Extracting neutron structure functions in the resonance region

Yonatan KahnNorthwestern University/JLab

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

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?

Nuclear structure functions• Impulse approximation –

virtual photon interacts with single nucleon inside nucleus

• Can write nuclear structure functions as convolutions of nucleon structure functions:

F x Q dy f y F xy

QA N A Nx

M M

N p n

A

22

0 22( , ) ( , ) ,//

,

xg x Q dy f y xg xy

QiA

ijN A

jN

x

M M

N p nj

A( , ) ( , ) ,//

,,

2 2

1 2

smearing functions

S. Kulagin and R. Petti, Nucl. Phys. A 765(126), 2006;

S. Kulagin and W. Melnitchouk, Phys. Rev. C77:015210, 2008

Smearing functions f (y, γ)

• Can be calculated from nuclear wavefunction

• parameterizes finite-Q2 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 f0 and fij

Effective smeared neutron function

• Subtract known proton contribution:

(For brevity, )

• Goal: extract neutron function from under the integral

~ ( , ) : ( , )F Fn Ax Q x Q2 2

~ ( , ) ( , ) ,//F x Q dy f y F x

yQn n A n

x

M MA

22

0 22

xg x Q dy f y xg xy

Q dy f y xg xy

Qin

in A n

x

M Min A n

x

M MA A~ ( , ) ( , ) , ( , ) ,// //2

1 12

2 22

(Note: system of 2 coupled equations for spin-dependent functions)

(F = F2, xg1,2)

( )( ) ( , ) / )/

f x dy f y x yx

M MA F F (

( Ff x Qp )( , )2

Extraction method – direct solution

• Need to solve an integral equation for single-variable function F(x) at fixed Q2

• Can put into standard form:

• This equation can be discretized, and solved by matrix inversion:

g x z y K x y dyx

y

( ) ( ) ( , )m ax

g K za abb a

N

b

z K g1

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…

K y y f y( , ) ( )m ax 0

This method fails because the smearing functions are sharply peaked

Iterative extraction method for F2n

~ ( ) ( ) ( ) ( , ))F x N F x x N dy f yn n2 2 0 (

F x F xN

F x f F xn n n n2

12

02 0 2

01( ) ( ) ( )( ) ( ) [ ~ ( ) ( )( )]

~ ( ) ( )( )F x f F xn n2 0 2 • Need to solve

→ takes advantage of sharply peaked form of f0: δ is small

• Treat δ(x) as a perturbation, solve iteratively with a first guess F2n(0)

for F2n(x) at fixed Q2

• Write

( ) ( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) [ ~ ( ) ( )]

10 2

02

0

21

211

x f F x N F x

F xN

F x x

n n

n n

→

YK, W. Melnitchouk, S. Kulagin, arXiv:0809.4308; to appear in Phys. Rev. C

Note: Because convolution involves f (y) F2n(x/y), F2

n(i+1) depends on F2n(i)

all the way up to x = 1, especially at large x

Iterative extraction method for xgn1,2

• Solve system of equations:

• Can ignore off-diagonal contributions, since f12 and f21 are so small

• In this approximation, extraction procedure same as for F2

Simulated xg1,2d looks identical with

or without off-diagonal terms

xg x f xg x f xg x

xg x f xg x f xg x

n n n

n n n

~ ( ) ( )( ) ( )( )~ ( ) ( )( ) ( )( )

1 11 1 1 2 2

2 2 1 1 2 2 2

Convergence of extraction method•Create deuteron “data” by smearing p and n parameterizations, try to recover input function

•Initial guesses: F2n(1) = 0, xg1

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)

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

~ ( ) ( ) ( )F Fn n nx S x x

divergences

Dependence on initial guess

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

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!

F2n extraction from data (1 iteration)

Data: Hall C experiment E00116 (S. Malace)

(preliminary)

F2n extraction from data (2 iterations)

Data: Hall C experiment E00116 (S. Malace)

(preliminary)

F2n extraction from data (2 iterations)

Errors have grown after two iterations More structure visible

Data: Hall C experiment E00116 (S. Malace)

(preliminary)

F2n 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)

gn1,2 extraction from 3He data (1 iteration)

Data: Hall A experiment E01-012 (P. Solvignon)

Sparse data points + bumpy input function → large errors!

xg1n extraction from CLAS deuteron data

Data: S. Kuhn, N. Guler

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

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?

Backup slides

Error tests (1)

Error tests (2)

1 iteration, calculate errors by shifting whole curve up or down by error bars

Quasi-elastic model

Data: CLAS experiments E1D, E6A. Note: not LT-separated

F2n at low Q2

Data: CLAS experiments E1D, E6A. Note: not LT-separated