SHADOWING EFFECT OF PHOTOABSORPTION AT INTERMEDIATE ENERGIES AND
NUCLEON-NUCLEON CORRELATION FUNCTION IN NUCLEAR MATTER
A.V. Stepanov, V. P. Zavarzina
Institute for Nuclear Research, Russian Academy of Sciences
60-th October Anniversary Prospect 7a, 117312 Moscow, Russia
The nature of the high-energy photon interaction with complex nuclei was actively
studies over a period of a few last decades. A dramatic feature of this interaction in the
shadowing effect observed in the dependence of the photoabsorption cross section on the
nuclear-target mass number, with is similar to the effect previously observed in the hadronnucleus scattering. A theoretical analysis of this experimental result, which is based on the
Vector Dominance Model (VDM), has revealed a possibility to gain information on hadronic
components of the photon distribution function from experimental data. From VDM vector
mesons are of crucial importance among these components, primarily - meson (for details see
[1] and the references therein).
The expression for the total cross section of -ray quantum interaction with a target
nucleus of mass number  can be written using the optical theorem:
σ γΑΤ E γ   AT σ γΝ (E γ )  Δσ ,
(1)
Here σ γΝ (E γ ) is the total interaction cross section for a -ray quantum of energy E , with a
nucleon and (   ) is the contribution of the shadowing effect.
Excluding the channels with vector mesons from consideration, we arrive at an equation
describing the propagation of -ray quantum in nuclear matter and including the photon
polarization operator ( the “effective optical potential”):


1
V  P HQ

(2)
QHP
E  QHQ iε
Here Ĥ is the Hamiltonian of a photon + target nucleus + vector meson compound system. The
projection operators have conventional values (see, for example, [2,4]). P selects the subspace of
the photon target nucleus system states and Q selects the subspace of the vector meson + target
nucleus system states.
For the term which takes into account the shadowing effect in (1) the following
expression is obtained [2]:
Δσ  2
Ω
Im Ψ (i  ) V Ψ (i  )
v0
(3)
Here i(  ) is the exact wave function of the photon + nucleus system in the initial state, Ω is the
normalization volume, and v0 is the relative velocity of the incident photon and the target
nucleus.
Hereafter, the following expression for the wave function i(  ) is used in calculation of
the cross section:
i(  )
Here 0 

0
k
= 
 .

is the ground-state wave function of the target nucleus and k 
is a plane wave
which describes the propagation of a photon with momentum k . In what follows ћ =c=1 is
assumed.
In the expression for the Green function of the intermediate state in (2)
1
G
(4)
E  QHQ i 
intranuclear motion of nucleons is usually ignored. This permits summation over a complete
system of wave functions of a target nucleus in the intermediate state ( the convolution
2
approximation), the eikonal approximation being used for the vector meson propagator in the
field g (v) ( r1 , r2 ) of static nuclear matter [5],[6]. The studies of the shadowing effect were
carried out not only for high photon energies but also for intermediate ones (1-3 GeV) [7].
As a result, it became possible to express the summand  in (1) in terms of a four-fold
integral over space coordinates ( r1 = ( b , z1), r2  (b , z2)) ( b is the impact parameter) of the
product
g (v) ( r1 , r2 ) K ( r1 , r2 ; 0),
where
K ( r1 , r2 ;0) =
 
0T ρ( r1 ,0) ρ( r2 ,0) 0T
(5)
 
is the static pair correlation function of the nuclear density ρ( r ,0) in the ground state,
 (r , t )  e
 iHT t
AT
 ( r  r ) e
 iHT t
l
l 1
is the nuclear density operator in the Heisenberg representation, t is time, r1 is the radius vector
of the l-th nucleon, and ĤT is the target-nucleus Hamiltonian.
We obtain an estimate of the
effect of the intranuclear motion of nucleons on the shadowing effect in the interaction of
intermediate-energy photons with complex nuclei. To do this, we invoke the temporalcorrelation-function (TCF) method. This method was previously used by the authors to calculate
the total interaction cross section of incident particles with nuclei and the first-order optical
potential when the interaction of a projectile particle with individual nucleon of the target
nucleus resulted in the resonance excitation in the intermediate state [8-10]. In the case of
complex nuclear target the excitation of such a resonance is followed by a change of the state of
motion relative to the internal degrees of freedom. To allow for the effect of this motion, one
should go beyond the convolution approximation. The TCF method offers such an opportunity.
In the problem under consideration, the vector meson, which moves in nonstatic nuclear
matter, is the intermediate-state resonance. Using as before ([8-10]) both the short-time t and
factorization approximations [11], we obtain for the central region of intermediate and heavy
nuclei, where inhomogeneity of nuclear medium can be neglected, the standard expression for
 [5,7], in which the static correlation function K ( r1 , r2 ; 0) (5) is replaced by TCF
K ( r1 , r2 ; t)=
 
0T ρ(r1 , t ) ρ(r2 ,0) 0T 

 
d
q
 iq( r1  r2 ) itq2 / 2 M N
 Ω1 
e
e
(2 π) 3
0T e
(6)

0 T FT (q) ,
 
iqpt/MN
(7)

where MN is the nucleon mass, p  i is the nucleon momentum operator in the target nucleus,
and

FT (q)  0T
==
AT
e
l,l1
   
0T ~
ρ (q) ~
ρ (  q) 0T 

iq rl
e

iq rl 
0T 
 
0T ~
ρ (q) 0T
2
.
 
~
ρ (q) is the
Here 0T ρ 0T is the static form factor of the target nucleus in the ground state and ~

Fourier transform of the density operator ρ( r ).
We shall assume the nucleon-nucleon correlations with a characteristic scale ~ 1/qF to be
the most significant for the intermediate -quantum (and vector meson) energies; here qF is the
2
3
Fermi momentum. Then the magnitude of the integral in (7) is determined by the region of
characteristic momentum values q ~ qF . Assuming q ~ qF in
exp(iq2t/2MN) and
 
exp(i qp t /M N ) , we come to
 
2
K ( r1 , r2 ; t) 0T exp(i q F pt/M N ) 0T exp(iq F t/(2 M N )) K ( r1 , r2 ; 0).
(8)
Thus we obtained two additional factors in comparison with the standard result. The first
 
correction factor 0T exp(i q F pt/M N 0T takes into account the effect of the Fermi motion (the
“Doppler broadening”) [8]-[10]. The second factor, which includes the nucleon “recoil”, can be
 
involved in the expression for the vector meson propagation function q (v) ( r1 , r2 ).
Correspondingly, this leads to a displacement of the pole position in the propagator q (v) owing to
an effective increase of the -meson mass.
We assume for estimations that
t~ 1/V ~ (0,15 GeV)-1 and the Fermi energy
2
EF =qF2/2MN is 30 MeV, so that q F /(2 M N )
1
 0, 2. Then it is easy to see that the “Doppler

broadening” effect may be neglected, and the -meson mass shift owing to the “recoil” of
nuclear nucleons in the intermediate state is of the order of EF. It is known [12] that the shift mv
of the -meson mass in a nuclear medium due to the meson interaction with nuclear nucleons is
of the order of or even less than 50 MeV. However, even such a low value of mv has a
pronounced effect on the magnitude of the shadowing effect [13].
Therefore, in accurate calculations of the shadowing effect with allowance for the
modification of the vector-meson mass owing to its interaction with a nuclear medium (the
dynamic effect), the intermediate-state excitation of the degrees of freedom of the nucleon
intranuclear motion (kinematic effect) should also be taken into consideration.
Another source of improvement of the standard result of calculation of the shadowing
effect in nuclear photoabsorption is noteworthy. This is the inclusion of noneiconal corrections
to the vector meson propagator in nuclear matter. Unfortunately, the expression for this Green
function in the first approximation of the noneikonal expansion [14] obtained by the authors in
[15] is rather cumbersome. One can hardly rely with confidence on the results obtained from an
analysis of the corresponding expressions for the total interaction cross section, reaction cross
sections, etc., because the magnitudes of noneikonal corrections may differ essentially for
different parameters, for example, for real and imaginary parts of zero-angle scattering amplitude
[16]. The employment of the small perameter [13] 1/kR (k and R are the nucleus momentum and
radius, respectively) which does not include the ratio of the potential characteristic value to the
kinetic energy of the particle moving in this potential, can be justified under conditions of strong
absorption of a projectile particle incident on the nucleus with a sharp boundary. However, it is
completely unjustified in the conditions of the problem under consideration. In the case of the
propagation of a -meson produced in the interaction of an intermediate-energy -quantum with
a nucleus, the ratio U/T of the characteristic value of the potential U to the -meson kinetic
energy is small compared to unity. This is favorable for the fulfillment of the applicability
condition of the eikonal approximation, kRU/T ≤ 1 for kR>>1. Here k is the -meson
momentum.
In conclusion we note that the result of the eikonal approximation can be refined in the
context of approaches other than the noneikonal expansion method [14]. The analysis of compact
expressions for the Green function of -meson in nuclear matter which can be obtained using
these methods [17] confirm the above conclusions on small corrections to the results of the
eikonal approximation for the description of the intermediate-energy -meson motion in nuclear
matter.
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