Relating F2p(x), F2n(x) and F2D(x) Using
a Relativistic Description of the Deuteron
Structure
G. Smirnov
Joint Institute for Nuclear Research, Dubna, Russia
and
Université Blaise Pascal, Clermont-Ferrand, France
ITEMS:
1. Changes of the nucleon structure in nuclei as a function of A
2. 4D structure of bound states (lightest nuclei: D, 3H, 3He and 4He)
3. Determination of the neutron structure function from experiments
G. Smirnov
INFN, Milano, 15.12.05
1
Motivation
Due to absence of free neutron targets the d quark distribution is poorely
constrained beyond x about 0.6.
The isoscalar and mirror nuclei have to be used as
the « neutron target ».
Also an advantage owing to the access to higher x.
Discussions of the « nuclear effects » in evaluating F2n(x) from nuclear
targets data often result in compromises and simplifications that distort
considerably F2n(x) , in particular in the region close to x = 1.
G. Smirnov
INFN, Milano, 15.12.05
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Introduction
Options for extracting neutron structure functions from deuteron data:
2F2D = F2p + F2n
1.
Naive assumption:
2.
Theoretical models
3.
Extrapolation of the EMC effect from heavy nuclei to A = 2
(Bodek - Yang correction)
Most dangerous is option (3) : one can approximately esimate the
amplitude of modification, but never the form of x dependence
G. Smirnov
INFN, Milano, 15.12.05
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EMC effect (BCDMS, EMC, SLAC)
 Data: BCDMS, SLAC
 Blue line: Burov,
Molochkov and Smirnov
G. Smirnov
INFN, Milano, 15.12.05
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Quantitative description of the EMC effect
G. Smirnov
INFN, Milano, 15.12.05
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Evolution of the nucleon structure in two stages
G. Smirnov
INFN, Milano, 15.12.05
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Topology of the interaction
( 1 – x3 ) ~ mean spacing of
nucleons inside a nucleus
A6:
 Distributions of quarks are not
sensitive to a nuclear structure

3 nucleons only can be seen by virtual
photon in the nucleus « in one shot »
 3He (3H) topology
G. Smirnov
INFN, Milano, 15.12.05
x3
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Lessons from the EMC effect studies
F2n(x) can not be evaluated from the data collected on medium and
heavy nuclear targets untill the realtion between
F2n(x), F2p(x) and F2D(x)
Is known.
Bodek-Yang correction is equivalent to assumption of
the same internucleon spacing in the deuteron and
heavy nuclei, which results in two errors:
(1) Wrong magnitude of the EMC effect
(2) Wrong x-dependence
G. Smirnov
INFN, Milano, 15.12.05
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Practical way of finding F2n(x)
We suggest an approach of extracting F2n(x) from the data collected in
deep inelastic scattering experiments, which relies on relativistic
theoretical description of F2D(x) and well defined assumptions on the
high x asymptotics for the ratio F2n(x) / F2p(x).
It is based on the covariant Bethe-Salpeter formalism and allows to
express the hadronic part of the nuclear deep inelastic amplitude W
in terms of the off-mass-shell nucleon and antinucleon amplitudes.
Work in collaboration with
V.V. Burov, A.V Molochkov and H. Toki
G. Smirnov
INFN, Milano, 15.12.05
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Hadronic Tensor ( bound states )
G. Smirnov
INFN, Milano, 15.12.05
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Diagrams
involved in evaluating of the forward Compton amplitude TA
W A(P,q) = 1/2 ImT A(P,q)
Impulse approximation
Interaction correction to G 4 
 contribution of 2-nucleon
propagators
G. Smirnov
INFN, Milano, 15.12.05
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The covariant Bethe-Salpeter formalism provides the integral equation
relating F2p, F2n and F2D.
It is solved iteratively by using input structure functions
F2p and F2D
Additionally, this allows extrapolation of F2D into a wide range of x and Q2
G. Smirnov
INFN, Milano, 15.12.05
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Deuteron structure function
(4)
Second term results from the Fermi motion along time axis
EMC-effect cannot be explained without changing the nucleon
structure in a nucleus –
4D – radius of a bound nucleon changes
G. Smirnov
INFN, Milano, 15.12.05
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Application of the Bethe-Salpeter formalism to the DIS of leptons
on the lightest nuclei
• BS amplitude for Compton scattering on the deuteron:
– V.V. Burov, A.V. Molochkov, G.I. Smirnov and H. Toki: Phys. Lett. B587
(2004) 175.
– V.V.Burov and A.V.Molochkov: Nucl. Phys. A637 (1998) 31.
• BS amplitude for Compton scattering on 3H, 3He and 4He:
– V.V. Burov, A.V. Molochkov and G.I. Smirnov: Phys. Lett. B466 (1999) 1.
– S.G. Bondarenko, V.V. Burov, A.V. Molochkov, G.I. Smirnov and H. Toki :
J. Prog. Part. Nucl. Phys. 48 (2002) 449—535.
G. Smirnov
INFN, Milano, 15.12.05
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Structure Functions of Light Nuclei:
A=3
G. Smirnov
INFN, Milano, 15.12.05
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Structure Functions of Light Nuclei:
A=4
G. Smirnov
INFN, Milano, 15.12.05
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Nuclear effects in 4He
« log » scale
« lin» scale
The ratio of the helium to deuteron structure functions as callculated by
Burov, Molochkov and Smirnov is shown by the solid line
G. Smirnov
INFN, Milano, 15.12.05
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Missing Data on the Structure Functions :
A=3
3H and 3He
G. Smirnov
INFN, Milano, 15.12.05
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Relating F2p , F2n and F2d
Use of SMC fit for F2p and its modification
Remember
about
derivative !
Modification suggested by BMST ( x  1 ) : 2  2 + (3 – 2 ) x15
G. Smirnov
INFN, Milano, 15.12.05
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Assumptions
Main assumption: F2p(x) is known in the range 3.5 10– 5 < x < 0.85
G. Smirnov
INFN, Milano, 15.12.05
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Deuteron structure function compared with data from SLAC and
NMC experiments (low x region)
Bjorken x
F2D(x)
G. Smirnov
Q 2 (GeV 2 )
approximated with Eq. (4) in the range 10-3 < x < 0.6 with constraints listed in
« Assumptions »
INFN, Milano, 15.12.05
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Deuteron structure function compared with data from SLAC and
NMC experiments (high x region)
Bjorken x
F2D(x)
G. Smirnov
Q 2 (GeV 2 )
approximated with Eq. (4) in the range 10-3 < x < 0.6 with constraints listed
in « Assumptions »
INFN, Milano, 15.12.05
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Results representing ratios of structure functions
versus Bjorken x
Squares and triangles — results
Squares and triangles — results of NMC and SLAC,
respectively, obtained in naive approach.
of NMC and SLAC, respectively.
G. Smirnov
INFN, Milano, 15.12.05
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Comparison with the naive approximation for the
F2n(x) evaluation
Naive approach:
F2n = 2 F2D – F2p
The ratio is virtually 1.0 below x = 0.7 due to cancellation of contributions
from 3D Fermi motion and the Fermi motion along time axis
G. Smirnov
INFN, Milano, 15.12.05
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Ratio of the neutron and proton structure functions
Three values of a2 correspond to three different assumptions on
F2n(x) / F2p(x) at x = 1
G. Smirnov
INFN, Milano, 15.12.05
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Conclusions
Theoretically justified and fully consistent procedure for extracting F2n(x) in
the kinematic range 10 –3 < x < 1 under three different assumptions on
F2n(x) / F2p(x) at x = 1 is proposed.
Increase in experimental accuracy in measurements of F2p(x) and F2D(x) in
the range 0.6 < x < 0.8 by factor of two will be sufficient for verification of
models suggested for the evaluation of the d/u ratio at x = 1 .
Technique relies on a good approximation of F2D(x) which is not sensitive to
different high x limits of the neutron structure function.
This also means that F2D(x) measured by already completed DIS
experiments ( x < 0.9 ) can be described without introducing nonbaryonic
degrees of freedom. The interval which remains unmeasured can in principle
accommodate dibaryon states or some other exotica.
G. Smirnov
INFN, Milano, 15.12.05
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