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,l1 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, kRU/T ≤ 1 for kR>>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. 3 4 REFERENCES 1. W. Weise. Phys. Rep. 13 (1974) 53; G. Piller, W. Weise. Phys. Rep. 330 (2000) 1. 2. M.L. Coldberger and K.М. Watson. Collision Theory, John Wiley& Sons, inc., New York – London – Sudney. 1964. (Moscow, Mir, 1967). 3. V.V. Balashov. Kvantovaya Teoriya Stolknovenii (Quantum Collision Theory). Moscow. MGU (Moscow State University). 1985. 4. J. Taylor. Scattering Theory: Theory of Nonrelativistic Collisions. Wiley, New York. 1972; (Moscow, Mir. 1975). 5. T.H. Bauer et al. Rev. of Mod. Phys. 50 (1978) 261; S. Boffi et al. Nucl.Phys.A 606 (1996) 421. 6. L.B. Bezrukov, E.V. Bugaev. Yadernaya fizika. V.33 (1981) 1195. 7. F. Falter et al. Phys. Rev.C 62 (2000) 031602; F. Falter et al. Phys. Rev.C 64 (2001) 024608. 8. V.P. Zavarzina, A.V. Stepanov. Bulletin of the Lebedev Physics Institute № 10 (1995) 4; Izv. RAS, seriya fizich. 61 (1997) 2170. 9. V.P. Zavarzina, A.V. Stepanov. Bulletin of the Lebedev Physics Institute. № 6 (1997) 39; ibid. №7 (1997) 20; Izv. RAS, seriya fizich. 62 (1998) 2195. 10. V.P. Zavarzina, A.V. Stepanov. Bulletin of the Lebedev Physics Institute. № 9 (1998) 36; Izv. RAS, seriya fizich. 64 (2000) 113. 11. V. Vineyard. Phys. Rev. 110 (1958) 999; A.V. Stepanov. Preprint INR. 1976. П -0046; A.A. Chumbalov et al. Z. Phys. A 328 (1987) 195; C. Bennhold, H.Tanabe. Nucl. Phys.A530. (1999) 625. 12. V.L. Eletsky, B.L. Ioffe. Phys. Rev. Lett.78 (1997) 1010; L.A. Kondratyk et al. Phys. Rev. C58 (1998) 1078. 13. Jan-C Alam et al. Phys. Rev. C66 ( 2002) 042202. 14. S.J. Wallace. Ann. of Phys. 78 (1973) 190. 15. V.P. Zavarzina, A.V. Stepanov. Soviet Physics Lebedev Institute Reports № 3 (1988) 14. 16. V.P. Zavarzina, A.V. Stepanov. Yadernaya Fizika. 43 (1988) 858; ibid. 54 (1991) 44; Izv. RAS, seriya fizich. 55 (1991) 960. 17. A. Baker. Phys. Rev.D 6 (1972) 3462; V.P. Zavarzina et al. Izv. AN SSSR, seriya fizich. 43 (1979) 2441. 4