GDH Q to zero

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Background and Objectives

Data obtained in electron scattering experiments correspond to exchange of virtual photons (Q2>0). It’s possible however to extrapolate such data to the real photon point (Q2=0), producing complementary data to experiments carried out with photon beams.

During the 6 GeV EG4 and E97-110 experiments, data were obtained for polarized proton, deuteron and 3He to very low Q2. These include polarized cross sections, asymmetries (A1), and/or the structure functions g1 and g2 covering the nucleon resonance region, particularly the Delta(1232) resonance. These results were tabulated in a 2D grid. Note that for electron scattering, three variables are needed to uniquely define the scattering kinematics, where the beam energy would be the 3rd variable in addition to the 2D grid. For EG4, the results were tabulated vs. the invariant mass W and the momentum transferred from beam to target Q2. For E97110, results were tabulated vs. the scattering angle theta and the scattered electron’s energy E’ (energy loss nu = E-E’).

Initially, the project involves extrapolating the proton results (specifically A1F1) from finite to zero Q2. This will allow them to be compared with data taken with real photons. Any inconsistency would be studied further. If the data are consistent, the method can be extended to other observables and they may be used to improve the precision of the real photon data. The project could then proceed by performing similar analysis with deuterium and 3He data, depending on the progress.

People/meetings

• Dr. Mark Dalton (JLab)
• Dr. Alexandre Deur (JLab)
• Darren Upton (ODU)
• Physics majors (undergrads) taking PHYS3995 at UVA
• Dr. Xiaochao Zheng (UVA)

Timeline

Summer 2024 (work by Blane)

1. setting up computing environment, understand the project and collect EG4 data, plot EG4 proton data
2. plot A1F1 vs. Q2 for fixed W values, then tried both linear and quadratic extrapolations to zero Q2, promising results, the two extrapolations give similar results, and the uncertainties are quite reasonable.
3. plot the extrapolated A1F1(Q2=0) vs. W.
4. Compare with real photon data from Mainz. Normalization:
   1. Mainz data are sigma_1/2-sigma_3/2 = 2 * sigma_TT
   2. Our A1F1 extropolation gives A1F1, and then we can calculate: sigma_TT = (4 pi^2 alpha)/(M nu) A_1F_1, where nu needs to be calculated point by point using W (for Q2=0), M is the proton mass, alpha = 1/137, and we also need hbarc somewhere for the dimension

Fall 2024 (work by Owen)

1. Wrap up proton A1F1 work and double-check normalization when comparing with real photon data, use correct normalizations, add chi2
2. Extend the work to EG4 A1F1 deuteron data
3. Extend the work to 3He, no A1F1 from E97110 but we have both sigma_TT and g1,2, see below
   1. For 3He, see sigma_TT in https://www.nature.com/articles/s41567-021-01245-9
   2. and g1,g2 in https://www.nature.com/articles/s41567-021-01245-9 (but figures only, tables only available from Alexandre)
   3. However, mid-semester we realized that the sigma_TT was interpolated data, while g1, g2 were “original” results. To avoid unnecessary uncertainty due to interpolation, we decided to work with g1, g2. See details below.
4. For 3He g1, g2 from E97110, the results were tabulated in equal nu spacing, for fixed Ebeam and theta. This means the table grid was neither constant Q2 nor constant W. (Unlike EG4 which were constant W). Thus, extra steps were taken as follows:
   1. the first step was to select a “common Q2 grid”.
   2. Owen then interpolated the g1 (and later g2) in W, using either a 2-point or a 3-point interpolation to obtain the g1 value at W_common, uncertainty of the interpolated g1 was estimated
   3. Once g1 for (W_common, Q2) grid was obtained, the code used for proton and deuteron was used to extrapolate g1 to zero Q2
   4. to be continued
5. Regarding EG4 p and d data, former group member Darren chimed in and provided estimate of extraction of the neutron using n ~ d – p, vs. full smearing model (de) convolution

Spring 2025

Systematic uncertainties

Our main goal in Spring 2025 is to finalize all p, d, and 3He extrapolation work, including evaluation of the full uncertainty. Here are some basic ideas for what to include in systematic uncertainty evaluation:

1. Extrapolation (in Q2) uncertainty
2. Vary Q2 range
   1. Difference between linear and quadratic results
   2. Vary photon flux convention (Hand vs Gilman)
   3. For 3He, interpolating in Q2
   4. anything else?
3. Propagation of correlated systematic uncertainty (including target and beam polarizations):
   • Since we are extrapolating, the propagation depends on whether the data changes sign or not, so I think the best think is to determine the uncertainty and then move all points up or down by that amount and see the effect on the result.
4. χ2 study and consequence on the uncorrelated error estimate
5. Neutron extraction:
   1. Naive vs less naive extractions
   2. anything else?
6. Do we need full-model neutron extraction? That is, do we assume A1F1(n) ~ A1F1(d) – A1F1(p), or do we need to run the full smearing code for a formal extraction?

More analysis?

Form the GDH sum for p and n with these extrapolated quantities, then compute gamma_0, the longitudinal spin polarizability. The only datum available is from Mainz. Do the same for E97110 3He sigma_LT

List of systematics for σ_TT (and σ_LT)

• Extrapolation uncertainty
  • Vary Q² range
  • Difference between linear and quadratic results
  • Vary photon flux convention (Hand vs Gilman)
  • ...anything else?
• Propagation of point-to-point correlated error
• χ² study and consequence on the uncorrelated error estimate
• Neutron extraction:
  • Naive vs less naive extractions
  • ...anything else?
• Do we need to worry about the error coming from the W-bin adjustment? (For 3He only)
• ...anything else?

References

1. EG4 proton paper and data table (see Refs_RealPhoton/ on Summer 2024 Google Drive)
2. EG4 deuteron data table (from Darren)
3. Real photon data (email from Mark D. These are Mainz/MAMI/ELSA data on sigma_3/2-sigma_1/2 which is equal to 2* sigma_TT, and may need to be converted to the proper unit)
4. Owen’s 3995 report (when done) and codes (once cleaned up)
5. GitLab repository:
   • Location: code.jlab.org:dalton/Q2to0
   • Access starts by making an account on https://code.jlab.org  This can be done without a jlab cue login using google or even email authentication.  Once the account exists send the username to Mark to get permission.
   • To access the repository it is suggested to authenticate via ssh as git.  The easiest way is to upload a public key to the site. Once access is achieved, then one can use the usual git tools to clone and update the repository.

git clone git@code.jlab.org:dalton/Q2to0.git

Documents

Link to the Overleaf draft paper