In this talk we present the first calculation of the gluon structure function
for nucleus in QCD. We discuss the Glauber formula for the gluon structure
function and the violation of this simple approach that we anticipate in QCD.
(Talk given by E. Levin at QCD and nuclear target session at the Workshop on
Deep Inelastic Scattering and QCD, Paris, April 1995).Comment: 10 pages(latex file),4 fig (eps.files

Ayala, Alvaro L.

Ducati, M. B. Gay

Levin, E.

English

arXiv

In this talk we present the first calculation of the gluon structure function for nucleus in QCD. We discuss the Glauber formula for the gluon structure function and the violation of this simple approach that we anticipate in QCD. (Talk given by E. Levin at QCD and nuclear target session at the Workshop on Deep Inelastic Scattering and QCD, Paris, April 1995)

Gay-Ducati, M B

Ayala, A P

Levin, E

CERN Document Server

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June 16 1995 CBPF-NF-010/95 hep - ph / 9503399
GLUON STRUCTURE FUNCTION
FOR DEEPLY INELASTIC SCATTERING
WITH NUCLEUS IN QCD
M. B. G A Y D U C A T I and A L V A R O L. A Y A L A F
o
Instituto de Fisica, Univ. Federal do Rio Grande do Sul
Caixa Postal 15051, 91500 Porto Alegre, RS, BRASIL
and
E U G E N E L E V I N
y)
LAFEX, Centro Brasileiro de Pesquisas Fsicas (CNPq)
Rua Dr. Xavier Sigaud 150, 22290 - 180 Rio de Janeiro, RJ, BRASIL
and
Theory Department, Petersburg Nuclear Physics Institute
188350, Gatchina, St. Petersburg, RUSSIA
Abstract: In this talk we present the rst calculation of the gluon structure function
for nucleus in QCD. We discuss the Glauber formula for the gluon structure function and
the violation of this simple approach that we anticipate in QCD

.
y
) Email: levin@lafex.cbpf.br; levin@fnalv.fnal.gov; levin@ccsg.tau.ac.il

Talk given by E. Levin at QCD and nuclear target session at the Workshop on Deep Inelastic Scat-
tering and QCD, Paris, April 1995
1 Introduction.
The subject of the talk is the gluon structure function for DIS with nucleus. The gluon
structure function is the most important physical observable that governs the physics
at high energy (low Bjorken x) in the DIS. Dealing with nucleus we have to take into
account the shadowing correction, which is the main point of interest in this talk. We
show that the shadowing correction in the region of small x can be treated theoretically
in QCD and can be reliably calculated using the information on the behaviour of the
gluon structure function for the nucleon. We organize the presentation in the following
way: rst, we discuss the Glauber approach to the nucleus gluon structure function and
answer the question what information on nucleon structure function we need to provide a
reliable calculation using the Glauber formula; second, we briey consider the corrections
to Glauber approach that have been anticipated in QCD. It should be stressed that this is
the rst presentation of our results and the lack of space does not allow us to discuss the
issue in details. This is why we are going to outline our strategy and to present the rst
estimates rather than to give the complete study of the problem which will be published
elsewhere.
2 Glauber approach in QCD .
The idea how to write the Glauber formula in QCD has been rst formulated in ref.
[1] and was carefully developed by Mueller in ref. [2]. It is easier to explain the idea
considering the penetration of quark-antiquark pair through the target. Indeed, during
the time of passage through the target the transverse distance r
t
between quark and
antiquark can vary by the amount r
t
/ R
k
t
E
, where E is the energy of the pair and R
is the size of the target (see Fig.1). The quark transverse momentum (k
t
) is k
t
/
1
r
t
due
to uncertainty principle. Therefore
 r
t
R
k
t
E
 r
t
(1)
holds if
r
2
t
 s  2mR (2)
In terms of Bjorken x the above condition looks as follows:
x 
1
2mR
(3)
1
It means that the transverse distance between quark and antiquark is a good degree of
freedom [1][2][3]. As has been shown by Mueller not only quark - antiquark pair can be
considered in such way. The propagation of a gluon through the target can be treated in
a similar way as the interaction of gluon - gluon pair with denite transverse separation
r
t
with the target. The total cross section of the absorbtion of gluon(G

) with virtuality
Q
2
and Bjorken x can be written in the form:

G

= (4)
Z
1
0
dz
Z
d
2
r
t
2
Z
d
2
b
t
2
	
G

?
(Q
2
; r
t
; x; z) 2  f1   exp[  (r
2
t
) S(b
2
t
) ] g  	
G

?

(Q
2
; r
t
; x; z)
where 	
G

?
is the wave function of the virtual gluon with transverse polarization. As was
σN(r2t )
N
G*(Q2)
r
→
⊥
A A
Figure 1: The structure of the parton cascade in the Glauber formula.
shown in ref. [2] within leading log approximation of perturbative QCD (pQCD) we can
2
safely replace this function by
1
r
2
t
after integration over z in eq.(3). Finally the Glauber
formula for the gluon structure function reads ( for N
c
= N
f
= 3):
xG(Q
2
; x) =
4

2
Z
1
x
dx
0
x
0
Z
d
2
b
t

Z
1
4
Q
2
d
2
r
t

1
r
4
t
2 f1   e
 
1
2

GG
(r
2
t
;x
0
)S(b
2
t
)
g (5)
where

GG
=
3
s
4

2
r
2
t
xG(
4
r
2
t
; x) (6)
and S(b
2
t
) is the prole function in impact parameter space for the interaction of the gluon-
gluon pair with the target. We use for the calculation the Gaussian parameterization for
S, namely:
S(b
2
t
) =
A
R
2
A
e
 
b
2
t
R
2
A
(7)
where A is the number of the nucleons in a nucleus and R
2
A
is the mean radius of a nucleus,
which is equal to
R
2
A
=
2
5
R
2
WS
R
WS
is the size of the nucleus in the Wood - Saxon parameterization, which we chose
R
WS
= r
0
A
1
3
with r
0
= 1:3fm. Using the Gaussian parameterization for S we can take
the integral over b
t
and get the answer:
xG(Q
2
; x) =
2

2
Z
1
x
dx
0
x
0
Z
1
1
Q
2
d
2
r
0
t

R
2
A
r
04
t
fC + ln
G
(x
0
; r
02
t
) + E
1
(
G
(x
0
; r
02
t
) g (8)
where C is the Euler constant and E
1
is the exponential integral (see ref.[4] 5.1.11) and

G
(x
0
; r
02
t
) =
3
s
A r
02
t
2R
2
A
x
0
G
N
(x
0
;
1
r
02
t
) (9)
3 Theory status of the Glauber formula.
In this section we would like to recall the main assumptions that have been made to get
the Glauber formula:
1.Energy (x) should be so high (small) to satisfy eqs.(2) and (3) and 
s
ln(1=x)  1.
The last condition means that we are doing the calculation in leading log(1/x) approxi-
mation of perturbative QCD (pQCD).
3
2.The GLAP [5] evolution equation holds in the region of small x. It means that

s
ln(1=r
2
t
)  1. One of the lessons that we have learned at this workshop is the fact
that the GLAP equation is able to describe the HERA data quite well.
3.Only the fastest partons ( GG pair) interacts with the target and there are no
correlations between partons from dierent parton cascades (see Fig.1).
4.There are no correlations between dierent nucleons in a nucleus.
5.The average b
t
for GG pair - nucleon interaction is much smaller than R
A
.
We are going to discuss how well all the above assumptions work in the last section of
the talk.
4 Results.
In our calculations we use the GRV parameterization [6] for the nucleon gluon structure
function. This parameterization describes the data quite well and it is suited for our
purpose because (i) the initial virtuality for the GLAP equation is small and we can discuss
the contribution of the large distances having some support in the experimental data; (ii)
the parameterization uses the GLAP equation and the most essential contribution comes
from the region where 
s
lnQ
2
 1 and 
s
ln(1=x)  1.
4.1 Where the shadowing corrections are big.
Fig.2 shows the kinematic region of the deeply inelastic scattering. The curves are the
solution of the equation 
G
= 1 for N (nucleon), Ca and Au. Above each of these curves
the value of 
G
> 1 and the shadowing correction (SC) are big, below 
G
< 1 and the
SC are rather small.
4.2 What we are able to calculate in QCD.
From the master equation (5) one can see that the large distances contribute to the value
of the gluon structure function. Such contributions we are not able to calculate in pQCD
and the value of the gluon structure function crucially depends on the hypothesis about
nonperturbative behaviour of the gluon structure function that we have to assume to treat
the large distances contribution. In pQCD we can safely calculate only the dierence
4
0.0 4.0 8.0 12.0
0.4
0.6
0.8
1.0
Q2=1.0 Gev2 Ca
Q2=10. Gev2 Ca
Q2=1.0 Gev2 Au
Q2=10. Gev2 Au
R1
ln(1/xB)
Figure 2: Solution for  = 1.
5
xG
A
(x;Q
2
)   xG
A
(x;Q
2
= Q
2
0
) where Q
2
0
is the initial virtuality. In Fig. 3 one can nd
the calculation for the ratio:
R
1
=
xG
A
(x;Q
2
)   xG
A
(x;Q
2
= Q
2
0
)
A (xG
N
(x;Q
2
)   xG
N
(x;Q
2
= Q
2
0
))
(10)
as function of x for Ca and Au (Q
2
0
= 0.25 GeV
2
).
0.0 5.0 10.0 15.0
0.2
0.5
0.8
1.1
1.4
Q2=1.0 Model 1
Q2=10. Model 1
Q2=1.0 Model 2
Q2=10. Model 2
R2
A=40 (Ca)
ln(1/xB)
Figure 3: R
1
as a function of ln(1=x
B
) for Ca and Au.
6
4.3 Contribution of the large distances.
As has been mentioned we are able to treat this problem only using some model for large
distance behaviour of xG
N
. Fig.4 shows the ratio:
R
2
=
xG
A
(x;Q
2
)
A (xG
N
(x;Q
2
))
(11)
for two models:
1. xG
N
(x;Q
2
< Q
2
0
) =
Q
2
Q
2
0
xG
N
(x;Q
2
= Q
2
0
). This model takes into account the
correct limit of the gluon structure function at small value of Q
2
, which follows from the
gauge invariance of QCD.
2.xG
N
(x;Q
2
< Q
2
0
) = xG
N
(x;Q
2
= Q
2
0
). In this model we assume that the scale
for the behaviour xG
N
/ Q
2
is much smaller than Q
2
0
. We look at the dierence in the
value of R
2
as the estimates of possible errors that originated from our poor knowledge of
long distance behaviour of the gluon structure function. The conclusion is that we cannot
calculate the value of R
2
even at x = 10
 3
at Q
2
= 1GeV
2
with better accuracy that
20%, while at larger value of Q
2
(Q
2
 10GeV
2
) the accuracy is better ( about 5%).
5 Correction to the Glauber formula.
To abandon the main assumptions which have been made in the Glauber formula we have
to develop a technique to include (i) the interactions of all partons ( not only the fastest
one) with a nucleus; (ii) the parton interaction inside a nucleon and (iii) the nucleon
correlation inside a nucleus. Such a technique has been suggested in ref.[8] and it is based
on new evolution equation that takes into account the parton interaction inside the parton
cascade as well as the parton interaction with the dierent nucleons. The lack of space
does not allow us to discuss this problem in details but we want to point out that the
Glauber formula shall be used as initial condition to the new evolution equation of ref.[8].
The numerical estimates [8] shows that the most essential contribution at least in
HERA kinematic region is generated by the interaction of all partons with the target
which corresponds to so called \fan" diagrams (see ref.[7]) while the dynamic correlation
(see refs.[9] [10]) due to the interaction inside the parton cascade remains to be rather
small.
However, reliable estimates can be done only after extracting from HERA data the
value of the scale for the SC for nucleon structure function. In our numerical estimates
7
0.0 5.0 10.0 15.0 20.0 25.0 30.0
0.0
5.0
10.0
15.0
nucleon
Ca
Au
ln(1/xB)
Q2
κ>1
κ<1
Figure 4: R
2
ratio for two models.
8
which will be published elsewhere we follow the following strategy: we neglect the dynamic
gluon correlations and iterate the master equation (5) several times. Since there is strong
ordering in rapidity of parton in each parton cascade the \i  th" iteration means that we
take into account the interaction with the target of all partons with the value of rapidity
(y) larger than y
i
. It turns out that we have to make only two iterations for x > 10
 3
to
get convergent result.
6 Conclusions.
We know the Glauber formula and the technique how to nd the corrections to the Glauber
formula in QCD. However we cannot provide reliable predictions for the gluon structure
function for nucleus until we will get more data on low Q
2
and low x behaviour of the
nucleon structure function. Unfortunately, we have to know not only the behaviour of the
nucleon structure function but also extract from the experimental data the scale for the
shadowing corrections to the value of the gluon structure function in the nucleon. We are
going to check how the information from DIS with nucleus can reduce these uncertainties.
At the moment we suggest to measure the ratio R
1
which can be calculated with much
better accuracy than the value of the gluon structure function.
Acknowledgements
It is a pleasure to thank the convenors of the QCD and nuclear target working group
for giving us time to discuss our results and the organizing committee of the conference
for the working atmosphere at the conference. Work partially nanced by CAPES and
CNPq, Brazil.
9
References
[1] E.M. Levin and M.G.Ryskin, Sov. J. Nucl.Phys 45 (1987) 150.
[2] A.H. Mueller, Nucl. Phys. B335 (1990) 115.
[3] A.H. Mueller, Nucl. Phys. B415 (1994) 373.
[4] Milton Abramowitz and Irene A. Stegen, Handbook of Mathematical Functions,
Dover Publications,Inc.,N.Y. 1970.
[5] V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438; L.N. Lipatov,
Yad. Fiz. 20 (1974) 181; G. Altarelli and G. Parisi, Nucl. Phys. B 126 (1977) 298;
Yu.L.Dokshitzer, Sov.Phys. JETP 46 (1977) 641.
[6] M.Gluck,E.Reya and A.Vogt, Z.Phys. C53 (1992) 127, DESY 94-206, Dec.1994.
[7] L.V.Gribov, E.M.Levin and M.G.Ryskin, Phys.Rep. 100 (1983) 1.
[8] E. Laenen and E. Levin, \A new evolution equation", CERN-TH/95-61, TAUP-
22226-95, CBPF NF-012/95, March 1995.
[9] E. Laenen and E. Levin, Ann. Rev. Nucl. Part. Sci. 44 (1995) 199.
[10] J.Bartels, Z.Phys C60 (1993) 471, Phys.Lett.B 298 (1993) 204;
E.M.Levin,M.G.Ryskin and A.G. Shuvaev, Nucl. Phys. B 387 (1992) 589.
10


Gluon structure function for deeply inelastic scattering with nucleus in QCD

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