 MeV (no con.)

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May 2, 2015P. Martel - Bosen 2009 34 Results The tabulated results for running with both the 0 and constraints: The tabulated results for running with no constraints:

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May 2, 2015P. 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!

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