Nuclear shadowing in the structure function $F_3$ is investigated as a
possible test of shadowing models. The $F_3$ shadowing is studied in two
different theoretical models: a parton-recombination model with $Q^2$ rescaling
and an aligned-jet model. We find that predictions in these models differ
completely in the small $x$ region. It suggests that the $F_3$ shadowing could
be used for discriminating among different models, which produce similar
shadowing behavior in the structure function $F_2$.Comment: 5 pages, Latex, Fig.1 is not included, Talk given at the IV
  International Symposium on ``Weak and Electromagnetic Interactions in Nuclei"
  Osaka, Japan, June 12 - 16, 1995, Complete postscript file including the
  figure is available at
  ftp://ftp.cc.saga-u.ac.jp/pub/paper/riko/quantum1/saga-he-82.ps.gz or at
  http://www.cc.saga-u.ac.jp/saga-u/riko/physics/quantum1/structure.htm

Kumano, S.

Miyama, M.

English

arXiv

Nuclear shadowing in the structure function F_3 is investigated as a possible test of shadowing models. The F_3 shadowing is studied in two different theoretical models: a parton-recombination model with Q^2 rescaling and an aligned-jet model. We find that predictions in these models differ completely in the small x region. It suggests that the F_3 shadowing could be used for discriminating among different models, which produce similar shadowing behavior in the structure function F_2

Kumano, S

Miyama, M

CERN Document Server

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SAGA-HE-82-95 June 20, 1995
Nuclear modication
in the structure function F
3
S. Kumano and M. Miyama
?
Department of Physics
Saga University
Saga 840, Japan
Talk given at the IV International Symposium on
\Weak and Electromagnetic Interactions in Nuclei"
Osaka, Japan, June 12 { 16, 1995
? Email: kumanos or 94sm10@cc.saga-u.ac.jp
Our group activities are listed at ftp://ftp.cc.saga-u.ac.jp/pub/paper/riko/quantum1
or at http://www.cc.saga-u.ac.jp/saga-u/riko/physics/quantum1/structure.html.
to be published in proceedings
NUCLEAR MODIFICATION
IN THE STRUCTURE FUNCTION F
3
S. Kumano and M. Miyama

Department of Physics, Saga University
Saga 840, Japan
E-mail: kumanos and 94sm10@cc.saga-u.ac.jp
ABSTRACT
Nuclear shadowing in the structure function F
3
is investigated as a possible test
of shadowing models. The F
3
shadowing is studied in two dierent theoretical
models: a parton-recombination model with Q
2
rescaling and an aligned-jet
model. We nd that predictions in these models dier completely in the small x
region. It suggests that the F
3
shadowing could be used for discriminating among
dierent models, which produce similar shadowing behavior in the structure
function F
2
.
1. Introduction
This talk is based on our investigation with R. Kobayashi in Ref. 1. Nuclear shad-
owing in the structure function F
2
has been studied extensively last several years.
2
There are various models in explaining the shadowing. These models include a tra-
ditional explanation in terms of vector-meson dominance and a new approach using
parton interactions in nuclei. These viewpoints seem to be very dierent. The former
model indicates that the virtual photon transforms into vector-meson states, which
then interact with a target nucleus. Because the mesons interact predominantly in
the nuclear surface, internal constituents are shadowed. On the other hand, the latter
model is based on a parton picture. At small x, the localization size of a parton could
exceed the average nucleon separation in a nucleus. Therefore, partons in dierent
nucleons could interact, and these interactions are called parton recombinations. The
recombinations are extra nuclear eects, which give rise to the nuclear shadowing.
These dierent models produce similar x dependence in the structure function F
2
,
so that we cannot distinguish among the models in comparison with experimental
data. Various shadowing models may be tested by other quantities such as sea-
quark and gluon distributions in nuclei.
3
In this paper, we propose that the structure
function F
3
could be useful in determining the appropriate shadowing description.
Namely, valence-quark distributions at small x are not well investigated and they
could be used for determining the shadowing model. In order to nd the possibility,
two dierent models are employed for the shadowing. The rst one is a parton model
with parton recombination and Q
2
rescaling eects,
2
and the other is an aligned-
jet model
4
which is an extension of the vector-meson-dominance model. Nuclear

(URL) http://www.cc.saga-u.ac.jp/saga-u/riko/physics/quantum1/structure.html,
(Anonymous FTP) ftp://ftp.cc.saga-u.ac.jp/pub/paper/riko/quantum1.
modication of F
3
(x) in these models is explained in section 2.1. Then, we discuss
whether or not the results are compatible with current experimental data. Experi-
mental restrictions are estimated in section 2.2 and the results are compared with the
theoretical predictions.
2. Shadowing in the structure function F
3
Nuclear modication of F
3
could be estimated from F
2
at medium and large
x because valence-quark distributions dominate both structure functions. Hence,
the essential point of our study is to investigate F
3
in the small x region. We ne-
glect next-to-leading order corrections in F
3
for simplicity, so that F
3
is given by the
valence-quark distributions: F
3
(x) = u
v
(x) + d
v
(x). Therefore, shadowing in F
3
indi-
cates nuclear modication of the valence-quark distributions at small x. We explain
theoretical and experimental situations in the following.
2.1. Model predictions
Because there is little theoretical study on the F
3
shadowing, existing theoretical
predictions are limited at this stage. Among the models, we employ two very dierent
ones for studying F
3
.
The rst one is a hybrid parton model with Q
2
rescaling and parton recombination
eects.
2
The rescaling model was originally proposed as a model for explaining the
medium x region, and the recombination as a model in the small x region. Combin-
ing these mechanism, we could explain nuclear structure functions F
A
2
from small x
( 0:01) to large x ( 0:8). This unied model can be applied to the structure func-
tion F
3
. According to the Q
2
rescaling model, nuclear structure functions F
A
3
(x;Q
2
)
are given by rescaling (increasing) Q
2
in the nucleon structure function F
N
3
(x;Q
2
).
Therefore, the ratio R
3
 F
A
3
(x)=F
D
3
(x) is smaller than unity at medium x as it ex-
plains the EMC eect in this region. Since the rescaling satises the baryon-number
conservation
Z
dx[u
v
(x) + d
v
(x)] = 3, the ratio R
3
becomes larger than unity at small
x. Parton-recombination contributions to F
3
(x) are rather contrary to those in the
Q
2
rescaling model in the sense that the recombinations decrease the ratio at small x
and increase it at medium and large x. The overall eects are shown by a solid line
(model 1) in Fig. 1. It is interesting to note that the model predicts antishadowing at
small x instead of shadowing. In this parton model, the F
3
shadowing diers distinc-
tively from the F
2
one: F
A
3
(x)=F
N
3
(x) 6= F
A
2
(x)=F
N
2
(x) at small x. In other words,
valence-quark modication is dierent from the sea-quark one. However, there is a
model which predicts F
3
shadowing similar to the F
2
one. An example is discussed
in the following.
The second model is the aligned jet model in Ref. 4. This model is based on the
vector-meson dominance model, which indicates that the virtual photon transforms
into vector meson states and they interact with the target. The propagation length of
the hadronic uctuation could exceed the average nucleon separation in a nucleus at
small x, and shadowing occurs due to multiple scatterings. In the aligned-jet model,
the virtual photon transforms into a qq pair, which then interacts with the target.
However, the only qq pair aligned in the direction of  (W) interacts in a similar way
to the vector-meson interactions with the target. The model predication for the F
3
shadowing is shown by a solid curve (model 2) in Fig. 1. This shadowing is very
similar to the F
2
shadowing: F
A
3
(x)=F
N
3
(x)  F
A
2
(x)=F
N
2
(x) at small x. It is because
the qq pair interacts with sea and valence quarks in the similar way. The model curve
is obtained by the aligned-jet-model together with the baryon-number conservation.
Their study to combine the shadowing mechanism with the medium x physics, such
as the nuclear binding, is still in progress.
As far as we know, these are only two papers which discuss the F
3
shadowing
explicitly. Obviously, other model predictions should be investigated in future in
comparison with the above estimates. We examine whether these model predications
are compatible with current experimental data in the following.
2.2. Experimental restriction and comparison
It is rather surprising that both model predications are opposite at small x even
though both results are very similar in the F
2
shadowing. This fact suggests that
the F
3
shadowing could be used for discriminating among various models. We can at
least rule out one of the above models by accurate experimental measurements.
The structure function F
3
is mea-
sured in neutrino interactions. Because
the process is a weak interaction, most
data are taken by using nuclear targets
with large mass number. In order to learn
about the nuclear modication, the ratio
F
A
3
(x)=F
D
3
(x) should be taken. Unfortu-
nately, available deuteron data are not ac-
curate enough to investigate 10{20% ef-
fects. There is little experimental infor-
mation on the F
3
shadowing from neutrino
data, so that we estimate a current experi-
mental restriction on the F
3
shadowing by
using F
A
2
=F
D
2
( R
2
) data at medium x
and the baryon number conservation.
le=g1.eps,width=6.5cm
Fig. 1 Shadowing in F
3
(x).
In nding the restriction, we neglect next-to-leading-order eects and assume
R
3
= R
2
[ F
A
2
(x)=F
D
2
(x)] in the region x  0:3 because valence-quark distributions
dominate both structure functions. We employ SLAC R
Ca
2
data, which are tted by a
smooth curve. This curve is extrapolated into the small x region. The only guideline
is to satisfy the baryon-number conservation. First, a straight line in the logarithmic
x is simply drawn from x=0.3 as shown by the dashed line in Fig. 1 so as to satisfy
the conservation. Second, the curve is smoothly extrapolated into the small x region
so as to satisfy the conservation by allowing about 6% antishadowing at x = 0:1 0:2.
The result is the dotted curve in Fig. 1. The rst line is roughly the upper limit of
nuclear modication, and the second curve is roughly the lower bound. The shaded
area between these curves is the area of possible nuclear modication, which is allowed
by present experimental data of R
2
and the baryon-number conservation.
It is noteworthy that the rst parton-model (model 1) prediction is roughly equal
to the upper bound curve, and the aligned-jet model (model 2) prediction is to the
lower bound curve. So the models are two extreme cases, which are both acceptable
in our present knowledge. We have not investigated the details of other model predic-
tions. However, it is very encouraging to investigate the (anti)shadowing phenomena
of F
3
in the sense that the observable could be useful in discriminating among dierent
models, which produce similar results in the F
2
shadowing.
3. Conclusions
We have investigated nuclear modication of the structure function F
3
. In par-
ticular, the F
3
shadowing at small x is studied in two dierent theoretical models:
the parton-recombination with Q
2
rescaling and the aligned-jet model. These models
predict very dierent shadowing behavior, namely antishadowing in the rst model
and shadowing in the second model. Therefore, it is in principle possible to rule out
one of these models by measuring the ratio F
A
3
=F
D
2
. However, current experimental
data are not accurate enough to probe the F
3
shadowing. Our investigation is merely
intended to shed light on the model dierence in F
3
at this stage. Much detailed
theoretical and experimental investigations have to be done in the near future.
Acknowledgment
This research was partly supported by the Grant-in-Aid for Scientic Research
from the Japanese Ministry of Education, Science, and Culture under the contract
number 06640406.
References
1. R. Kobayashi, S. Kumano, and M. Miyama, preprint SAGA-HE-78-95, to be
published in Phys. Lett. B.
2. S. Kumano, Phys. Rev. C 48 (1993) 2016; Phys. Rev. C 50 (1994) 1247.
3. S. Kumano, Phys. Lett. B 298 (1993) 171; Phys. Lett. B 342 (1994) 339.
4. L. L. Frankfurt, M. I. Strikman, and S. Liuti, Phys. Rev. Lett. 65 (1990)
1725.


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