Jastrow Two-nucleon Overlap Functions and Cross Sections of $^{16}$O$(e,e^{\prime}NN)^{14}$C Reactions

Abstract

Using the relationship between the two-particle overlap functions (TOF's) and the two-body density matrix (TDM), the TOF's for $^{16}$O$(e,e^{\prime}pp)^{14}$C reaction are calculated on the basis of TDM obtained with a Jastrow-type approach. The main contributions of the removal of $^1S$ and $^3P$ $pp$-pairs from $^{16}$O are taken into account in the calculations of the cross sections of the $^{16}$O$(e,e^{\prime}pp)^{14}$C reaction using the Jastrow TOF's. The contributions of the one-body and two-body delta currents are considered. The results are compared with the calculations using TOF's from other approaches.

Full text

Page 1

arXiv:nucl-th/0401032v1 15 Jan 2004
c ? submitted to 6thWorkshop on “e-m induced Two-Hadron Emission”,Pavia, 2003. February 8, 2008 – 1
Jastrow Two-nucleon Overlap Functions and Cross Sections of
16O(e,e′NN)14C Reactions∗
D. N. Kadrev,1,2M. V. Ivanov,1A. N. Antonov,1, 3C. Giusti,2,4and F. D. Pacati2, 4
1Institute for Nuclear Research and Nuclear Energy,
Bulgarian Academy of Sciences, Sofia 1784, Bulgaria
2Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Pavia, Italy
3Departamento de Fisica Atomica, Molecular y Nuclear, Facultad de Ciencias Fisicas,
Universidad Complutense de Madrid, E-28040 Madrid, Espa˜ na
4Dipartimento di Fisica Nucleare e Teorica, Universit` a di Pavia, Pavia, Italy
Using the relationship between the two-particle overlap functions (TOF’s) and the two-
body density matrix (TDM), the TOF’s for16O(e,e′pp)14C reaction are calculated on the
basis of TDM obtained with a Jastrow-type approach. The main contributions of the removal
of1S and3P pp-pairs from16O are taken into account in the calculations of the cross sections
of the16O(e,e′pp)14C reaction using the Jastrow TOF’s. The contributions of the one-body
and two-body delta currents are considered. The results are compared with the calculations
using TOF’s from other approaches.
1.INTRODUCTION
As known, two nucleons can be ejected from the nucleus by two-body currents due to meson
exchanges and delta-isobar excitation. But also, the real or virtual photon can hit, through a one-
body current either nucleon of a correlated pair and both nucleons are then ejected simultaneously
from the nucleus. The role and relevance of these two competing processes can be different in
different reactions and kinematics. It is thus possible to envisage situations where either process is
dominant and various specific effects can be disentangled and separately investigated. This gives
ground for studies of short-range correlations (SRC) [1, 2, 3, 4] in a nucleus by means of the
two-nucleon knockout processes.
Various theoretical models for cross section calculations have been developed in recent years in
order to explore the effects of ground-state NN correlations on (e,e′NN) [5, 6, 7, 8, 9, 10] and
(γ,NN) [11, 12, 13, 14, 15, 16, 17] knockout reactions. It appears from these studies that the most
promising tool for investigating SRC in nuclei is represented by the (e,e′pp) reaction, where the
effect of the two-body currents is less dominant as compared to the (e,e′pn) and (γ,NN) processes.
Measurements of the exclusive16O(e,e′pp)14C reaction performed at NIKHEF in Amsterdam [18,
19, 20] and MAMI in Mainz [21, 22] have confirmed, in comparison with the theoretical results, the
validity of the direct knockout mechanism for transitions to low-lying states of the residual nucleus
and have given clear evidence of SRC for the transition to the ground state of14C.
One of the main ingredients in the transition matrix elements of exclusive two-nucleon knockout
reactions is the two-nucleon overlap function (TOF). The TOF contains information on nuclear
structure and correlations and allows one to write the cross section in terms of the two-hole spectral
function [2]. The TOF’s and their properties are widely reviewed, e.g., in [23].
In [8] the TOF’s for the16O(e,e′pp)14C reaction are given by the product of a coupled and fully
antisymmetrized pair function of the shell model and a Jastrow-type correlation function which
incorporates SRC. A more sophisticated treatment is used in [9], where the TOF’s are obtained
from an explicit calculation of the two-proton spectral function of16O [24], which includes, with
some approximations but consistently, both SRC and long-range correlations (LRC).
A different method to calculate the TOF’s has been suggested in [25] using the established
general relationships connecting TOF’s with the ground state two-body density matrix (TDM). The
∗The extended paper was published in Phys. Rev. C 68, 014617 (2003)

Page 2

c ? submitted to 6thWorkshop on “e-m induced Two-Hadron Emission”,Pavia, 2003. February 8, 2008 – 2
procedure is based on the asymptotic properties of the TOF’s in coordinate space, when the distance
between two of the particles and the center of mass of the remaining core becomes very large. This
procedure can be considered as an extension of the method suggested in [26], where the relationship
between the one-body density matrix and the one-nucleon overlap function is established. The latter
has been applied [27, 28, 29, 30, 31, 32, 33, 34, 35] to calculate the one-nucleon overlap functions,
spectroscopic factors and to make consistent calculations of the cross sections of different one-
nucleon removal reactions, such as (p,d), (e,e′p), and (γ,p) [27, 30, 31, 32, 33, 34, 35] on16O
[31, 32, 34] and40Ca [33, 34], (p,d) on24Mg,28Si and32S [35], as well as (e,e′p) on32S [35] within
various correlation methods.
The first aim of the present work is to apply the procedure suggested in [25] to calculate TOF’s
for16O using the TDM calculated in [36] with the Jastrow correlation method (JCM), which
incorporates the nucleon-nucleon SRC. As a second aim, the resulting two-proton overlap functions
are used to calculate the cross section of the16O(e,e′pp) reaction for the transition to the 0+ground
and the 1+excited states of14C. The cross sections are calculated on the basis of the theoretical
approach developed in [5, 8, 9].
2.TWO-BODY DENSITY MATRIX AND OVERLAP FUNCTIONS
In this Section we present shortly the definitions and some properties of the TDM and related
quantities in the overlap function representation. The method to extract the TOF’s from the TDM
[25] used in this work is also given.
The TDM is defined in coordinate space as:
ρ(2)(x1,x2;x′
1,x′
2) = ?Ψ(A)|a†(x1)a†(x2)a(x′
2)a(x′
1)|Ψ(A)?, (1)
where |Ψ(A)? is the antisymmetric A-fermion ground state wave function normalized to unity and
a†(x), a(x) are creation and annihilation operators at position x. The coordinate x includes the
spatial coordinate r and spin and isospin variables. The TDM ρ(2)is trace-normalized to the
number of pairs of particles:
Tr ρ(2)=1
2
?
ρ(2)(x1,x2)dx1dx2=A(A − 1)
2
.(2)
Of direct physical interest is the decomposition of the TDM in terms of the overlap functions
between the A-particle ground state and the eigenstates of the (A−2)-particle systems, since TOF’s
can be probed in exclusive knockout reactions.
The TOF’s are defined as the overlap between the ground state of the target nucleus Ψ(A)and
a specific state Ψ(C)
α
of the residual nucleus (C = A − 2) [23]:
Φα(x1,x2) = ?Ψ(C)
α|a(x1)a(x2)|Ψ(A)?. (3)
Inserting a complete set of (A − 2) eigenstates |α(A − 2)? into Eq. (1) one gets
ρ(2)(x1,x2;x′
1,x′
2) =
?
α
Φ∗
α(x1,x2)Φα(x′
1,x′
2). (4)
The norm of the two-body overlap functions defines the spectroscopic factor
S(2)
α = ?Φα|Φα?. (5)
A procedure for obtaining the TOF’s on the basis of the TDM suggested in [25] uses the par-
ticular asymptotic properties of the TOF’s.

Page 3

c ? submitted to 6thWorkshop on “e-m induced Two-Hadron Emission”,Pavia, 2003. February 8, 2008 – 3
In the case when two like nucleons (neutrons or protons) unbound to the rest of the system
are simultaneously transferred, the following hyperspherical type of asymptotics is valid for the
two-body overlap functions [23, 37, 38]

¯ h2
Φ(r,R) −→ N exp

−
?
4m|E|
?
R2+1
4r2
?


?
R2+1
4r2
?−5/2
, (6)
where r and R are the magnitudes of the relative and center-of-mass (c.m.) coordinates, r = r1−r2
and R = (r1+ r2)/2, respectively, m is the nucleon mass and E = E(A)− E(C)is the two-nucleon
separation energy.
For a target nucleus with Jπ
ν0(corresponding to the smallest two-nucleon separation energy) of the radial part of the TOF
Φν0JSLlLR(r,R) can be expressed in terms of the TDM as
tar.= 0+and for large r′= a and R′= b a single term with
Φν0JSLlLR(r,R) =ρ(2)
JSLlLR(r,R;a,b)
Φν0JSLlLR(a,b)
=
ρ(2)
JSLlLR(r,R;a,b)
??
N exp
?
−k
b2+1
4a2???
b2+1
4a2?−5/2,(7)
where k = (4m|E|/¯ h2)1/2is constrained by the experimental values of the two-nucleon separation
energy E. The relationship obtained in Eq. (7) makes it possible to extract TOF’s with quantum
numbers JSLlLRfrom a given TDM. The coefficient N and the constant k can be determined from
the asymptotics of ρ(2)
JSLlLR(r,R;r,R).
3. RESULTS
A.The two-proton overlap functions
The procedure described briefly in Section 2 has been applied to calculate the two-proton overlap
functions in the16O nucleus for the transition to the 0+ground and the 1+excited states of14C.
The TDM obtained in [36] in the framework of the low-order approximation (LOA) of the Jastrow
correlation method has been used [39].
The TOF’s for the1S0 and3P1 states are obtained in the JCM. The result for1S0 state is
presented in Fig. 1. It is compared with the uncorrelated TOF’s obtained applying the same
procedure to the uncorrelated TDM. The notation for the partial waves in our case is2S+1lL.
The spectroscopic factors corresponding to the1S0 and3P1 overlap functions are 0.958 and
0.957, respectively.
As a next step, we derive the total TOF ΦνJM(x,X) in terms of a sum over all possible partial
components, i.e.
ΦνJM(x,X) =
?
LSlLR
ΦνJSLlLR(r,R)AJM
SLlLR(σ1,σ2; ? r,?R). (8)
We integrate the squared modulus of the total TOF in Eq. (8) over the angles and sum over the
spin variables. The result can be written in the form (for the smallest value of ν = ν0):
|ΦJM(x,X)|2≡|?ΦJM(r,R)|2=
?
LSlLR
ρ(2)
JLSlLR(r,R),(9)

Page 4

c ? submitted to 6thWorkshop on “e-m induced Two-Hadron Emission”,Pavia, 2003. February 8, 2008 – 4
FIG. 1: The1S0 two-proton overlap functions for the nucleus16O leading to the 0+ground state of14C
extracted from the JCM (left) and uncorrelated (right) two-body density matrices.
where the bar denotes the integration over the angles and summation over the spin variables, and
?ΦJM(r,R) is the radial part of the total TOF obtained after the integration and summation. Using
?
N exp−k
b2+1
the asymptotics of?ΦJM(r,R) at r −→ a, R −→ b one can write:
?ΦJM(r,R) =
LSlLR
??
ρ(2)
JSLlLR(r,R;a,b)
?
4a2???
b2+1
4a2?−5/2
.(10)
The results for the1S0and3P1partial components have a similar behaviour as previous ones,
the main difference is that they are somewhat reduced in magnitude. The spectroscopic factor
corresponding to the total TOF is equal to unity in the uncorrelated case and 0.965 in the Jastrow
case.
The Jastrow TDM (including only SRC) is not “rich” enough to be able to explain realistically
transitions to all the excited states of14C. Therefore, only the transition to the 1+state is considered
in the present paper as an example of the applicability of the method.
In the case of the transition to the 1+excited state of14C pp-pairs in the states3P0,1,2give
main contributions to the process. The value of the spectroscopic factor e.g. for3P1is 0.967 in the
Jastrow case and unity in the uncorrelated one.
B.The16O(e,e′pp)14C reaction
The TOF’s obtained from the TDM within the Jastrow correlation method have been used
to calculate the cross section of the16O(e,e′pp)14C knockout reaction in one-photon exchange
approximation [2, 5].
As an example, the differential cross section calculated for the transition to the 0+ground state
of14C is shown in Fig. 2 for the kinematical setting considered in the experiment performed at
MAMI [21, 22].
The results are compared with the cross sections already shown in [9], where the TOF is taken
from a calculation of the two-proton spectral function (SF) [24], where a two-step procedure has
been adopted to include both SRC and LRC.
In the figure are also shown for a comparison the results obtained with a simpler approach,
where the two-nucleon wave function is given by the product of the pair function of the shell model
and of a Jastrow type central and state independent correlation function (SM+CORR).

Page 5

c ? submitted to 6thWorkshop on “e-m induced Two-Hadron Emission”,Pavia, 2003. February 8, 2008 – 5
FIG. 2: The differential cross section of the16O(e,e′pp) reaction as a function of the recoil momentum pBfor
the transition to the 0+ground state of14C in the super-parallel kinematics with E0= 855 MeV, ω = 215
MeV and q = 316 MeV/c. The curves are obtained with different treatments of the TOF:1S0(dashed line)
and3P1 (dotted line) as “independent” TOF’s in the JCM in the left panel and as partial components in
the right panel.
SRC are quite strong and even dominant for the1S0state and much weaker for the3P1state.
The role of the isobar current is strongly reduced for1S0pp knockout, since there the magnetic
dipole NN ↔ N∆ transition is suppressed [17, 40]. As a consequence, in the figures the1S0results
are dominated by the one-body current and thus by SRC, while the ∆ current gives the main
contribution to3P1pp knockout.
It can be seen from Fig. 2 that the cross section calculated with the Jastrow TOF for the1S0
state is close to the SF and also to the SM+CORR results at low values of pB, up to ∼ 150 − 200
MeV/c. For pB≥ 200 MeV/c3P1knockout becomes dominant with all the different treatments of
the TOF. The results with the3P1TOF from the Jastrow TDM is however much larger than the
SF result and also larger than the SM+CORR cross section.
The cross section calculated with the total TOF, obtained from the combination of the1S0
and3P1partial components, are shown in the right panel of Figs. 2. In both kinematical settings
the1S0 component dominates at low values of pB, while the3P1 component produces a strong
enhancement at high momenta.
Although obtained from a calculation of the TDM within the JCM where only SRC are included,
the TOF used in our calculations are able to reproduce the main qualitative features which were
found in previous theoretical investigations. This means that the procedure suggested in [25] to
calculate the TOF’s from the TDM can be applied and exploited in the study of two-nucleon
knockout reactions.
The differential cross section calculated for the transition to the 1+state of14C, at 11.3 MeV
excitation energy, is shown in Fig. 3 in the same kinematical setting already considered for the 0+
state in Fig. 2. With respect to the other results, the Jastrow TOF produces in the super-parallel
kinematics a strong enhancement at high momenta, which makes the shape of the cross sections
larger and flatter than that with the SF and SM+CORR TOF’s.
4.CONCLUSIONS
The results of the present work can be summarized as follows:
i) The two-nucleon overlap functions (and their norms, the spectroscopic factors) corresponding

Page 6

c ? submitted to 6thWorkshop on “e-m induced Two-Hadron Emission”,Pavia, 2003. February 8, 2008 – 6
FIG. 3: The differential cross section of the16O(e,e′pp) reaction as a function of the recoil momentum pB
for the transition to the 1+excited state of14C in the same kinematics as in Fig. 2.
to the knockout of two protons from the ground state of16O and the transition to the ground
and 1+excited states of14C are calculated using the recently established relationship [25]
between the TOF’s and the TDM. In the calculations the TDM obtained within the JCM
[36] is used. Though only SRC are accounted for in the Jastrow TDM, the results can be
considered as a first attempt to use an approach which fulfils the general necessity the TOF’s
to be extracted from theoretically calculated TDM’s corresponding to realistic wave functions
of the nuclear states.
ii) The TOF’s extracted from the Jastrow TDM are included in the theoretical approach of
[5, 8, 9] to calculate the cross section of the16O(e,e′pp)14C knockout reaction. Numerical
results in different kinematics are compared with the cross sections calculated, within the
same theoretical model for the reaction mechanism, with different treatments of the TOF, in
particular with the more refined approach of [9, 24], where the TOF’s are obtained from a
calculation of the two-proton spectral function of16O where both SRC and LRC are included.
The cross sections calculated in the present work, where the TOF’s are extracted from the
Jastrow TDM, confirm the dominant contribution of1S0pp knockout at low values of recoil
momentum, up to ≃ 150 − 200 MeV/c. The3P1contribution is mainly responsible for the
high-momentum part of the cross section at pB≥ 200 MeV/c.
iii) Our method is applied in the present work only to the 0+ground and the 1+excited states of
14C. The main aim was to check the practical application of all steps of the method to a given
state of the residual nucleus. Therefore, the results obtained for the16O(e,e′pp)14C reaction,
which are able to reproduce the main qualitative features of the cross sections calculated
with different treatments of the TOF’s, can serve as an indication of the reliability of the
method, that can be applied to a wider range of situations and, as an alternative to an
explicit calculation of the two-hole spectral function, to more refined approaches of the TDM
[4, 41, 42, 43].
Acknowledgments
One of the authors (D. N. K.) would like to thank the Pavia Section of the INFN for the warm
hospitality and for providing the necessary fellowship. A.N.A. is grateful for support during his stay
at the Complutense University of Madrid to the State Secretariat of Education and Universities of

Page 7

c ? submitted to 6thWorkshop on “e-m induced Two-Hadron Emission”,Pavia, 2003. February 8, 2008 – 7
Spain (N/Ref. SAB2001-0030). Three of the authors (A.N.A., M.V.I. and D.N.K.) are thankful
to the Bulgarian National Science Foundation for partial support under the Contracts Nos. Φ-809
and Φ-905.
[1] K. Gottfried, Nucl. Phys. 5, 557 (1958); Ann. of Phys. 21, 63 (1963).
[2] S. Boffi, C. Giusti, F. D. Pacati, and M. Radici, Electromagnetic Response of Atomic Nuclei, Oxford
Studies in Nuclear Physics (Clarendon Press, Oxford, 1996).
[3] A.N. Antonov, P.E. Hodgson and I.Zh. Petkov, Nucleon Correlations in Nuclei (Springer–Verlag, Berlin,
1993).
[4] H. M¨ uther and A. Polls, Prog. Part. Nucl. Phys. 45, 243 (2000).
[5] C. Giusti and F. D. Pacati, Nucl. Phys. A535, 573 (1991).
[6] J. Ryckebusch, M. Vanderhaeghen, K. Heyde, and M. Waroquier, Phys. Lett. B 350, 1 (1995).
[7] J. Ryckebusch, V. Van der Sluys, K. Heyde, H. Holvoet, W. Van Nespen, M. Waroquier, and M. Van-
derhaeghen, Nucl. Phys. A624, 581 (1997).
[8] C. Giusti and F. D. Pacati, Nucl. Phys. A615, 373 (1997).
[9] C. Giusti, F. D. Pacati, K. Allaart, W. J. W. Geurts, W. H. Dickhoff, and H. M¨ uther, Phys. Rev. C
57, 1691 (1998).
[10] C. Giusti, H. M¨ uther, F. D. Pacati, and M. Stauf, Phys. Rev. C 60, 054608 (1999).
[11] C. Giusti, M. Radici, and F. D. Pacati, Nucl. Phys. A546, 607 (1992).
[12] S. Boffi, C. Giusti, M. Radici, and F. D. Pacati, Nucl. Phys. A564, 473 (1993).
[13] J. Ryckebusch, M. Vanderhaeghen, L. Machenil, and M. Waroquier, Nucl. Phys. A568, 828 (1994).
[14] J. Ryckebusch, L. Machenil, M. Vanderhaeghen, V. Van der Sluys and M. Waroquier, Phys. Rev. C49,
2704 (1994).
[15] J. Ryckebusch, Phys. Lett. B 383, 1 (1996).
[16] P. Wilhelm, H. Arenh¨ ovel, C. Giusti, and F. D. Pacati, Z. Phys. A 359, 467 (1997).
[17] C. Giusti and F. D. Pacati, Nucl. Phys. A641, 297 (1998).
[18] C. J. Onderwater et al., Phys. Rev. Lett. 78, 4893 (1997).
[19] C. J. Onderwater et al., Phys. Rev. Lett. 81, 2213 (1998).
[20] R. Starink et al., Phys. Lett. B 474, 33 (2000).
[21] G. Rosner, in Proceedings of 10thMini-Conference on Studies of Few-Body Systems with High Duty-
Factor Electron Beams, NIKHEF, Amsterdam, 1999, p.95.
[22] G. Rosner, Prog. Part. Nucl. Phys. 44, 99 (2000).
[23] J. M. Bang, F. A. Gareev, W. T. Pinkston, and J. S. Vaagen, Phys. Rep. 125(6), 253 (1985).
[24] W. J. W. Geurts, K. Allaart, W. H. Dickhoff, and H. M¨ uther, Phys. Rev. C 54, 1144 (1996).
[25] A. N. Antonov, S. S. Dimitrova, M. V. Stoitsov, D. Van Neck, and P. Jeleva, Phys. Rev. C 59, 722
(1999).
[26] D. Van Neck, M. Waroquier, and K. Heyde, Phys. Lett. B 314, 255 (1993).
[27] M. V. Stoitsov, A. N. Antonov, and S. S. Dimitrova, Phys. Rev. C 53, 1254 (1996).
[28] D. Van Neck, A. E. L. Dieperink, and M. Waroquier, Phys. Rev. C 53, 2231 (1996); Z. Phys. A 355,
107 (1996).
[29] D. Van Neck, L. Van Daele, Y. Dewulf, and M. Waroquier, Phys. Rev. C 56, 1398 (1997).
[30] S. S. Dimitrova, M. K. Gaidarov, A. N. Antonov, M. V. Stoitsov, P. E. Hodgson, V. K. Lukyanov,
E. V. Zemlyanaya, and G. Z. Krumova, J. Phys. G 23, 1685 (1997).
[31] M. K. Gaidarov, K. A. Pavlova, S. S. Dimitrova, M. V. Stoitsov, A. N. Antonov, D. Van Neck, and
H. Muether, Phys. Rev. C 60 024312 (1999).
[32] M. K. Gaidarov, K. A. Pavlova, A. N. Antonov, M. V. Stoitsov, S. S. Dimitrova, M. V. Ivanov, and
C. Giusti, Phys. Rev. C 61, 014306 (2000).
[33] M. V. Ivanov, M. K. Gaidarov, A. N. Antonov, and C. Giusti, Phys. Rev. C 64, 014605 (2001).
[34] M. V. Ivanov, M. K. Gaidarov, A. N. Antonov, and C. Giusti, Nucl. Phys. A699, 336 (2002).
[35] M. K. Gaidarov, K. A. Pavlova, A. N. Antonov, C. Giusti, S. E. Massen, Ch. C. Moustakidis, and
K. Spasova, Phys. Rev. C 66, 064308 (2002).
[36] S. S. Dimitrova, D. N. Kadrev, A. N. Antonov, and M. V. Stoitsov, Eur. Phys. J. A 7, 335 (2000).

Page 8

c ? submitted to 6thWorkshop on “e-m induced Two-Hadron Emission”,Pavia, 2003. February 8, 2008 – 8
[37] S. P. Merkuriev, Sov. J. Nucl. Phys. 19, 447 (1974).
[38] J. Bang, Physica Scripta 22, 324 (1980).
[39] R. Jastrow, Phys. Rev. 98, 1479 (1955).
[40] P. Wilhelm, J. A. Niskanen and H. Arenh¨ ovel, Nucl. Phys. A597, 613 (1996).
[41] S. C. Pieper, R. B. Wiringa, and V. R. Pandharipande, Phys. Rev. C 46, 1741 (1992).
[42] J. H. Heisenberg and B. Mihaila, Phys. Rev. C 59, 1440 (1999).
[43] A. Fabrocini, F. Arias de Saavedra, and G. Co’, Phys. Rev. C 61, 044302 (2000).