1. Extracting neutron structure functions in the resonance region Yonatan Kahn Northwestern University/JLab

2. Why are (new) extraction methods needed? No free neutron targets – use light nuclei as effective targets EMC effect – nucleus not just a sum of free protons and neutrons! Previous extraction methods only reliable for positive-definite functions

3. Difficulties in the resonance region Fermi motion smears out resonance structure Is it possible to reconstruct full resonance structure of neutron structure functions from nuclear data?

4. Nuclear structure functions Impulse approximation – virtual photon interacts with single nucleon inside nucleus Can write nuclear structure functions as convolutions of nucleon structure functions: smearing functions S. Kulagin and R. Petti, Nucl. Phys. A 765(126), 2006; S. Kulagin and W. Melnitchouk, Phys. Rev. C77:015210, 2008

5. Smearing functions f (y, γ) Can be calculated from nuclear wavefunction parameterizes finite-Q 2 effects; for most kinematics, γ ≤ 2 For γ = 1, interpret as nucleon light-cone momentum distributions Note sharp peak at y=1, similar shapes for f 0 and f ij

6. Effective smeared neutron function Subtract known proton contribution: (For brevity, ) Goal: extract neutron function from under the integral (Note: system of 2 coupled equations for spin-dependent functions) (F = F 2, xg 1,2 )

7. Extraction method – direct solution Need to solve an integral equation for single- variable function F (x) at fixed Q 2 Can put into standard form: This equation can be discretized, and solved by matrix inversion:

8. Extraction method – direct solution However, kernel vanishes on diagonal, so matrix is singular and inversion fails Strong physical reasons: single nucleon has vanishing probability of carrying entire momentum of nucleus In particular, no existence/uniqueness theorems for this kind of integral equation –may be several free neutron functions that all give the same smeared neutron function… This method fails because the smearing functions are sharply peaked

9. Iterative extraction method for F 2 n Need to solve → takes advantage of sharply peaked form of f 0 : δ is small Treat δ(x) as a perturbation, solve iteratively with a first guess F 2 n(0) for F 2 n (x) at fixed Q 2 Write → YK, W. Melnitchouk, S. Kulagin, arXiv:0809.4308; to appear in Phys. Rev. C Note: Because convolution involves f (y) F 2 n (x/y), F 2 n(i+1) depends on F 2 n(i) all the way up to x = 1, especially at large x

10. Iterative extraction method for xg n 1,2 Solve system of equations: Can ignore off-diagonal contributions, since f 12 and f 21 are so small In this approximation, extraction procedure same as for F 2 Simulated xg 1,2 d looks identical with or without off-diagonal terms

11. Convergence of extraction method Create deuteron “data” by smearing p and n parameterizations, try to recover input function Initial guesses: F 2 n(1) = 0, xg 1 n(1) = 0 (not a great first guess!) Nearly perfect convergence after 30 iterations, despite initial guess! (But don’t really need 30 iterations in practice)

12. Comparison with smearing-factor method Instead of assuming an additive correction, can assume a multiplicative “smearing factor”: Works fine for positive-definite functions, but can diverge for spin- dependent functions, while our method has no such problems divergences

13. Dependence on initial guess → Eventual convergence regardless of initial guess, but resonance peaks converge quicker when first guess is better

14. Estimating errors Vary deuteron data points by Gaussians (ignore proton errors, since smeared) Run 50 sample extractions, calculate RMS error on neutron function same size as deuteron error bars!

15. F 2 n extraction from data (1 iteration) Data: Hall C experiment E00116 (S. Malace) (preliminary)

16. F 2 n extraction from data (2 iterations) Data: Hall C experiment E00116 (S. Malace) (preliminary)

17. F 2 n extraction from data (2 iterations) Errors have grown after two iterationsMore structure visible Data: Hall C experiment E00116 (S. Malace) (preliminary)

18. F 2 n extraction from data (5 iterations) Convergence with two different initial guesses, but error bars are quite large (preliminary) Data: Hall C experiment E00116 (S. Malace)

19. g n 1,2 extraction from 3 He data (1 iteration) Data: Hall A experiment E01-012 (P. Solvignon) Sparse data points + bumpy input function → large errors!

20. xg 1 n extraction from CLAS deuteron data Data: S. Kuhn, N. Guler

21. Limitations of extraction method Discontinuities in input data are sharply magnified in output – worse for sparse data sets Error bars grow after each iteration, so convergence after 10 iterations not practical –Some dependence on initial guess (faster convergence with a better first guess, so smaller errors) Method currently limited to convolution representation of nuclear structure functions –Needs to be extended for off-shell effects (done), final- state interactions (more difficult) –Quasi-elastic peak can provide constraints on convolution model

22. Open questions How to address dependence on initial guess and eventual convergence? Better ways to estimate errors on extracted neutron structure function? Best way to deal with sparse data?

23. Backup slides

24. Error tests (1)

25. Error tests (2) 1 iteration, calculate errors by shifting whole curve up or down by error bars

26. Quasi-elastic model Data: CLAS experiments E1D, E6A. Note: not LT-separated

27. F 2 n at low Q 2 Data: CLAS experiments E1D, E6A. Note: not LT-separated