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Final-state interaction in electromagnetic response functions
Nuclear Physics North-Holland
A524 (1991) 681-705
FINAL-STATE
INTERACTION
F. CAPUZZI,
IN ELECTROMAGNETIC FUNCTIONS
C. GIUSTI
RESPONSE
and F.D. PACATI
Dipartimento
di Fisica Nucleare e Teorica dell’ Universitci, Pavia and Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Italy
Received
2 April 1990
Abstract: The longitudinal
and transverse response functions of the inclusive quasi-elastic (e, e’) scattering are calculated in a many-body approach where the Green function is expanded over the eigenfunctions of a non-hermitian optical potential. Direct and interference contributions of all the channels are included in a simple way and final-state interaction is treated consistently with the exclusive (e, e’p) reaction. The response functions of ‘*C and 40Ca are evaluated for momentum transfers between 400 and 550 MeV/c.
1. Introduction The inclusive
electron
scattering
in the quasi-elastic
region
is dominated
direct single-nucleon emission, which suggests the one-body mechanism interpretation. In fact, the cross section seemed fairly well reproduced Fermi-gas model ‘): the shift of the maximum from the free-scattering associated
with the average
binding
energy
of the nucleons,
by the
as a natural by a simple value was
and the width of the
peak with the Fermi motion of the particles. However, this simple model is not able to simultaneously describe quasi-elastic longitudinal and transverse responses, which, in recent years, have been experimentally separated on a variety of nuclei, over a large range of momentum transfers 2W4).Individual responses display peculiar and interesting features, whose interpretation requires a more sophisticated treatment. Some properties can be explained by conventional nuclear models, but others defy an accurate description. Therefore, the theoretical analysis of the nuclear responses aroused a great interest, providing a severe test both of the nuclear structure description and of the reaction mechanism. The transverse response shows a pronounced maximum, whose position and size seem reasonably described already by the Fermi-gas model. Furthermore, a massnumber scaling is observed, which points to nuclear matter properties. Indeed, theoretical descriptions based on infinite nuclear matter can reasonably account for the data, when also subnuclear degrees of freedom, such as meson-exchange currents and isobar excitation, are included 5). 0375-9474/91/%03.50 @
1991 - Elsevier
Science
Publishers
B.V. (North-Holland)
682
F. Capuzzi et al. / Final-state
The longitudinal shape
response
with respect
overestimated
is more difficult
to the prediction
by single-particle
interaction
to understand.
of the Fermi-gas
models
and a sizeable
It displays
a change
of
model.
Its size is largely
violation
of the Coulomb
sum rule results. Many different
approaches
have been attempted
in order to solve these problems,
with partial success, but a completely consistent and satisfactory the data does not yet exist. A number of calculations have been carried out, in order to particle-hole correlations, in the context of Tamm-Dancoff “) approximation (RPA) [ref. ‘)I. These calculations are successful
description
of all
take into account or random-phase in improving the
description of the charge response at low momentum transfers, but at momenta higher than 400 MeV/c a small effect is obtained. Relativistic corrections in the frame of Dirac phenomenology were evaluated in the mean field theory 8.9) and including RPA correlations lo). They momentum transfers. The longitudinal
give a sizeable suppression, response of nuclear matter
even at high was calculated
in a variational approach with a correlated basis, obtaining a fair agreement with the data, except at high values of both momentum and energy transfer I’). The effects of neutron-proton tensor correlations were also studied “). The longitudinal structure function in ‘*C was fitted in the frame of a Glauber-like multiple-scattering theory with not completely understood parameters 13). Medium modification of nucleon properties was advocated as a possible source of explanation 14); a connection of this effect with 2p2h correlations was presented in ref. 15). Scaling properties of the electromagnetic responses were used in order to analyse the relevance of different approximations ‘6). Finally, the sum rule saturation was also considered I’). The role of final-state interaction is a crucial point in the analysis of inclusive electron scattering. Its effect is essential to fit the exclusive (e, e’p) cross section, but the current absorption produced by the imaginary part of the optical potential seems inconsistent with an inclusive measurement. A method in which these two aspects are reconciled was presented in the frame of the many-body Green function theory
I’). However,
in ref. “) the determination
task, as the actual calculation is performed coordinate space, and a very large number
of the Green
function
is a difficult
in an angular momentum basis and in of partial waves is needed.
Here, we propose a direct and simple method for taking into account the effect of a non-hermitian optical potential in the many-body Green function approach j9). The final-state interaction is treated consistently with the exclusive (e, e’p) reaction, whose contribution to the inclusive scattering can be easily separated. The formalism concerning the spectral representation of the optical-model Green function is developed in sect. 2. The effect of the interference between different channels is discussed and an approximate way to take into account its contribution is proposed. The preservation of the Coulomb sum rule is also demonstrated. In sect. 3 the results of calculations carried out for “C and 40Ca are presented. Some conclusions are drawn in sect. 4.
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F. Capuzzi et al. / Final-state interaction
2. The Green function approach In the one-photon exchange approximation, the inclusive differential cross section for the quasi-elastic (e, e’) scattering on a nucleus is given by da p= dR dE:,
K(ZE&+
RT) ,
where K is a kinematical factor and 2 EL=-fg
c l-2
4
2 3
-1
tan’ $0 ) V
measures the polarization of the virtual photon. In eq. (2) 8 is the scattering angle of the electron and qt = w* - q2, where (w, q) is the four-momentum transfer. All nuclear structure information is contained in the longitudinal and transverse response functions, R,_ and RT, defined by RL(W, 9) = WOO(w,4)) in terms of the components
RT(w, q) = W”(w
q)+ W**b,
cl)
(3)
of the hadronic tensor WC””
Here .P is the nuclear charge-current operator, which connects the antisymmet~zed initial state of the nucleus j&J, of energy E,, with a final A-nucleon state II+!+) of energy Ef. The sum runs over the scattering states corresponding to all of the allowed asymptotic configurations and includes possible discrete states. Degeneracy indices are suppressed for sake of simplicity. The diagonal elements of the hadronic tensor can be equivalently expressed in terms of the Green function G+, related to the nuclear hamiltonian H, by WLLcL(o,q) =iIm
(t,bo~JF+G~(EC)Jp(JIO)
(5)
1 E-H-i?’
(6)
where EC= E,+ o and G+(E) =
The limit for q + i-0 is understood in all the equations involving G+. It must be performed after the calculation of the matrix element and is considered in the sense of a limit of distributions of the variable E. The hadronic tensor in eq. (5) contains the full A-body propagator of the nuclear system. As such, it is an extremely complicate object, which defies a practical evaluation. Only an approximate treatment allows to reduce the problem to a tractable form. To this end, the nuclear response is written in terms of the opticalmodel Green function 18%9). Th e resulting expression is derived in the following,
684
under
F. Capuzzi et al. / Final-state
suitable
formalism
approximations,
the
Feshbach
projection
operator
*“).
Only the one-body definite,
using
interaction
the proton
part of the charge-current case is exemplified,
operator
is retained.
To be more
by setting
(7) Therefore,
eq. (5) becomes
The next approximation
consists
in neglecting
terms i # 1.There-
the nondiagonal
fore, eq. (8) takes the form
where, in the final step, the symmetry of G’ for exchange exploited. Now let us denote by C, the channel subspace spanned by set of vectors (r, n), corresponding to the proton 1 at the point (A - 1) nucleus in the discrete eigenstate In). Likewise, let us analogous
vectors
continuous particles
representing
spectrum.
the residual
Antisymmetrization
only. Therefore
the projection
orthogonal
and decompose
the orthonormalized r and to the residual indicate by Ir, E) the
in the eigenstate
is here performed
has been
1s) of the
on the residual-nucleus
operators P,=
p, = c (r, n)(r, nl dr, J are mutually
nucleus
of protons
(r, E)(r, El d& dr
(10)
J
the identity
in the last matrix
element
of
eq. (9) as l=cP,+Pc. n
(11)
between Then eq. (11) is inserted on both sides of G+ in eq. (9) and interference processes involving intermediate states in different channel subspaces is neglected. Therefore the nondiagonal terms P,,G’(E)Pm and P,,G+(E)P, are suppressed and in the matrix element of eq. (9) it results G+(E) -C
P,G+(E)P, n
+ P,G+(E)P,
.
(12)
E Capuzri et al. / Final-state
685
interaciion
Finally, one assumes that jy connects the initial state (JI,J only with the discrete channel subspaces C,,. Therefore P,G’(E)P, is dropped. Under the above approximations, W’” is decomposed as (13) where Gz = P,G+P, is simply related to the unsymmetrized Feshbach A-body hamiltonian H:(E), which is able to describe, under the incoming-wave boundary condition, elastic scattering of a nucleon by an (A - 1)-nucleus in the discrete state In). In fact, applying th e projection operator method, one has G:(E)
where E, is the residual-nucleus
=
PII
(14)
E-en-Hz(E)-iv’
energy and
H:(E)
= P,,TP, + V:(E)
(1%
includes the kinetic energy T of the proton 1 and the unsymmetrized A-body optical potential V,,(E) = P,VP, + P,VQ,
E-Q,,AQ,,+is
Feshbach
QnVPn7
where Q,, = 1 -P, and V describes the interaction between the proton 1 and the residual nucleus. The definition of P,, in eq. (10) can be exploited to reduce eq. (13) to the single-particle expression W’+(o,
q)=i;
A,, Im
I
4z(r)jr+(r)%z(Ef;
r,
r’)j@(r’)&(r’)
dr dr’,
(17)
where A, is the spectral strength 2’) of the hole state &,, which is the normalized overlap integral between It&) and In), j” is the one-body current and %,,(E; r, r’) is the integral kernel associated to the Green function of the single-particle Feshbach potential Sm(E) related to V,,(E). A similar result can be derived for neutrons. The above approximation scheme was extensively discussed in refs. ‘**‘).We shall only add a few remarks. The approximation of neglecting nondiagonal terms in eq. (8) seems adequate only for enough high values of momentum transfer. It is needed if we want to write the nuclear response in terms of single-particle matrix elements. The contributions arising from interference between different channels, which are suppressed in eq. (12), are due to correlations in the final state and vanish in an independent-particle model.
686
F. Capuzzi et al. / Final-state interaction
The approximation of neglecting P,G’P, in eq. (12) cannot be adopted for the contributions to P, due to resonances of the residual nucleus. However, in appendix A, the contribution of isolated resonances is extracted from PC through projection operators onto states characterized by the residual nucleus in a bound state. Since such projection operators can be treated on the same footing as P,, we state to remove their contribution from PC and to include them in the sum over n.
2.1. SPECTRAL
REPRESENTATION
The complexity of a practical caiculation of %z(E; r, 8) in eq. (17) can be avoided by means of its spectral representation, based on a bio~hogonal expansion in terms of eigenvectors of ‘Yf: and ‘Vn. In fact, the incoming-wave scattering solutions of the eigenvalue equations [E-T-V;(E,)]#(E,)=O,
(18)
[E-T-Y’-~(E,)]X”;‘(E,)=O,
(19)
satisfy the biorthogonahty
relation
,&‘*( Ef, r)igf(Ef,
r) dr = 6(E - E’) .
i
Disregarding the bound-state eigenfunctions, which in practice do not contribute to W”‘, we assume the completeness relation as &‘(Ef,
r)&)*(Ef,
r’) dE = S(r-
r’)
.
(21)
In general the completeness relation for a biorthogonal set of eigenfunctions is a very delicate point, but in our case it is supported by the fact that it was rigorously proved for complex local potentials *‘) and appears to be reliable also for nonlocal potentials. From eq. (21) the spectral representation of the single-panicle Green function is obtained as
and eq. (17) becomes +CC W’*“(w, q)=+z
where
Im
I0
1
E,-E,-E--iv
TY(E,
EC) dE,
(23)
687
F. Capuzzi et al. / Final-state inieraet~on
T:*(E,
Ef) = A,
J
,f&)*(Ef, r)~~(r~~~(r) dr
r)~~(~)#~(~) dr [Ix’k?&, I*a (24)
The Cauchy integral in eq. (23) can be calculated by exploiting the standard symbolic relation 1 =P ; x - in 0
-t-I’77S(x),
(25)
where P denotes the principal value. Then eq. (23) becomes W“‘(w,
q) =C
+* J
Re TZ”(Ef-
n +IP
‘IT
0
E,, E,)
1 Ef-e,-E
Im Tc’“(E, E,) dE
1 .
(26)
It is conceptually interesting to observe that eqs. (24) and (26), although containing eigenfunctions of a complex optical potential, are not affected by a spurious attenuation of the strength due to flux absorption. In fact, the flux lost in a particular A-body final state contributes to excite the other final states, which are all contained in the inclusive response. This is clearly seen in eq. (24), by considering the contribution of a particular channel n. The loss of flux in x’-’ is compensated by a corresponding gain in it-’ due to the flux lost, towards the channel n, by the A-body final states which are asymptotically originated from different channels. We want to stress here that in the Green function approach it is just the imaginary part of the optical potential which accounts for the redistribution of the strength among different channels. Before concluding this section, we want to add a remark about the implications of the Pauli principle. In the present approach the initial state I+,,)is antisymmetrized, but the projection operators of eq. (10) are unsymmetrized. Therefore, in eqs. (24) and (26) the Pauli principle effects are considered in the hole states &, which are overlap integrals between antisymmetrized wavefunctions, but not in the final states, as the optical potential W;, does not account for these effects. Actually, x’-’ is affected by an unphysical loss of flux, since V: accounts for scattering processes involving exchange of two protons as an absorption, but this effect is compensated by the contributions to 2(-i due to the unphysical exchange degeneration of the final states, which are implicitly included in G. Therefore, in the inclusive reaction this treatment of antisymmetrization is not incorrect. Nevertheless, eq. (24) is exposed to the criticism of both introducing and removing an unphysical effect. Furthermore, V;, has to be related to the phenomenological potentials, which include the Pauliprinciple effects. Therefore, from a theoretical point of view, it is worthwhile to modify the present treatment, in order to deal with an optical potential that includes antisymmetrization effects. A proper treatment of antisymmetrization is outlined in appendix B.
688 2.2.
E Capuzzi INTERFERENCE
A delicate
consider
DIFFERENT
point in the approach
effects between and P,,G+P,.
BETWEEN
et al. / Final-state
different
interaction
CHANNELS
of the previous
channels,
section
which are related
In order to estimate
is to neglect
to nondiagonal
the consequences
interference
terms P,,G’P,,,
of this approximation,
let us
the exact relation dG+( E) dE
which characterizes
the A-body
Green
= -G+*(E) function,
,
(27)
and then compare
the effect of the
approximation on both sides of eq. (27). Understanding that the Green functions are included into the appropriate matrix elements, the derivatives of eqs. (12) and (14) give dG+ dE where,
under
disregarded.
the same Likewise,
- V;‘)G;
G;(l
---C
,
(28)
n
approximation
as in the preceding in Gt2 the nondiagonal
by suppressing
G +*,C
section, P,G+P, terms, it results
G;‘.
is
(29)
n
The two approximate expressions in eqs. (28) and (29) do not satisfy eq. (27) and the resulting discrepancy indicates that the effects due to interference between different channels are directly related to the energy slope of V,(E). Phenomenological calculations 13) of the Feshbach potential, based on dispersion relations, show an appreciable slope in the quasi-elastic region of interest and an almost linear behaviour of the energy dependence. A simple way of improving the approximation is to replace G, in eq. (13) by an effective Green function G,,, still operating in C,, which embodies contribution of interference
with the other channels
(29). Under requirements
the assumption of an almost linear energy are fulfilled by the symmetric operator h;(E)
=Jl-
and restores
V;‘(E)
consistency
G;(E)&
between
dependence
eqs. (28) and
of V,,(E),
- V;‘(E),
these
(30)
since from eq. (28) it follows dG+( E) nzz
-&“(E)
.
(31)
dE We want to notice
that 6,
is really a Green 6;(E)
=
function,
pfl E-.c,,-Hz-iv’
since eq. (30) gives (32)
F. Capuzzi et al. / Final-state
689
interaction
where A:=[~-V~‘(E)]-“*[H~(E)-(E-E,)V~‘(E)][~-V:’(E)]-“* =P,,TP,+
c;
(33)
is energy independent, under the condition VE( E) = 0. Therefore we expect that replacement of the Feshbach hamiltonian H,,(E) by Gn is a simple and consistent prescription which improves the approximation of eq. (13). As a consequence, the single-particle eigenfunctions ,&)( E,) [ ig’( E,)] of Yi( E,) [ ‘Y,,(E,)] are replaced by the corresponding eigenfunctions ig) [,&)I of the single-particle potential e(FO), related to i/i( en). Therefore, eqs. (24) and (26) are replaced by Tsp(E)
= A,
dr dr 1 [5i~‘*(r)Y‘(rM,(r) J,i%‘*~rW~~h(r) 1 *,
+7=
Re T:“(E,-q,)+LP
?T I ,,
1
E,--E,-E
Im T;*(E)
dE
(34)
.
(35)
From eq. (33) the wavefunctions i’,-’ and ,&) are connected with the eigenfunctions of the Feshbach potential by the phase conserving relation ~~‘(r)=Jl-~:‘(E+&,)x~‘(E+&,,r), $‘,-‘(~)=J~-YL(E+E,)~~)(E+~,,,~).
(36)
We have shown that the effective Green function 6, of eq. (30) cures the inconsistency due to the energy dependence of the Feshbach potential and takes into account contribution of interference between different channels, under the assumption that V,(E) and, therefore, Yn,(E) is linearly dependent on the energy. Although the Feshbach potential is not well known in practice, the above assumption is supported by phenomenological analyses 23), as we have already observed. However, as the Feshbach potential is not exactly a linear function of E, 9, maintains a residual energy dependence. There are no available treatments which take in due account the effects of a deviation from linearity in 7fn(E). These effects are expected to modify 6, and therefore the related potential $“‘,,but cannot change the funda* mental requisite that v,, must be energy independent. Thus the eigenfunctions ig’ and x”(;‘, in eq. (34), must be preferably determined on the basis of such a requisite. Unfortunately, the existing phenomenological analyses ‘“) based on an energy independent optical-model potential are not suitable for calculations on specific nuclei. In sect. 3 we will follow a method, based on the requisite of energy independence, which allows to use more adequate phenomenological potentials. 2.3. SUM
RULE
Energy sum rules represent a useful test to check the approximations introduced in the previous sections. Let us consider, e.g., the integrated longitudinal response
690
F. Capuzzi et al. / Final-state
interaction
function +‘X E(9)
&(w,
=
9) dw =
_-oci l(~fIJ”(q)lrLo)1*6(EoI f
&+a)
do,
(37)
where, according to eq. (7), Jo(q) is the one-body charge operator for protons. As a consequence of the completeness of the set of the final states, the Coulomb sum rule results
E(4) = z(~ol_c+mo)
(38)
2
where, as in eq. (9), the terms involving charge operators been disregarded. In the Green function formalism, Z(q) is written as
of different
particles
have
(39) where the understood theory, i.e. P(q)zZ
One recovers
limit
for 7 + +0 is considered
lim ?T R-+cc
in the sense
lim Im
(40)
,,++O
eq. (38) from eq. (40) by using the general
1
expansion
1
dE,
E-A-iv-E-A+i7 which holds
of distribution
for any energy-independent
self-adjoint
operator
theorem
25)
(41)
A.
Let us now consider the effect on the sum rule of the approximate treatments given in the previous sections, Eq. (13), where interference terms and continuous channel
contribution
are neglected, E(4)
is integrated
by means
of eq. (41) and gives
n(rLoli?+~njNo) =Z(~oIX+j~I~o), = Z C
(42)
which shows that the sum rule is not affected by neglecting interference terms. In sect. 2.2 the contribution of interference between different channels is accounted for by expressing G+ as a sum of effective Green functions 6:. A criterion to check consistency of this treatment requires that the result of eq. (42) is unchanged. The integrated longitudinal response is expressed in terms of the effective Green functions as +R E(q)
--zC
r n
lim R-tot
lim Im r)++O
OR (+olj?‘&(E)j?lcCro)
dE.
(43)
F. Capurzi et al. / Final-state i~te?a~tion
691
However, 6: is related by eq. (32) to the non-hermitian hamiltonian &, and eq. (41) cannot be applied. Therefore, a different approach, based on the analyticity properties of &, must be used to verify the Coulomb sum rule. Let us consider the second resolvent equation &(E)=GO,+(E)+GO,f(E)$;(E)&;(E),
where G?(E)
(44)
is the free Green function G:+(E)
=
P” E-E,-T-iq’
(45)
By substituting eq, (44) in eq. (43) one obtains
[I (~olj~‘G0,‘(E)O:(E:)8:( dE I, JSR 4-R
lim X;(q)--zC Ii- n R++m +
lim Im
_R (&Ij?*GOn’(E)j$M
dE
T,-.+O
(46)
where, in the last term, the contour of integration is extended to the semicircle SR, of radius R, lying in the lower complex half-plane. We first consider the case of Gr, given by eq. (33), without disregarding its weak energy dependence. From eqs. (30) and (33) we can observe that the integrand in the last term of eq. (46) is an analytic function without poles inside SR and gives no contributions. The first integral in eq. (46) contains the Green function of an energy-independent self-adjoint operator, and can be evaluated by means of eq. (41), leading to eq. (42). The same result holds if 6, is chosen to be the Green function related to a complex potential that is rigorously energy independent, provided that no complex eigenvalues exist.
3. Results In this section, numerical results for the response functions are presented and discussed. The theoretical approach outlined in sect. 2.1 allows to calculate the components of the nuclear response in a straightforward and relatively simple way. In fact, as a result of the spectral representation of the optical-model Green function in terms of the biorthogonal set of eigenfunctions defined in eqs. (18) and (19), the ingredients of the nuclear response are the integrals in eq. (24), which are of the same kind as those giving the transition amplitudes of the electron-induced nucleon knockout reaction in the distorted-wave impulse approximation (DWIA) framework of ref. 26). The complexity of a practical calculation of the optical-model Green function is thus avoided in the present approach and the evaluation of the inclusive response is reduced to a problem of the same difficulty as the calculation of single-nucleon knockout contributions.
E Capuzzi et al. / Final-state
692
interaction
The integrals in eq. (24) have been therefore calculated adopting the same treatment as in ref. 26), which was successfully applied to describe exclusive (e, e’p) data. This implies the use of a non-relativistic approach, where the one-body nuclear current operator is given by the non-relativistic approximation of McVoy and Van Hove *‘). Both initial and final state wavefunctions are in principle eigenfunctions of the Feshbach optical potential. However, as a practical expression of the Feshbach potential is unavailable, the single-particle bound state wavefunction (P,, is taken from a phenomenological Woods-Saxon potential and the distorted wavefunctions X (-I and i’ .-’ are obtained through a local equivalent spin-dependent optical potential, given by a fit to elastic proton-nucleus scattering data. An improvement of the Green function approach is represented by the prescription proposed in sect. 2.2. Replacement of the eigenfunctions x(-’ by the corresponding eigenfunctions f’-’ of the energy independent potential q’:, allows to take into account in a simple way the contribution of interference between different channels and also provides a theoretical justification for using a phenomenological optical potential. In fact, the eigenfunctions of the energy-independent potential e are connected with the eigenfunctions xl;:, belonging to the eigenvalue E, of the local phenomenologi~al optical potential Y:(E) by the “de Forest factor” [ref. ““)I f;)(r)
=Jl-
V:'(E)X\TA(r)
(47)
,
where terms involving spatial derivatives of Yz, leading to small corrections, have been neglected. Likewise, denoting by 2 L-L , the corresponding eigenfunctions of ‘V,(E),
one has /$‘(r)
= m/$:$(r)
.
(48)
In eqs. (47) and (48), following the usual convention for phenomenological potentials, the residual-nucleus energy has been subtracted in the energy dependence of “u;(E). Furthermore, the same local optical potential, obtained from elastic scattering and, therefore, By means
related
to the ground
state, is assumed
for all the channels.
of eqs. (47) and (48), eq. (34) becomes
(49) where the de Forest factors take into account interference contributions. The use of a phenomenological optical potential is now justified on the basis of eqs. (47) and (48) and not only by our poor practical knowledge of the Feshbach potential. In the calculations residual-nucleus states In) are restricted to be one-hole states in the target and the sum over n in eq. (35) includes both discrete states and isolated resonances embedded in the continuous spectrum of the (A- 1)-body system. In fact, as it is shown in appendix A, they can be treated in a similar way.
F. Capuzzi et al. / Final-state
A pure shell model spectral
strength
and neutron
is assumed
for the nuclear
inreraction
structure,
for each n, and the sum over n is limited
states. Even if such an assumption
the details of the inclusive
response,
693
i.e. we take a unitary to the occupied
may not be fully adequate
it allows to perform
enough
proton
to describe
simple calculations,
on a conceptually clear basis and seems justified as a first approximation. In the following, results will be presented for “C and 4oCa. Having in mind the limitations and the ingredients of the present calculations, suitable values of the momentum transfer have been considered, i.e. enough large to keep the quasi-free scattering condition far from giant resonances and collective excited states, in order to make particle-hole rescattering less important reduce relativistic effects (q = 400-550 MeV/ c).
3.1. RESPONSE
FUNCTIONS
‘), but not too large, in order to
FOR “C
The longitudinal and transverse response functions for ‘*C, calculated without the de Forest factor, at q = 400 MeV/ c, are shown in fig. 1 and compared with the Saclay data 3).The contribution resulting from all the integrated single-nucleon knockout channels is also drawn. The difference between the two curves gives an
idea of the relevance of multi-nucleon removal. For the longitudinal response, the result obtained by retaining only the real part of the optical potential is also presented in the figure; in this case integration of single nucleon knockout channels gives the same result as the inclusive (e, e’) calculation. The difference among the three curves stresses the role of the imaginary part of the optical potential and clearly shows that the present approach is significantly different from a calculation with a real optical potential. In fact, a positive contribution is added to the single nucleon knockout, but different and smaller than the one resulting by neglecting the imaginary part of the optical
potential.
The underestimate of the transverse response is a systematic result of our calculations. It may be attributed to physical mechanisms, not included in the considered approach, e.g. meson-exchange currents together with 2p2h excitations. Therefore, this result is not surprising. However, we are not particularly interested in obtaining a fit to experimental data, our main aim being to understand the role and the relevance
of final state interaction
in quasi-free
electron
scattering.
As those mechan-
isms are expected to modify mainly the transverse response, in the following we will focus our attention on the longitudinal response. In fig. la the longitudinal response is overestimated. This result is not surprising, as it is comparable with the one given in fig. la of ref. 6), which was calculated in the approach of ref. 18). When the contribution of interference between different channels is taken into account through the de Forest factor, an overall reduction of both responses is obtained. A reduction is also obtained when a Perey factor 3’) is included in all of the analyses where a local equivalent potential replaces the nonlocal one. However,
694
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40
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120
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200
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40
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fd
(MeV)
0.005 0
Fig, 1. longitudinal (a) and transverse jb) response functions for “C at q =400 MeVJc a5 a functian of energy transfer. The data are from ref. 3). Hole states are taken from ref. Z9f and optical potential from ref. 30]. Solid line7gives the inclusive response without the de Forest factor and dashed line contains contributions of integrated singfe-nucleon knockout onty. Dot-dashed line in [af gives the result of a calculation where the imaginary part of the optical potential has been neglected.
both the nature and the physical meaning of Percy and de Forest factors are different. The de Forest factor removes the energy dependence of the phenomenologjca~ local potential, thus taking into account the interference between different channe‘ts, The Perey factor recovers the wavefunctions of the Feshbach potential from the corresponding phenomenological one. Thus it only removes the spurious energy dependence due to locality, without changing the dynamical one. Therefore, it cannot be expressed in terms of the tocal potential, but requires a model of the original nonloeality. The effect of the de Forest factor on the longitudinal response is shown in fig. 2 for two different choices of bound-state wavef~nction~ and optical potentials. A
695
0
40
60
120
160
200
240
260
w (MeV)
120
160
200
240
260
w (MeV) Fig. 2. Longitudinal response function for “C at q =400 MeV/c as a function of energy transfer. The data are from ref. ‘). Solid line gives the inclusive response with the de Forest factor, dashed line is without the de Forest factor. In (a) hole states and optical potential as in fig. 1, in (b) they are taken from ref. 32).
significant
reduction
is obtained,
whereas
the shape
is only slightly
changed.
The
de Forest factor produces a larger reduction in fig. 2b, where hole states and optical potentials are taken from ref. 32). The experimental data are better reproduced in fig. 2a, where the parameters of refs. 29730),which also give the best description of quasi-free (e, e’p) data 33), are adopted. Low values of energy transfer are not drawn in figs. 1 and 2a, as the optical potential of ref. 30) cannot be extrapolated at low energies. In general, available phenomenological optical potentials are given in a restricted range of energy values, and cannot be extrapolated far beyond this range, which imposes a restriction to the kinematical conditions that can be considered.
696
E Capuzzi
For the same reason eq. (35) is practically
a precise
impossible,
tions of the continuous of response
functions
et al. / Final-state
and reliable
calculation
of the dispersive
as it requires
integration
over all of the eigenfunc-
spectrum presented
interaction
of the optical in this paper
first term in eq. (35). The uncertainty relevant for our present purposes.
potential. contain
Therefore,
term in
all the results
only the contribution
of the
due to this approximation is neither large nor In fact, for all of the considered situations,
Im T:@(E) < Re TY(E), and a rough calculation of the neglected term, obtained by extrapolating the value of the function in the dispersive integral beyond the allowed energy range, gives only a small contribution is shown in fig. 3, for the bound state and the optical choice of phenomenological but does not significantly
ingredients affect the
to the final result. An example potential of ref. 32). A different
may change the results conclusion about the
presented in fig. 3, small size of the
contribution. The longitudinal response at q = 500 MeV/c is shown in fig. 4. The result is similar to the one obtained at q = 400 MeV/c and the fair agreement with the data is confirmed for the curve with the de Forest factor and the optical potential of ref. 30). A similar trend as in fig. 2 is also found in the dependence on the phenomenological ingredients.
3.2. RESPONSE
FUNCTIONS
FOR
?Ia
A similar analysis as the one presented for “C has been performed for 40Ca. In fig. 5 the longitudinal response function calculated without the de Forest factor for hole states
and optical
-0.005
potential
from ref. ‘*) is given and compared
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 40 00 120 160 200 240
with the
280
w (MeV) Fig. 3. Contribution to the longitudinal line), for 12C at 9 = 400 MeV/ c, obtained in dashed line. Hole states and optical
response function from the dispersive integral of eq. (35) (solid by extrapolating the quantity Im I,,~,‘(E), which is also shown potential as in fig. 2b. The de Forest factor is included in the calculations.
F. Capuzzi et al. / Final-state
0.008
-
0.004
7
0
40
80
120
160
interaction
200
697
240
280 w
320
(MeV)
Fig. 4. Longitudinal response function for ‘*C at 9 = 500 MeV/c as a function data are from ref. ‘). The inclusive response with the de Forest factor is shown and optical potential as in fig. 2a (solid line) and 2b (dashed
of energy transfer. The for the same hole states line).
Saclay data “) at q = 410 MeV/ c. The figure shows that experimental data are overestimated by the considered approach. As in “C, the disagreement is comparable with the one obtained in fig. lb of ref. ‘j), where a similar theoretical approach, with different phenomenological ingredients, is used. The effect of the de Forest factor and the dependence on the choice of the optical potential is shown in fig. 6. The reduction given by the de Forest factor is relevant, but unable, in this case, to resolve the disagreement with data. The dependence on the optical potential is significant, but not really crucial. The potential of ref. 34) seems preferable, however it is still inadequate to produce a good fit to the data. c
iJ
S ‘= [r
0.07
L
0.06 0.05 0.04
0
~,...‘.*..‘,...‘,,,.‘,,.,I,.
0
40
60
120
160
200
240
w (MeV) Fig. 5. Longitudinal response function for 40Ca at 9 = 410 MeV/c. The data are from ref. 4). Hole states and optical potential are taken from ref. “). Solid and dashed line as in fig. 1.
698
F. Capuzzi et al. / Final-state ;
7
interaction
0.07 0.06
r”
0
0
40
80
120
160
200
240
0
(MeV)
Fig. 6. Longitudinal response function for 4”Ca at 9 = 410 MeV/c. The inclusive response with (solid line) and without (dot-dashed line) the de Forest factor is shown for hole states and optical potential from ref. s2). Dashed line gives the inclusive response with the de Forest factor for hole states taken from ref. 29) and optical potential from ref. 34).
The longitudinal response function at q = 550 MeV/c, calculated with the optical potential of ref. 34) with and without the de Forest factor, is given in fig. 7 and compared with the Saclay data “). The effect of the de Forest factor is smaller than the one resulting when the optical potential of ref. 32) is adopted. Comparison with data has a similar trend as in fig. 6. Therefore in the present approach longitudinal response are not so well described as “C data. However,
0
40
80
120
160
200
240
280
4oCa data of the resulting
320
w (MeV) Fig. 7. Longitudinal response function for ?a at q = 550 MeV/c as a function of the energy transfer. The data are from ref. 4). Hole states from ref. 29) and optical potential from ref. s4). Solid line gives the inclusive response with the de Forest factor, dashed line is without the de Forest factor.
I? Capurri
disagreement mations. nuclear
et al. / Final-state
interaction
is not too large and may be explained
A better result could presumably state a more sophisticated
under
be obtained,
description
699
the considered
approxi-
e.g., by using for the initial
than a pure shell model.
4. Summary and conclusions The electromagnetic response functions for the inclusive (e, e’) scattering in the quasi-elastic region have been calculated in a many-body approach, where the Green function is expanded over the complete biorthogonal system of the eigenfunctions of the Feshbach optical potential and of its hermitian conjugate. complication of an explicit determination of the Green function nuclear
responses
are expressed
in terms
of matrix
elements
In such a way, the is avoided and the similar
to the ones
appearing in the exclusive (e, e’p) cross section. Therefore, the calculation of the inclusive response involves no more difficulties than the calculation of single-nucleon knockout. All the final states contributing to the inclusive reaction, and not only one-nucleon emission, are contained in the Green function. However, the final state interaction is treated consistently with the exclusive reaction, where it plays a crucial fact, both the real and the imaginary parts are used in the optical potential, results are not affected by a spurious absorption. The effect of the interference between de Forest factor, which also provides a phenomenological optical potential,
attenuation different
of the strength channels
is included
role. In but the
due to flux through
the
a justification for using the eigenfunctions of instead of the ones of the Feshbach potential.
Results have been presented, in comparison with experimental data, for “C and 40Ca at different values of momentum transfer, between 400 and 550 MeV/c. The effect
of final
state
interaction
is large
and
important.
Including
in the
calculation the contribution of all the final states is essential, as the inclusive response functions are significantly different from the ones obtained by integrating all singlenucleon knockout channels. The imaginary part of the optical potential originates such a difference. Therefore, it plays a crucial role and cannot be neglected. It decreases the height of the peak both in the longitudinal and in the transverse response and is responsible for the redistribution of the total flux among various channels. The de Forest factor gives an overall reduction, which significantly improves comparison with the data of longitudinal response. In ‘*C the agreement is satisfactory; the lack of strength at low energy transfers could be, at least partly, explained by particle rescattering, which otherwise is expected to be small at the considered momentum transfers. Final state interaction has a similar effect on the two components of the nuclear response and cannot reproduce their relative strength. However, the systematic underestimate of the transverse response points to non-nucleonic degrees of freedom
700
F. Capuzzi et al. / Final-state
and correlation in the inclusive The
present
quasi-elastic physics
effects, which should transverse response. results
region
seem
give an enhancement
to indicate
can be reasonably
and to confirm
interaction
that
inelastic
explained
= [ Y(r)[
evidence
spectrum
In this appendix we extract from eq. (A.l) the contribution Let us consider the spectroscopic strength, defined by
where
the
spectrum
dr.
(A.1)
of isolated
resonances.
I(r,sl&Jl’dr.
A(E)
of the residual
(A.2)
(r, The positions
by means
El&J=
A(E) can be expressed
r(E) =-I_ 2?7 [E--Eo+F(E)]2+[fr(E)]2r
where F(E) and @(E) are the average part of the hole self-energy operator. +(a, r) can be defined
nucleus,
as *‘) (A.3)
values of the hermitian and anti-hermitian The normalized spectroscopic amplitudes
of eq. (A.3) as
r(F) dZZ
of resonances
(A - 1)
is given by
I In the continuous
in the nuclear
of exotic mechanisms.
due to channels
1 k, a)(r, a/&) de]
h(s)=2
scattering
A
The contribution to the response function system is in an eigenstate 1s) of the continuous pC_CkJ
electron
be neglected
in terms of conventional
that there is no compelling
Appendix
and cannot
e-E,+F:.c)-fir(E)
4(E’r)’
(A.4)
are given by the roots E, of the equation E,-Eo+F(~,)=O.
(A.5)
For each root E,, we choose, as suggested by phenomenological analyses, an energy interval I,, where the spectroscopic strength A(E) has appreciable values and the functions F(E), T(E) and 4(~, r) have approximately constant values, i.e. F(E) = F(G),
r(E)=r,,
HE, r)=&(r),
VEEI,.
(‘4.6)
We also assume that energy intervals I,, corresponding to different roots, are disjoint. By use of eqs. (A.4), (A.5) and (A.6), we can thus extract from eq. (A.1) the contribution of isolated resonances as (A.7)
F. Capuzzi
et al. / Final-state
interaction
701
where the residual nucleus states Ia), corresponding to each resonance LY,can be expressed in terms of the eigenfunctions of the continuous spectrum in the corresponding energy interval I, as (E -
E,
-$I-,)-‘I&) ds.
L
(A.@
We notice that (aIa’)=O
if a#a’,
(r+Y)=O.
(A-9)
Therefore the projection operators (A.lO) are mutually orthogonal and orthogonal to P,. The overlap integral between I+J and Ir, a) is given by means of eqs. (A.8), (A.4) and (A.6) as (r,
alW=-&bla)&(r).
(A.ll)
After substituting eq. (A.1 1) in eq. (A.7) we obtain
(A.12) which shows that isolated resonances can be extracted from the continuous spectrum and treated in a similar way as discrete eigenstates. Therefore, their contribution can be included in the framework of sect. 2. Appendix B
In this appendix, the inclusive cross section is written in terms of final state wavefunctions which are eigenfunctions of an optical potential that includes antisymmetrization effects. To this end, and without modifying the involved approximations, some changes are needed in the formal treatment of sect. 2. In order to deal with antisymmetrized quantities as far as possible, we do not assume, for the moment, the approximation of eq. (9). By exploiting the symmetry of G+ for exchange of protons, eq. (5) becomes
Wpp(m,q) =f h (hljYfG+(E~)JF Itid.
(B.1)
Inserting eq. (11) in eq. (B.l) and then neglecting PC, one has W“w((w,q) =f;
Im
I
(+oI_C%, n)(r, nl G+(Er)J*“I&J dr.
(B.2)
702
F. Capuzzi et al. / Final-state
Notice strongly
that
it would
connects
If we want function
be incorrect
to neglect
II,&) with the continuous
to write
which
the nuclear
includes
interaction
PC directly
in eq. (5), since
Jp
channels.
response
the full implications
in terms
of a single-particle
of the Pauli
principle,
Green
G’ must be
kept into matrix elements involving antisymmetrized wavefunctions. To this aim, we replace the matrix elements involving G+ in eq. (B.2), where only one wavefunction is antisymmetrized, by the equivalent ones where the two wavefunctions are antisymmetrized. Using the more convenient formalism of the second quantization, we have
where
a, destroys
a proton
at the point
Wp”l”(w, 4) =$E In sect. 2 the diagonality
n
Im
I
r. Therefore,
n)(n(a,G+(Ef)J~“I~o)dr.
(&IjP’lr,
approximation
eq. (B.2) becomes
of eq. (12) consists
P,,G+(&)Q,
(B.4)
in assuming
==O.
(B.5)
An analogous approximation can be applied to eq. (B.4) by introducing the operators Pt and Qt, which project onto the subspace spanned by the vectors aTIn) and onto the complementary orthogonal one. These operators were first introduced by Feshbach 35) to express the optical potential in the antisymmetrized case. In the second quantization formalism they read as Pf =
a:ln)K(r,
r’)(nla,, dr dr’,
Q$=l-P;,
(B.6)
with K(r, r’)=fY(r-r’)+C where u, and rty are the normalized tion numbers,
defined
Y
natural
ny u,(r) 4Xr’), 1 -nv orbitals
(B.7)
and the corresponding
by
I
u,(r)(nla:u,,In)u~,(r’)
dr dr’=
03.8)
nuSvu,.
The sum in eq. (B.7) is over those values of Y corresponding The diagonality approximation is now formulated as
to n, # 1.
P:G+(E,)Q;=O. Therefore
occupa-
(B-9)
eq. (B.4) becomes WFF(w, q) =$Z;
”
Im
(&Ij~+[r, n)(nlu,P~G+(Ef)P;;lJ’“I~o)dr.
(B.lO)
F. Capuzzi et al. / Final-state
703
interaction
By means of eq. (B.6) and the relation (nlcY
I&) =Jz(r,
nlJ%kJ,
(B.ll)
eq. (B.lO) results
xh, nP‘l+ddrIdr2h,
(B.12)
%?(E;
(B.13)
where rl, 4 = (nla,,G+(W$d
is the one-body optical-model Green function, which includes antisymmetrization effects. However, eq. (B.12) is not yet reduced to a one-body expression, since we have not exploited the approximation of neglecting terms involving charge-current operators of different protons. This can be performed now by setting in eq. (B.12) (r, nlJ’
Ih>= (6 4.Kl+J,
(B.14)
as, for enough high values of momentum transfer, jp does not contribute to the overlap integral if i # 1. When eq. (B.14) is inserted in eq. (B.12), one has
xi”(rdcbn(r3)dr, dr2 dr3.
(B.15)
Spectral decomposition of eq. (B.15) is then obtained, along the same lines as in sect. 2.1, by means of the AGS-based optical potential “Irt+, which is related to the one-body Green function 3;’ [refs. 36V3779)]. Therefore eq. (26) is recovered, with
X
[I
x!&)*(J%, r)_?‘(r)&(r) dr
* 1 ,
(B.16)
where x’-’ and f(-’ here denote the eigenfunctions of Y;;‘+ and V;; and therefore contain the full implications of the Pauli principle. Also the hole wavefunctions #, are eigenfunctions of the same potential Vf, evaluated at the energy E,,. The operator IS is necessary in eq. (B.16) to preserve the sum rule. In fact, the proof given in sect. 2.3, which was based on integration over a complex path, can be again applied here, since the AGS-based potential Vt is hermitian analytic. However, at high energies, ‘V;;’includes a term which is linearly dependent on the energy. This produces a contribution from the contour at infinity and hence a violation of the sum rule, which is exactly compensated by the operator K.
E Capuzzi et al. / Final-stare inreraction
704
The approximate be developed
treatment
of interference
as in sect. 2.2 without
further
between
different
channels
can then
comments.
References 1) E.J. Moniz et al, Phys. Rev. Lett. 26 (1971) 455 2) R. Altemus er al, Phys. Rev. Lett. 44 (1980) 965; M. Deady et al., Phys. Rev. C28 (1983) 631; C33 (1986) 1897; R. Hotta et al.. Phys. Rev. C30 (1984) 87; C. Marchand et al., Phys. Lett. 8153 (1985) 29; C.C. Blatchley et al., Phys. Rev. C34 (1986) 1243; S.A. Dytman et nl., Phys. Rev. C38 (1988) 800 3) P. Barreau et al., Nucl. Phys. A402 (1983) 515 4) Z.E. Meziani et al., Phys. Rev. Lett. 52 (1984) 2130; 54 (1985) 1233 5) W.M. Alberico, M. Ericson and A. Molinari, Nucl. Phys. A379 (1982) 429; Ann. of Phys. 154 (1984) 356; W.M. Alberico et a/., Phys. Lett. B233 (1989) 37 6) A. Dellafiore, F. Lenz and F.A. Brieva, Phys. Rev. C31 (1985) 1088 7) M. Cavinato er a/., Nucl. Phys. A423 (1984) 376; M. Cavinato, M. Marangoni and A.M. Saruis, Phys. Lett. B235 (1990) 15; W.M. Alberico er al., Phys. Rev. C34 (1986) 977; Nucl. Phys. A462 (1987) 269; F.A. Brieva and A. Dellafiore, Phys. Rev. C36 (1987) 899; S. Drozdz et al., Phys. Lett. B185 (1987) 287; G. Co’ et al., Nucl. Phys. A485 (1988) 61; U. Stroth, R.W. Hasse and P. Schuck, Nucl. Phys. A462 (1987) 45; J. Jaenicke, P. Schuck and R.W. Hasse, Phys. Lett. 8214 (1988) 1 8) G. Do Dang and M. Van Giai, Phys. Rev. C30 (1984) 731; S. Nishizaki er a/., Nucl. Phys. A485 (1988) 515; C.J. Horowitz and J. Pikarewicz, Phys. Rev. Lett. 62 (1989) 391; R. Brockmann et al., Z. Phys. A332 (1989) 51 9) C.R. Chinn, A. Picklesimer and J.W. Van Orden, Phys. Rev. C40 (1989) 790 10) H. Kurasawa and T. Suzuki, Nucl. Phys. A454 (1986) 527; K. Wehrberger and F. Beck, Phys. Rev. C35 (1987) 299; C37 (1988) 1148 11) S. Fantoni and V.R. Pandharipande, Nucl. Phys. A473 (1987) 234; A. Fabrocini and S. Fantoni, Nucl. Phys. AS03 (1989) 375 12) M. Ericson, W. Leidemann and G. Orlandini, Z. Phys. A331 (1988) 369; W. Leidemann and G. Orlandini, Nucl. Phys. AS06 (1990) 447 13) R. Rosenfelder, Nucl. Phys. A459 (1986) 452 14) J.V. Noble, Phys. Rev. Lett. 46 (1981) 412; L.S. Celenza et al., Phys. Rev. C31 (1985) 946; P.J. Mulders, Nucl. Phys. A459 (1986) 525 15) M. Ericson and M. Rosa-Clot, 2. Phys. A324 (1986) 373 16) R. Cenni, C. Ciofi degli Atti and G. Salmi, Phys. Rev. C39 (1989) 1485 17) G. Orlandini and M. Traini, Phys. Rev. C31 (1985) 280; C.R. Chinn, A. Picklesimer and J.W. Van Orden, Phys. Rev. C40 (1989) 1159 18) Y. Horikawa, F. Lenz and N.C. Mukhopadhyay, Phys. Rev. C22 (1980) 1680 19) F. Capuzzi, C. Giusti and F.D. Pacati, in Perspectives on theoretical nuclear physics, ed. L. Bracci et al. (ETS, Pisa, 1990) p. 156 20) H. Feshbach, Ann. of Phys. 5 (1958) 357 21) S. Boffi and F. Capuzzi, Nucl. Phys. A351 (1981) 219 22) M. Bertero and G. Dillon, Nuovo Cim. 2A (1971) 1024 23) H. Fiedeldey and C.A. Engelbrecht, Nucl. Phys. A128 (1969) 673 24) C.A. Engelbrecht and H. Fiedeldey, Ann. of Phys. 42 (1967) 262
I? Capuzzi et al. / Final-state 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37)
interaction
K. Yosida, Functional analysis, 2nd ed., (Springer, Berlin, 1968) p. 324 S. Boffi, C. Giusti and F.D. Pacati, Nucl. Phys. A336 (1980) 437; A386 (1982) 25 K.W. McVoy and L. Van Hove, Phys. Rev. 125 (1962) 1034 T. de Forest Jr., Nucl. Phys. Al63 (1971) 237 L.R.B. Elton and A. Swift, Nucl. Phys. A94 (1967) 52 D.F. Jackson and I. Abdul-Jalil, J. of Phys. G6 (1980) 481 F.G. Perey, in Direct reactions and nuclear reaction mechanism, ed. E. Clementel (Gordon and Breach, New York, 1963) 125 M.M. Giannini and G. Ricco, Ann. of Phys. 102 (1976) 458 M. Bernheim et al, Nucl. Phys. A375 (1982) 381 P. Schwandt et a!., Phys. Rev. C26 (1982) 55 H. Feshbach, Ann. of Phys. 19 (1962) 287 R. Goldflam and K.L. Kowalski, Phys. Rev. C22 (1980) 949 K.L. Kowalski and A. Picklesimer, Nucl. Phys. A369 (1981) 336
105
and C. Villi,
A524 (1991) 681-705
FINAL-STATE
INTERACTION
F. CAPUZZI,
IN ELECTROMAGNETIC FUNCTIONS
C. GIUSTI
RESPONSE
and F.D. PACATI
Dipartimento
di Fisica Nucleare e Teorica dell’ Universitci, Pavia and Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Italy
Received
2 April 1990
Abstract: The longitudinal
and transverse response functions of the inclusive quasi-elastic (e, e’) scattering are calculated in a many-body approach where the Green function is expanded over the eigenfunctions of a non-hermitian optical potential. Direct and interference contributions of all the channels are included in a simple way and final-state interaction is treated consistently with the exclusive (e, e’p) reaction. The response functions of ‘*C and 40Ca are evaluated for momentum transfers between 400 and 550 MeV/c.
1. Introduction The inclusive
electron
scattering
in the quasi-elastic
region
is dominated
direct single-nucleon emission, which suggests the one-body mechanism interpretation. In fact, the cross section seemed fairly well reproduced Fermi-gas model ‘): the shift of the maximum from the free-scattering associated
with the average
binding
energy
of the nucleons,
by the
as a natural by a simple value was
and the width of the
peak with the Fermi motion of the particles. However, this simple model is not able to simultaneously describe quasi-elastic longitudinal and transverse responses, which, in recent years, have been experimentally separated on a variety of nuclei, over a large range of momentum transfers 2W4).Individual responses display peculiar and interesting features, whose interpretation requires a more sophisticated treatment. Some properties can be explained by conventional nuclear models, but others defy an accurate description. Therefore, the theoretical analysis of the nuclear responses aroused a great interest, providing a severe test both of the nuclear structure description and of the reaction mechanism. The transverse response shows a pronounced maximum, whose position and size seem reasonably described already by the Fermi-gas model. Furthermore, a massnumber scaling is observed, which points to nuclear matter properties. Indeed, theoretical descriptions based on infinite nuclear matter can reasonably account for the data, when also subnuclear degrees of freedom, such as meson-exchange currents and isobar excitation, are included 5). 0375-9474/91/%03.50 @
1991 - Elsevier
Science
Publishers
B.V. (North-Holland)
682
F. Capuzzi et al. / Final-state
The longitudinal shape
response
with respect
overestimated
is more difficult
to the prediction
by single-particle
interaction
to understand.
of the Fermi-gas
models
and a sizeable
It displays
a change
of
model.
Its size is largely
violation
of the Coulomb
sum rule results. Many different
approaches
have been attempted
in order to solve these problems,
with partial success, but a completely consistent and satisfactory the data does not yet exist. A number of calculations have been carried out, in order to particle-hole correlations, in the context of Tamm-Dancoff “) approximation (RPA) [ref. ‘)I. These calculations are successful
description
of all
take into account or random-phase in improving the
description of the charge response at low momentum transfers, but at momenta higher than 400 MeV/c a small effect is obtained. Relativistic corrections in the frame of Dirac phenomenology were evaluated in the mean field theory 8.9) and including RPA correlations lo). They momentum transfers. The longitudinal
give a sizeable suppression, response of nuclear matter
even at high was calculated
in a variational approach with a correlated basis, obtaining a fair agreement with the data, except at high values of both momentum and energy transfer I’). The effects of neutron-proton tensor correlations were also studied “). The longitudinal structure function in ‘*C was fitted in the frame of a Glauber-like multiple-scattering theory with not completely understood parameters 13). Medium modification of nucleon properties was advocated as a possible source of explanation 14); a connection of this effect with 2p2h correlations was presented in ref. 15). Scaling properties of the electromagnetic responses were used in order to analyse the relevance of different approximations ‘6). Finally, the sum rule saturation was also considered I’). The role of final-state interaction is a crucial point in the analysis of inclusive electron scattering. Its effect is essential to fit the exclusive (e, e’p) cross section, but the current absorption produced by the imaginary part of the optical potential seems inconsistent with an inclusive measurement. A method in which these two aspects are reconciled was presented in the frame of the many-body Green function theory
I’). However,
in ref. “) the determination
task, as the actual calculation is performed coordinate space, and a very large number
of the Green
function
is a difficult
in an angular momentum basis and in of partial waves is needed.
Here, we propose a direct and simple method for taking into account the effect of a non-hermitian optical potential in the many-body Green function approach j9). The final-state interaction is treated consistently with the exclusive (e, e’p) reaction, whose contribution to the inclusive scattering can be easily separated. The formalism concerning the spectral representation of the optical-model Green function is developed in sect. 2. The effect of the interference between different channels is discussed and an approximate way to take into account its contribution is proposed. The preservation of the Coulomb sum rule is also demonstrated. In sect. 3 the results of calculations carried out for “C and 40Ca are presented. Some conclusions are drawn in sect. 4.
683
F. Capuzzi et al. / Final-state interaction
2. The Green function approach In the one-photon exchange approximation, the inclusive differential cross section for the quasi-elastic (e, e’) scattering on a nucleus is given by da p= dR dE:,
K(ZE&+
RT) ,
where K is a kinematical factor and 2 EL=-fg
c l-2
4
2 3
-1
tan’ $0 ) V
measures the polarization of the virtual photon. In eq. (2) 8 is the scattering angle of the electron and qt = w* - q2, where (w, q) is the four-momentum transfer. All nuclear structure information is contained in the longitudinal and transverse response functions, R,_ and RT, defined by RL(W, 9) = WOO(w,4)) in terms of the components
RT(w, q) = W”(w
q)+ W**b,
cl)
(3)
of the hadronic tensor WC””
Here .P is the nuclear charge-current operator, which connects the antisymmet~zed initial state of the nucleus j&J, of energy E,, with a final A-nucleon state II+!+) of energy Ef. The sum runs over the scattering states corresponding to all of the allowed asymptotic configurations and includes possible discrete states. Degeneracy indices are suppressed for sake of simplicity. The diagonal elements of the hadronic tensor can be equivalently expressed in terms of the Green function G+, related to the nuclear hamiltonian H, by WLLcL(o,q) =iIm
(t,bo~JF+G~(EC)Jp(JIO)
(5)
1 E-H-i?’
(6)
where EC= E,+ o and G+(E) =
The limit for q + i-0 is understood in all the equations involving G+. It must be performed after the calculation of the matrix element and is considered in the sense of a limit of distributions of the variable E. The hadronic tensor in eq. (5) contains the full A-body propagator of the nuclear system. As such, it is an extremely complicate object, which defies a practical evaluation. Only an approximate treatment allows to reduce the problem to a tractable form. To this end, the nuclear response is written in terms of the opticalmodel Green function 18%9). Th e resulting expression is derived in the following,
684
under
F. Capuzzi et al. / Final-state
suitable
formalism
approximations,
the
Feshbach
projection
operator
*“).
Only the one-body definite,
using
interaction
the proton
part of the charge-current case is exemplified,
operator
is retained.
To be more
by setting
(7) Therefore,
eq. (5) becomes
The next approximation
consists
in neglecting
terms i # 1.There-
the nondiagonal
fore, eq. (8) takes the form
where, in the final step, the symmetry of G’ for exchange exploited. Now let us denote by C, the channel subspace spanned by set of vectors (r, n), corresponding to the proton 1 at the point (A - 1) nucleus in the discrete eigenstate In). Likewise, let us analogous
vectors
continuous particles
representing
spectrum.
the residual
Antisymmetrization
only. Therefore
the projection
orthogonal
and decompose
the orthonormalized r and to the residual indicate by Ir, E) the
in the eigenstate
is here performed
has been
1s) of the
on the residual-nucleus
operators P,=
p, = c (r, n)(r, nl dr, J are mutually
nucleus
of protons
(r, E)(r, El d& dr
(10)
J
the identity
in the last matrix
element
of
eq. (9) as l=cP,+Pc. n
(11)
between Then eq. (11) is inserted on both sides of G+ in eq. (9) and interference processes involving intermediate states in different channel subspaces is neglected. Therefore the nondiagonal terms P,,G’(E)Pm and P,,G+(E)P, are suppressed and in the matrix element of eq. (9) it results G+(E) -C
P,G+(E)P, n
+ P,G+(E)P,
.
(12)
E Capuzri et al. / Final-state
685
interaciion
Finally, one assumes that jy connects the initial state (JI,J only with the discrete channel subspaces C,,. Therefore P,G’(E)P, is dropped. Under the above approximations, W’” is decomposed as (13) where Gz = P,G+P, is simply related to the unsymmetrized Feshbach A-body hamiltonian H:(E), which is able to describe, under the incoming-wave boundary condition, elastic scattering of a nucleon by an (A - 1)-nucleus in the discrete state In). In fact, applying th e projection operator method, one has G:(E)
where E, is the residual-nucleus
=
PII
(14)
E-en-Hz(E)-iv’
energy and
H:(E)
= P,,TP, + V:(E)
(1%
includes the kinetic energy T of the proton 1 and the unsymmetrized A-body optical potential V,,(E) = P,VP, + P,VQ,
E-Q,,AQ,,+is
Feshbach
QnVPn7
where Q,, = 1 -P, and V describes the interaction between the proton 1 and the residual nucleus. The definition of P,, in eq. (10) can be exploited to reduce eq. (13) to the single-particle expression W’+(o,
q)=i;
A,, Im
I
4z(r)jr+(r)%z(Ef;
r,
r’)j@(r’)&(r’)
dr dr’,
(17)
where A, is the spectral strength 2’) of the hole state &,, which is the normalized overlap integral between It&) and In), j” is the one-body current and %,,(E; r, r’) is the integral kernel associated to the Green function of the single-particle Feshbach potential Sm(E) related to V,,(E). A similar result can be derived for neutrons. The above approximation scheme was extensively discussed in refs. ‘**‘).We shall only add a few remarks. The approximation of neglecting nondiagonal terms in eq. (8) seems adequate only for enough high values of momentum transfer. It is needed if we want to write the nuclear response in terms of single-particle matrix elements. The contributions arising from interference between different channels, which are suppressed in eq. (12), are due to correlations in the final state and vanish in an independent-particle model.
686
F. Capuzzi et al. / Final-state interaction
The approximation of neglecting P,G’P, in eq. (12) cannot be adopted for the contributions to P, due to resonances of the residual nucleus. However, in appendix A, the contribution of isolated resonances is extracted from PC through projection operators onto states characterized by the residual nucleus in a bound state. Since such projection operators can be treated on the same footing as P,, we state to remove their contribution from PC and to include them in the sum over n.
2.1. SPECTRAL
REPRESENTATION
The complexity of a practical caiculation of %z(E; r, 8) in eq. (17) can be avoided by means of its spectral representation, based on a bio~hogonal expansion in terms of eigenvectors of ‘Yf: and ‘Vn. In fact, the incoming-wave scattering solutions of the eigenvalue equations [E-T-V;(E,)]#(E,)=O,
(18)
[E-T-Y’-~(E,)]X”;‘(E,)=O,
(19)
satisfy the biorthogonahty
relation
,&‘*( Ef, r)igf(Ef,
r) dr = 6(E - E’) .
i
Disregarding the bound-state eigenfunctions, which in practice do not contribute to W”‘, we assume the completeness relation as &‘(Ef,
r)&)*(Ef,
r’) dE = S(r-
r’)
.
(21)
In general the completeness relation for a biorthogonal set of eigenfunctions is a very delicate point, but in our case it is supported by the fact that it was rigorously proved for complex local potentials *‘) and appears to be reliable also for nonlocal potentials. From eq. (21) the spectral representation of the single-panicle Green function is obtained as
and eq. (17) becomes +CC W’*“(w, q)=+z
where
Im
I0
1
E,-E,-E--iv
TY(E,
EC) dE,
(23)
687
F. Capuzzi et al. / Final-state inieraet~on
T:*(E,
Ef) = A,
J
,f&)*(Ef, r)~~(r~~~(r) dr
r)~~(~)#~(~) dr [Ix’k?&, I*a (24)
The Cauchy integral in eq. (23) can be calculated by exploiting the standard symbolic relation 1 =P ; x - in 0
-t-I’77S(x),
(25)
where P denotes the principal value. Then eq. (23) becomes W“‘(w,
q) =C
+* J
Re TZ”(Ef-
n +IP
‘IT
0
E,, E,)
1 Ef-e,-E
Im Tc’“(E, E,) dE
1 .
(26)
It is conceptually interesting to observe that eqs. (24) and (26), although containing eigenfunctions of a complex optical potential, are not affected by a spurious attenuation of the strength due to flux absorption. In fact, the flux lost in a particular A-body final state contributes to excite the other final states, which are all contained in the inclusive response. This is clearly seen in eq. (24), by considering the contribution of a particular channel n. The loss of flux in x’-’ is compensated by a corresponding gain in it-’ due to the flux lost, towards the channel n, by the A-body final states which are asymptotically originated from different channels. We want to stress here that in the Green function approach it is just the imaginary part of the optical potential which accounts for the redistribution of the strength among different channels. Before concluding this section, we want to add a remark about the implications of the Pauli principle. In the present approach the initial state I+,,)is antisymmetrized, but the projection operators of eq. (10) are unsymmetrized. Therefore, in eqs. (24) and (26) the Pauli principle effects are considered in the hole states &, which are overlap integrals between antisymmetrized wavefunctions, but not in the final states, as the optical potential W;, does not account for these effects. Actually, x’-’ is affected by an unphysical loss of flux, since V: accounts for scattering processes involving exchange of two protons as an absorption, but this effect is compensated by the contributions to 2(-i due to the unphysical exchange degeneration of the final states, which are implicitly included in G. Therefore, in the inclusive reaction this treatment of antisymmetrization is not incorrect. Nevertheless, eq. (24) is exposed to the criticism of both introducing and removing an unphysical effect. Furthermore, V;, has to be related to the phenomenological potentials, which include the Pauliprinciple effects. Therefore, from a theoretical point of view, it is worthwhile to modify the present treatment, in order to deal with an optical potential that includes antisymmetrization effects. A proper treatment of antisymmetrization is outlined in appendix B.
688 2.2.
E Capuzzi INTERFERENCE
A delicate
consider
DIFFERENT
point in the approach
effects between and P,,G+P,.
BETWEEN
et al. / Final-state
different
interaction
CHANNELS
of the previous
channels,
section
which are related
In order to estimate
is to neglect
to nondiagonal
the consequences
interference
terms P,,G’P,,,
of this approximation,
let us
the exact relation dG+( E) dE
which characterizes
the A-body
Green
= -G+*(E) function,
,
(27)
and then compare
the effect of the
approximation on both sides of eq. (27). Understanding that the Green functions are included into the appropriate matrix elements, the derivatives of eqs. (12) and (14) give dG+ dE where,
under
disregarded.
the same Likewise,
- V;‘)G;
G;(l
---C
,
(28)
n
approximation
as in the preceding in Gt2 the nondiagonal
by suppressing
G +*,C
section, P,G+P, terms, it results
G;‘.
is
(29)
n
The two approximate expressions in eqs. (28) and (29) do not satisfy eq. (27) and the resulting discrepancy indicates that the effects due to interference between different channels are directly related to the energy slope of V,(E). Phenomenological calculations 13) of the Feshbach potential, based on dispersion relations, show an appreciable slope in the quasi-elastic region of interest and an almost linear behaviour of the energy dependence. A simple way of improving the approximation is to replace G, in eq. (13) by an effective Green function G,,, still operating in C,, which embodies contribution of interference
with the other channels
(29). Under requirements
the assumption of an almost linear energy are fulfilled by the symmetric operator h;(E)
=Jl-
and restores
V;‘(E)
consistency
G;(E)&
between
dependence
eqs. (28) and
of V,,(E),
- V;‘(E),
these
(30)
since from eq. (28) it follows dG+( E) nzz
-&“(E)
.
(31)
dE We want to notice
that 6,
is really a Green 6;(E)
=
function,
pfl E-.c,,-Hz-iv’
since eq. (30) gives (32)
F. Capuzzi et al. / Final-state
689
interaction
where A:=[~-V~‘(E)]-“*[H~(E)-(E-E,)V~‘(E)][~-V:’(E)]-“* =P,,TP,+
c;
(33)
is energy independent, under the condition VE( E) = 0. Therefore we expect that replacement of the Feshbach hamiltonian H,,(E) by Gn is a simple and consistent prescription which improves the approximation of eq. (13). As a consequence, the single-particle eigenfunctions ,&)( E,) [ ig’( E,)] of Yi( E,) [ ‘Y,,(E,)] are replaced by the corresponding eigenfunctions ig) [,&)I of the single-particle potential e(FO), related to i/i( en). Therefore, eqs. (24) and (26) are replaced by Tsp(E)
= A,
dr dr 1 [5i~‘*(r)Y‘(rM,(r) J,i%‘*~rW~~h(r) 1 *,
+7=
Re T:“(E,-q,)+LP
?T I ,,
1
E,--E,-E
Im T;*(E)
dE
(34)
.
(35)
From eq. (33) the wavefunctions i’,-’ and ,&) are connected with the eigenfunctions of the Feshbach potential by the phase conserving relation ~~‘(r)=Jl-~:‘(E+&,)x~‘(E+&,,r), $‘,-‘(~)=J~-YL(E+E,)~~)(E+~,,,~).
(36)
We have shown that the effective Green function 6, of eq. (30) cures the inconsistency due to the energy dependence of the Feshbach potential and takes into account contribution of interference between different channels, under the assumption that V,(E) and, therefore, Yn,(E) is linearly dependent on the energy. Although the Feshbach potential is not well known in practice, the above assumption is supported by phenomenological analyses 23), as we have already observed. However, as the Feshbach potential is not exactly a linear function of E, 9, maintains a residual energy dependence. There are no available treatments which take in due account the effects of a deviation from linearity in 7fn(E). These effects are expected to modify 6, and therefore the related potential $“‘,,but cannot change the funda* mental requisite that v,, must be energy independent. Thus the eigenfunctions ig’ and x”(;‘, in eq. (34), must be preferably determined on the basis of such a requisite. Unfortunately, the existing phenomenological analyses ‘“) based on an energy independent optical-model potential are not suitable for calculations on specific nuclei. In sect. 3 we will follow a method, based on the requisite of energy independence, which allows to use more adequate phenomenological potentials. 2.3. SUM
RULE
Energy sum rules represent a useful test to check the approximations introduced in the previous sections. Let us consider, e.g., the integrated longitudinal response
690
F. Capuzzi et al. / Final-state
interaction
function +‘X E(9)
&(w,
=
9) dw =
_-oci l(~fIJ”(q)lrLo)1*6(EoI f
&+a)
do,
(37)
where, according to eq. (7), Jo(q) is the one-body charge operator for protons. As a consequence of the completeness of the set of the final states, the Coulomb sum rule results
E(4) = z(~ol_c+mo)
(38)
2
where, as in eq. (9), the terms involving charge operators been disregarded. In the Green function formalism, Z(q) is written as
of different
particles
have
(39) where the understood theory, i.e. P(q)zZ
One recovers
limit
for 7 + +0 is considered
lim ?T R-+cc
in the sense
lim Im
(40)
,,++O
eq. (38) from eq. (40) by using the general
1
expansion
1
dE,
E-A-iv-E-A+i7 which holds
of distribution
for any energy-independent
self-adjoint
operator
theorem
25)
(41)
A.
Let us now consider the effect on the sum rule of the approximate treatments given in the previous sections, Eq. (13), where interference terms and continuous channel
contribution
are neglected, E(4)
is integrated
by means
of eq. (41) and gives
n(rLoli?+~njNo) =Z(~oIX+j~I~o), = Z C
(42)
which shows that the sum rule is not affected by neglecting interference terms. In sect. 2.2 the contribution of interference between different channels is accounted for by expressing G+ as a sum of effective Green functions 6:. A criterion to check consistency of this treatment requires that the result of eq. (42) is unchanged. The integrated longitudinal response is expressed in terms of the effective Green functions as +R E(q)
--zC
r n
lim R-tot
lim Im r)++O
OR (+olj?‘&(E)j?lcCro)
dE.
(43)
F. Capurzi et al. / Final-state i~te?a~tion
691
However, 6: is related by eq. (32) to the non-hermitian hamiltonian &, and eq. (41) cannot be applied. Therefore, a different approach, based on the analyticity properties of &, must be used to verify the Coulomb sum rule. Let us consider the second resolvent equation &(E)=GO,+(E)+GO,f(E)$;(E)&;(E),
where G?(E)
(44)
is the free Green function G:+(E)
=
P” E-E,-T-iq’
(45)
By substituting eq, (44) in eq. (43) one obtains
[I (~olj~‘G0,‘(E)O:(E:)8:( dE I, JSR 4-R
lim X;(q)--zC Ii- n R++m +
lim Im
_R (&Ij?*GOn’(E)j$M
dE
T,-.+O
(46)
where, in the last term, the contour of integration is extended to the semicircle SR, of radius R, lying in the lower complex half-plane. We first consider the case of Gr, given by eq. (33), without disregarding its weak energy dependence. From eqs. (30) and (33) we can observe that the integrand in the last term of eq. (46) is an analytic function without poles inside SR and gives no contributions. The first integral in eq. (46) contains the Green function of an energy-independent self-adjoint operator, and can be evaluated by means of eq. (41), leading to eq. (42). The same result holds if 6, is chosen to be the Green function related to a complex potential that is rigorously energy independent, provided that no complex eigenvalues exist.
3. Results In this section, numerical results for the response functions are presented and discussed. The theoretical approach outlined in sect. 2.1 allows to calculate the components of the nuclear response in a straightforward and relatively simple way. In fact, as a result of the spectral representation of the optical-model Green function in terms of the biorthogonal set of eigenfunctions defined in eqs. (18) and (19), the ingredients of the nuclear response are the integrals in eq. (24), which are of the same kind as those giving the transition amplitudes of the electron-induced nucleon knockout reaction in the distorted-wave impulse approximation (DWIA) framework of ref. 26). The complexity of a practical calculation of the optical-model Green function is thus avoided in the present approach and the evaluation of the inclusive response is reduced to a problem of the same difficulty as the calculation of single-nucleon knockout contributions.
E Capuzzi et al. / Final-state
692
interaction
The integrals in eq. (24) have been therefore calculated adopting the same treatment as in ref. 26), which was successfully applied to describe exclusive (e, e’p) data. This implies the use of a non-relativistic approach, where the one-body nuclear current operator is given by the non-relativistic approximation of McVoy and Van Hove *‘). Both initial and final state wavefunctions are in principle eigenfunctions of the Feshbach optical potential. However, as a practical expression of the Feshbach potential is unavailable, the single-particle bound state wavefunction (P,, is taken from a phenomenological Woods-Saxon potential and the distorted wavefunctions X (-I and i’ .-’ are obtained through a local equivalent spin-dependent optical potential, given by a fit to elastic proton-nucleus scattering data. An improvement of the Green function approach is represented by the prescription proposed in sect. 2.2. Replacement of the eigenfunctions x(-’ by the corresponding eigenfunctions f’-’ of the energy independent potential q’:, allows to take into account in a simple way the contribution of interference between different channels and also provides a theoretical justification for using a phenomenological optical potential. In fact, the eigenfunctions of the energy-independent potential e are connected with the eigenfunctions xl;:, belonging to the eigenvalue E, of the local phenomenologi~al optical potential Y:(E) by the “de Forest factor” [ref. ““)I f;)(r)
=Jl-
V:'(E)X\TA(r)
(47)
,
where terms involving spatial derivatives of Yz, leading to small corrections, have been neglected. Likewise, denoting by 2 L-L , the corresponding eigenfunctions of ‘V,(E),
one has /$‘(r)
= m/$:$(r)
.
(48)
In eqs. (47) and (48), following the usual convention for phenomenological potentials, the residual-nucleus energy has been subtracted in the energy dependence of “u;(E). Furthermore, the same local optical potential, obtained from elastic scattering and, therefore, By means
related
to the ground
state, is assumed
for all the channels.
of eqs. (47) and (48), eq. (34) becomes
(49) where the de Forest factors take into account interference contributions. The use of a phenomenological optical potential is now justified on the basis of eqs. (47) and (48) and not only by our poor practical knowledge of the Feshbach potential. In the calculations residual-nucleus states In) are restricted to be one-hole states in the target and the sum over n in eq. (35) includes both discrete states and isolated resonances embedded in the continuous spectrum of the (A- 1)-body system. In fact, as it is shown in appendix A, they can be treated in a similar way.
F. Capuzzi et al. / Final-state
A pure shell model spectral
strength
and neutron
is assumed
for the nuclear
inreraction
structure,
for each n, and the sum over n is limited
states. Even if such an assumption
the details of the inclusive
response,
693
i.e. we take a unitary to the occupied
may not be fully adequate
it allows to perform
enough
proton
to describe
simple calculations,
on a conceptually clear basis and seems justified as a first approximation. In the following, results will be presented for “C and 4oCa. Having in mind the limitations and the ingredients of the present calculations, suitable values of the momentum transfer have been considered, i.e. enough large to keep the quasi-free scattering condition far from giant resonances and collective excited states, in order to make particle-hole rescattering less important reduce relativistic effects (q = 400-550 MeV/ c).
3.1. RESPONSE
FUNCTIONS
‘), but not too large, in order to
FOR “C
The longitudinal and transverse response functions for ‘*C, calculated without the de Forest factor, at q = 400 MeV/ c, are shown in fig. 1 and compared with the Saclay data 3).The contribution resulting from all the integrated single-nucleon knockout channels is also drawn. The difference between the two curves gives an
idea of the relevance of multi-nucleon removal. For the longitudinal response, the result obtained by retaining only the real part of the optical potential is also presented in the figure; in this case integration of single nucleon knockout channels gives the same result as the inclusive (e, e’) calculation. The difference among the three curves stresses the role of the imaginary part of the optical potential and clearly shows that the present approach is significantly different from a calculation with a real optical potential. In fact, a positive contribution is added to the single nucleon knockout, but different and smaller than the one resulting by neglecting the imaginary part of the optical
potential.
The underestimate of the transverse response is a systematic result of our calculations. It may be attributed to physical mechanisms, not included in the considered approach, e.g. meson-exchange currents together with 2p2h excitations. Therefore, this result is not surprising. However, we are not particularly interested in obtaining a fit to experimental data, our main aim being to understand the role and the relevance
of final state interaction
in quasi-free
electron
scattering.
As those mechan-
isms are expected to modify mainly the transverse response, in the following we will focus our attention on the longitudinal response. In fig. la the longitudinal response is overestimated. This result is not surprising, as it is comparable with the one given in fig. la of ref. 6), which was calculated in the approach of ref. 18). When the contribution of interference between different channels is taken into account through the de Forest factor, an overall reduction of both responses is obtained. A reduction is also obtained when a Perey factor 3’) is included in all of the analyses where a local equivalent potential replaces the nonlocal one. However,
694
0
40
80
120
160
200
240
280
0
40
80
120
160
200
240
280
fd
(MeV)
0.005 0
Fig, 1. longitudinal (a) and transverse jb) response functions for “C at q =400 MeVJc a5 a functian of energy transfer. The data are from ref. 3). Hole states are taken from ref. Z9f and optical potential from ref. 30]. Solid line7gives the inclusive response without the de Forest factor and dashed line contains contributions of integrated singfe-nucleon knockout onty. Dot-dashed line in [af gives the result of a calculation where the imaginary part of the optical potential has been neglected.
both the nature and the physical meaning of Percy and de Forest factors are different. The de Forest factor removes the energy dependence of the phenomenologjca~ local potential, thus taking into account the interference between different channe‘ts, The Perey factor recovers the wavefunctions of the Feshbach potential from the corresponding phenomenological one. Thus it only removes the spurious energy dependence due to locality, without changing the dynamical one. Therefore, it cannot be expressed in terms of the tocal potential, but requires a model of the original nonloeality. The effect of the de Forest factor on the longitudinal response is shown in fig. 2 for two different choices of bound-state wavef~nction~ and optical potentials. A
695
0
40
60
120
160
200
240
260
w (MeV)
120
160
200
240
260
w (MeV) Fig. 2. Longitudinal response function for “C at q =400 MeV/c as a function of energy transfer. The data are from ref. ‘). Solid line gives the inclusive response with the de Forest factor, dashed line is without the de Forest factor. In (a) hole states and optical potential as in fig. 1, in (b) they are taken from ref. 32).
significant
reduction
is obtained,
whereas
the shape
is only slightly
changed.
The
de Forest factor produces a larger reduction in fig. 2b, where hole states and optical potentials are taken from ref. 32). The experimental data are better reproduced in fig. 2a, where the parameters of refs. 29730),which also give the best description of quasi-free (e, e’p) data 33), are adopted. Low values of energy transfer are not drawn in figs. 1 and 2a, as the optical potential of ref. 30) cannot be extrapolated at low energies. In general, available phenomenological optical potentials are given in a restricted range of energy values, and cannot be extrapolated far beyond this range, which imposes a restriction to the kinematical conditions that can be considered.
696
E Capuzzi
For the same reason eq. (35) is practically
a precise
impossible,
tions of the continuous of response
functions
et al. / Final-state
and reliable
calculation
of the dispersive
as it requires
integration
over all of the eigenfunc-
spectrum presented
interaction
of the optical in this paper
first term in eq. (35). The uncertainty relevant for our present purposes.
potential. contain
Therefore,
term in
all the results
only the contribution
of the
due to this approximation is neither large nor In fact, for all of the considered situations,
Im T:@(E) < Re TY(E), and a rough calculation of the neglected term, obtained by extrapolating the value of the function in the dispersive integral beyond the allowed energy range, gives only a small contribution is shown in fig. 3, for the bound state and the optical choice of phenomenological but does not significantly
ingredients affect the
to the final result. An example potential of ref. 32). A different
may change the results conclusion about the
presented in fig. 3, small size of the
contribution. The longitudinal response at q = 500 MeV/c is shown in fig. 4. The result is similar to the one obtained at q = 400 MeV/c and the fair agreement with the data is confirmed for the curve with the de Forest factor and the optical potential of ref. 30). A similar trend as in fig. 2 is also found in the dependence on the phenomenological ingredients.
3.2. RESPONSE
FUNCTIONS
FOR
?Ia
A similar analysis as the one presented for “C has been performed for 40Ca. In fig. 5 the longitudinal response function calculated without the de Forest factor for hole states
and optical
-0.005
potential
from ref. ‘*) is given and compared
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 40 00 120 160 200 240
with the
280
w (MeV) Fig. 3. Contribution to the longitudinal line), for 12C at 9 = 400 MeV/ c, obtained in dashed line. Hole states and optical
response function from the dispersive integral of eq. (35) (solid by extrapolating the quantity Im I,,~,‘(E), which is also shown potential as in fig. 2b. The de Forest factor is included in the calculations.
F. Capuzzi et al. / Final-state
0.008
-
0.004
7
0
40
80
120
160
interaction
200
697
240
280 w
320
(MeV)
Fig. 4. Longitudinal response function for ‘*C at 9 = 500 MeV/c as a function data are from ref. ‘). The inclusive response with the de Forest factor is shown and optical potential as in fig. 2a (solid line) and 2b (dashed
of energy transfer. The for the same hole states line).
Saclay data “) at q = 410 MeV/ c. The figure shows that experimental data are overestimated by the considered approach. As in “C, the disagreement is comparable with the one obtained in fig. lb of ref. ‘j), where a similar theoretical approach, with different phenomenological ingredients, is used. The effect of the de Forest factor and the dependence on the choice of the optical potential is shown in fig. 6. The reduction given by the de Forest factor is relevant, but unable, in this case, to resolve the disagreement with data. The dependence on the optical potential is significant, but not really crucial. The potential of ref. 34) seems preferable, however it is still inadequate to produce a good fit to the data. c
iJ
S ‘= [r
0.07
L
0.06 0.05 0.04
0
~,...‘.*..‘,...‘,,,.‘,,.,I,.
0
40
60
120
160
200
240
w (MeV) Fig. 5. Longitudinal response function for 40Ca at 9 = 410 MeV/c. The data are from ref. 4). Hole states and optical potential are taken from ref. “). Solid and dashed line as in fig. 1.
698
F. Capuzzi et al. / Final-state ;
7
interaction
0.07 0.06
r”
0
0
40
80
120
160
200
240
0
(MeV)
Fig. 6. Longitudinal response function for 4”Ca at 9 = 410 MeV/c. The inclusive response with (solid line) and without (dot-dashed line) the de Forest factor is shown for hole states and optical potential from ref. s2). Dashed line gives the inclusive response with the de Forest factor for hole states taken from ref. 29) and optical potential from ref. 34).
The longitudinal response function at q = 550 MeV/c, calculated with the optical potential of ref. 34) with and without the de Forest factor, is given in fig. 7 and compared with the Saclay data “). The effect of the de Forest factor is smaller than the one resulting when the optical potential of ref. 32) is adopted. Comparison with data has a similar trend as in fig. 6. Therefore in the present approach longitudinal response are not so well described as “C data. However,
0
40
80
120
160
200
240
280
4oCa data of the resulting
320
w (MeV) Fig. 7. Longitudinal response function for ?a at q = 550 MeV/c as a function of the energy transfer. The data are from ref. 4). Hole states from ref. 29) and optical potential from ref. s4). Solid line gives the inclusive response with the de Forest factor, dashed line is without the de Forest factor.
I? Capurri
disagreement mations. nuclear
et al. / Final-state
interaction
is not too large and may be explained
A better result could presumably state a more sophisticated
under
be obtained,
description
699
the considered
approxi-
e.g., by using for the initial
than a pure shell model.
4. Summary and conclusions The electromagnetic response functions for the inclusive (e, e’) scattering in the quasi-elastic region have been calculated in a many-body approach, where the Green function is expanded over the complete biorthogonal system of the eigenfunctions of the Feshbach optical potential and of its hermitian conjugate. complication of an explicit determination of the Green function nuclear
responses
are expressed
in terms
of matrix
elements
In such a way, the is avoided and the similar
to the ones
appearing in the exclusive (e, e’p) cross section. Therefore, the calculation of the inclusive response involves no more difficulties than the calculation of single-nucleon knockout. All the final states contributing to the inclusive reaction, and not only one-nucleon emission, are contained in the Green function. However, the final state interaction is treated consistently with the exclusive reaction, where it plays a crucial fact, both the real and the imaginary parts are used in the optical potential, results are not affected by a spurious absorption. The effect of the interference between de Forest factor, which also provides a phenomenological optical potential,
attenuation different
of the strength channels
is included
role. In but the
due to flux through
the
a justification for using the eigenfunctions of instead of the ones of the Feshbach potential.
Results have been presented, in comparison with experimental data, for “C and 40Ca at different values of momentum transfer, between 400 and 550 MeV/c. The effect
of final
state
interaction
is large
and
important.
Including
in the
calculation the contribution of all the final states is essential, as the inclusive response functions are significantly different from the ones obtained by integrating all singlenucleon knockout channels. The imaginary part of the optical potential originates such a difference. Therefore, it plays a crucial role and cannot be neglected. It decreases the height of the peak both in the longitudinal and in the transverse response and is responsible for the redistribution of the total flux among various channels. The de Forest factor gives an overall reduction, which significantly improves comparison with the data of longitudinal response. In ‘*C the agreement is satisfactory; the lack of strength at low energy transfers could be, at least partly, explained by particle rescattering, which otherwise is expected to be small at the considered momentum transfers. Final state interaction has a similar effect on the two components of the nuclear response and cannot reproduce their relative strength. However, the systematic underestimate of the transverse response points to non-nucleonic degrees of freedom
700
F. Capuzzi et al. / Final-state
and correlation in the inclusive The
present
quasi-elastic physics
effects, which should transverse response. results
region
seem
give an enhancement
to indicate
can be reasonably
and to confirm
interaction
that
inelastic
explained
= [ Y(r)[
evidence
spectrum
In this appendix we extract from eq. (A.l) the contribution Let us consider the spectroscopic strength, defined by
where
the
spectrum
dr.
(A.1)
of isolated
resonances.
I(r,sl&Jl’dr.
A(E)
of the residual
(A.2)
(r, The positions
by means
El&J=
A(E) can be expressed
r(E) =-I_ 2?7 [E--Eo+F(E)]2+[fr(E)]2r
where F(E) and @(E) are the average part of the hole self-energy operator. +(a, r) can be defined
nucleus,
as *‘) (A.3)
values of the hermitian and anti-hermitian The normalized spectroscopic amplitudes
of eq. (A.3) as
r(F) dZZ
of resonances
(A - 1)
is given by
I In the continuous
in the nuclear
of exotic mechanisms.
due to channels
1 k, a)(r, a/&) de]
h(s)=2
scattering
A
The contribution to the response function system is in an eigenstate 1s) of the continuous pC_CkJ
electron
be neglected
in terms of conventional
that there is no compelling
Appendix
and cannot
e-E,+F:.c)-fir(E)
4(E’r)’
(A.4)
are given by the roots E, of the equation E,-Eo+F(~,)=O.
(A.5)
For each root E,, we choose, as suggested by phenomenological analyses, an energy interval I,, where the spectroscopic strength A(E) has appreciable values and the functions F(E), T(E) and 4(~, r) have approximately constant values, i.e. F(E) = F(G),
r(E)=r,,
HE, r)=&(r),
VEEI,.
(‘4.6)
We also assume that energy intervals I,, corresponding to different roots, are disjoint. By use of eqs. (A.4), (A.5) and (A.6), we can thus extract from eq. (A.1) the contribution of isolated resonances as (A.7)
F. Capuzzi
et al. / Final-state
interaction
701
where the residual nucleus states Ia), corresponding to each resonance LY,can be expressed in terms of the eigenfunctions of the continuous spectrum in the corresponding energy interval I, as (E -
E,
-$I-,)-‘I&) ds.
L
(A.@
We notice that (aIa’)=O
if a#a’,
(r+Y)=O.
(A-9)
Therefore the projection operators (A.lO) are mutually orthogonal and orthogonal to P,. The overlap integral between I+J and Ir, a) is given by means of eqs. (A.8), (A.4) and (A.6) as (r,
alW=-&bla)&(r).
(A.ll)
After substituting eq. (A.1 1) in eq. (A.7) we obtain
(A.12) which shows that isolated resonances can be extracted from the continuous spectrum and treated in a similar way as discrete eigenstates. Therefore, their contribution can be included in the framework of sect. 2. Appendix B
In this appendix, the inclusive cross section is written in terms of final state wavefunctions which are eigenfunctions of an optical potential that includes antisymmetrization effects. To this end, and without modifying the involved approximations, some changes are needed in the formal treatment of sect. 2. In order to deal with antisymmetrized quantities as far as possible, we do not assume, for the moment, the approximation of eq. (9). By exploiting the symmetry of G+ for exchange of protons, eq. (5) becomes
Wpp(m,q) =f h (hljYfG+(E~)JF Itid.
(B.1)
Inserting eq. (11) in eq. (B.l) and then neglecting PC, one has W“w((w,q) =f;
Im
I
(+oI_C%, n)(r, nl G+(Er)J*“I&J dr.
(B.2)
702
F. Capuzzi et al. / Final-state
Notice strongly
that
it would
connects
If we want function
be incorrect
to neglect
II,&) with the continuous
to write
which
the nuclear
includes
interaction
PC directly
in eq. (5), since
Jp
channels.
response
the full implications
in terms
of a single-particle
of the Pauli
principle,
Green
G’ must be
kept into matrix elements involving antisymmetrized wavefunctions. To this aim, we replace the matrix elements involving G+ in eq. (B.2), where only one wavefunction is antisymmetrized, by the equivalent ones where the two wavefunctions are antisymmetrized. Using the more convenient formalism of the second quantization, we have
where
a, destroys
a proton
at the point
Wp”l”(w, 4) =$E In sect. 2 the diagonality
n
Im
I
r. Therefore,
n)(n(a,G+(Ef)J~“I~o)dr.
(&IjP’lr,
approximation
eq. (B.2) becomes
of eq. (12) consists
P,,G+(&)Q,
(B.4)
in assuming
==O.
(B.5)
An analogous approximation can be applied to eq. (B.4) by introducing the operators Pt and Qt, which project onto the subspace spanned by the vectors aTIn) and onto the complementary orthogonal one. These operators were first introduced by Feshbach 35) to express the optical potential in the antisymmetrized case. In the second quantization formalism they read as Pf =
a:ln)K(r,
r’)(nla,, dr dr’,
Q$=l-P;,
(B.6)
with K(r, r’)=fY(r-r’)+C where u, and rty are the normalized tion numbers,
defined
Y
natural
ny u,(r) 4Xr’), 1 -nv orbitals
(B.7)
and the corresponding
by
I
u,(r)(nla:u,,In)u~,(r’)
dr dr’=
03.8)
nuSvu,.
The sum in eq. (B.7) is over those values of Y corresponding The diagonality approximation is now formulated as
to n, # 1.
P:G+(E,)Q;=O. Therefore
occupa-
(B-9)
eq. (B.4) becomes WFF(w, q) =$Z;
”
Im
(&Ij~+[r, n)(nlu,P~G+(Ef)P;;lJ’“I~o)dr.
(B.lO)
F. Capuzzi et al. / Final-state
703
interaction
By means of eq. (B.6) and the relation (nlcY
I&) =Jz(r,
nlJ%kJ,
(B.ll)
eq. (B.lO) results
xh, nP‘l+ddrIdr2h,
(B.12)
%?(E;
(B.13)
where rl, 4 = (nla,,G+(W$d
is the one-body optical-model Green function, which includes antisymmetrization effects. However, eq. (B.12) is not yet reduced to a one-body expression, since we have not exploited the approximation of neglecting terms involving charge-current operators of different protons. This can be performed now by setting in eq. (B.12) (r, nlJ’
Ih>= (6 4.Kl+J,
(B.14)
as, for enough high values of momentum transfer, jp does not contribute to the overlap integral if i # 1. When eq. (B.14) is inserted in eq. (B.12), one has
xi”(rdcbn(r3)dr, dr2 dr3.
(B.15)
Spectral decomposition of eq. (B.15) is then obtained, along the same lines as in sect. 2.1, by means of the AGS-based optical potential “Irt+, which is related to the one-body Green function 3;’ [refs. 36V3779)]. Therefore eq. (26) is recovered, with
X
[I
x!&)*(J%, r)_?‘(r)&(r) dr
* 1 ,
(B.16)
where x’-’ and f(-’ here denote the eigenfunctions of Y;;‘+ and V;; and therefore contain the full implications of the Pauli principle. Also the hole wavefunctions #, are eigenfunctions of the same potential Vf, evaluated at the energy E,,. The operator IS is necessary in eq. (B.16) to preserve the sum rule. In fact, the proof given in sect. 2.3, which was based on integration over a complex path, can be again applied here, since the AGS-based potential Vt is hermitian analytic. However, at high energies, ‘V;;’includes a term which is linearly dependent on the energy. This produces a contribution from the contour at infinity and hence a violation of the sum rule, which is exactly compensated by the operator K.
E Capuzzi et al. / Final-stare inreraction
704
The approximate be developed
treatment
of interference
as in sect. 2.2 without
further
between
different
channels
can then
comments.
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105
and C. Villi,