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On the Dirac structure of the nucleon self-energy in nuclear matter
NUCLEAR PHYSICS A Nuclear Physics A 640 (1998) 471489
On the Dirac structure of the nucleon self-energy in nuclear matter A. Trasobares”,
A. Polls a, A. Ramosa, H. Miitherb
a Departament d’Estructura i Constituents de la Matkia Universitat de Barcelona, E-08028 Barcelona, Spain b Institut fir Theoretische Physik, Vniversitiit Tiibingen, D-72076 Tiibingen, Germany
Received 10 November 1997; revised 22 June 1998; accepted 14 August 1998
Abstract The relativistic structure of the self-energy of a nucleon in nuclear matter is investigated including the imaginary and real components which arise from the terms of first and second order in the NN interaction. A parameterized form of the Brueckner G-matrix is used for the NN interaction. The effects of the terms beyond the DBHF approximation on quasiparticle energies and the optical potential for nucleon-nucleus scattering are discussed. @ 1998 Elsevier Science B.V.
1. Introduction During
the last few years the attempts
clear systems
from a realistic
to derive the ground-state
nucleon-nucleon
(NN)
interaction,
properties
of nu-
have been promoted
very much by the understanding that relativistic effects may be non-negligible in such investigations. These ideas were originally developed within the various versions of the phenomenological Walecka model [ 11. The Dirac structure of a strong repulsive component originating from the exchange of an attractive component of medium range described in terms leads to a self-energy of the nucleon in the nuclear medium
the NN interaction with the w vector meson and of a scalar meson ((T) which contains a large
scalar component 2” and a large time-like vector component 2?. These two components compensate each other to a large extent if one calculates the single-particle energy. This leads to the well-known fact that the binding energy of nuclei are very small as compared to the rest mass M of the nucleon, a fact which has often been used to argue that relativistic effects should be small in the many-body problem of nuclear physics. The individual components of the self-energy 27 are of the order of the nucleon rest mass 0375-9474/98/$ - seefront matter @ 1998 Elsevier Science B.V. All rights reserved. PIISO375-9474(98)00471-O
A. Trasobares et al. /Nuclear
412
(typically
one third of M) . Therefore
medium
is modified
nucleon.
This medium
the NN interaction
Physics A 640 (1998) 471-489
the structure of the Dirac spinors in the nuclear
to quite some extent as compared dependence
in the medium,
These more or less empirical
to the Dirac spinor for a free
of the Dirac spinors, which affects the evaluation leads to a saturation
mechanism
models received support from calculations
from a realistic One-Boson-Exchange
model of the NN interaction
realistic NN interaction means that the parameters contained to describe the experimental data of free NN scattering.
of
for nuclear matter. which start
[ 2,3]. In this context
in these models are adjusted Using such a realistic NN
interaction in a many-body calculation of the Dirac-Brueckner-Hartree-Fock (DBHF) type, results were obtained for the saturation point of nuclear matter which were in quite a good agreement with the empirical saturation point [4,5]. Such a DBHF calculation accounts for the effect of correlations on the level of the BHF approximation, i.e. the lowest order in the hole line expansion, and allows for the relativistic effects which we just outlined. This success of the relativistic features contained in the DBHF approach could be a solution of an old problem: the so-called Coester band phenomenon [6], which stands for the fact that many-body calculations based on various realistic models of the NN interaction lead to predictions for the saturation point of nuclear matter which either fail to yield enough binding or predict a saturation density twice as large as the empirical value. Such a Coester band can also be observed in non-relativistic studies of finite nuclei [ 71. Therefore it was quite obvious that attempts have been made to include the relativistic features also in DBHF calculations of finite nuclei. Indeed the inclusion of relativistic effects gave a substantial improvement in the calculated binding energies and radii of finite nuclei [ 8,9]. However, there remains still a discrepancy between the DBHF results and the experimental values. On the other hand it was observed that an extension of the non-relativistic BHF approach to a definition of the nucleon self-energy, which accounts in a symmetric way not only for the particle-particle ladders included in the Brueckner G-matrix but also for the corresponding hole-hole scattering term, may lead to a larger binding energy and a larger radius than obtained in the BHF approach [ lo]. A similar feature, moving the saturation point away from the Coester band by including hole-hole scattering terms, has also been observed in particle-particle hole-hole RPA calculation of nuclear matter [ 11,121. Taking a very optimistic point of view one may argue that the combination of relativistic effects and hole-hole scattering terms may lead to an improved microscopic understanding of ground state properties of nuclear systems. In the work presented here we would like to investigate the effects of particle-particle and hole-hole scattering terms on the relativistic structure of the nucleon self-energy. For that purpose we consider the nucleon self-energy defined with all terms of first and second order in the NN interaction (see Fig. 1). For the NN interaction we use a parameterization of a G-matrix, evaluated within the DBHF approach, in terms of effective (+ and w mesons [ 81. Note that this parameterization does not include the pion exchange explicitly, therefore we may miss part of the second-order tensor contribution, which is included only in an average way. Since this effective interaction has been
A. Trasobares et al./Nuclear
Physics A 640 (1998) 471-489
413
Fig. 1. Graphical representation of the Hartree (a), Fock (b), 2plh direct (c), 2plh exchange (d), 2hlp direct (e) and 2hlp exchange (f) contributions to the self-energy of the nucleon. The momenta labeling the various contractions in (c) correspond to the nomenclature used in Eq. (15). constructed from a DBHF G-matrix we have to use a subtracted dispersion relation to avoid a double counting of the 2 particle-l hole contributions to the real part of the self-energy. In contrast to earlier investigations [ 13-161 we evaluate the imaginary contributions to the nucleon self-energy allowing for all possible combinations of momentum k and energy k”. Therefore we can use dispersion relations to evaluate also the real part of the second-order diagrams. We investigate in detail the effect of these higher order diagrams on the Dirac structure of the self-energy. Furthermore we derive from this relativistic self-energy
an optical potential to be used in a Schradinger
equation for nucleon-nucleus
scattering. The technical details for the calculation of the imaginary components in the self-energy are presented in Section 2. The dispersion relations used to evaluate the corresponding real components are given in Section 3. That section also contains a detailed discussion of the results. The main conclusions
2. Imaginary
are summarized
in the final section.
part of the nucleon self-energy
The physical system well as rotational
of
or nuclear it
by definition, to
In general,
as be
written ’ as [l] kxY(k),
(1)
’ Notation: within formulae we shall use reman type (p,q, .) for 4-vectors, boldface type (p. q, 3-vectors and normal italic type (p,q, .) for the norm of 3-vectors.
.) for
Z(k) = -Z’(k) -rap(k)
+y.
A. Trasobares et al. /Nuclear Physics A 640 (1998) 471-489
474
where the functions
2:’ (r = s, 0, u) can be projected
out by taking the appropriate
traces: (2) Ae = $Tr[yaP],
(3) (4)
As mentioned Hartree-Fock
in the Introduction,
approximation
in this work we want to go beyond
to the self-energy,
visualized
by the diagrams
the usual a and b of
Fig. 1, and pay special attention to the 2 hole-l particle (2hlp) and 2 particle-l hole (2plh) diagrams displayed in Figs. lc-f. Therefore, the different components of the self-energy
are complex functions
and, in general, we will write them as
2:’ = V’ + iW’.
(5)
Previous works have focused on the imaginary part of the self-energy at the on-shell energy of the propagating nucleon. In Ref. [ 131 the 2plh and 2hlp direct contributions where studied, while the work of Ref. [ 151 considered also the Fock-exchange terms. This study, however, was restricted to the on-shell 2hlp contributions. In the present work we extend these calculations and evaluate the direct and exchange terms for both 2hlp and 2plh contributions to the nucleon self-energy considering offshell effects, i.e. investigating the self-energy .Z(ke, k) for all combinations of energy ka and momentum k. In a first step we calculate the imaginary part of the self-energy, as described below, from which we later obtain the real part by means of a dispersion relation. The nucleon-nucleon interaction is derived from a G-matrix evaluated within the Dirac-Brueckner-Hartree-Fock approach. This G-matrix is parameterized in terms of an exchange of effective g and w mesons [ 8,171. The starting point of our discussion will be the Hartree-Fock (HF) approach. Therefore, and for the sake of simplifying the notation, we will define the following effective quantities in terms of the Hartree-Fock components
of the self-energy:
M*(k)
= M+ Z&(k),
(6)
k* = k(1 + a&(k)),
(7)
E*(k)
= di&%,
(8)
k*-(k;,k*)=(ko+ZO,,(k),k*), W(k)
= E*(k) - p&(k).
The nucleon G(k)
=
HF propagator
(9) (10)
is then
‘*2E*(k) +M*(k)g(k) ’
(11)
where
B(E*(k)-E;)
+
g(k)=k; - E*(k) +iq
1
e(E; -E*(k))
with EF = dv.
The first two terms on the right-hand
refer to the propagation
of solutions
of the Dirac equation
positive energies. These two terms distinguish i.e. states with energies
(12) side of this equation in nuclear
between the propagation
matter
with
of particle states,
above the Fermi energy, and hole states in the Fermi sea. The
last term in Eq. ( 12) refers to the propagation of solutions of the Dirac equation negative energies. In calculating the 2plh and 2hlp contributions to the nucleon
with self-
energy, this propagation of negative energy solutions will be ignored. This is done to be consistent with the propagators used in solving the NN scattering equations, where the terms are ignored as well [ 31. Recent studies of de Jong and Lenske [ 181
corresponding
demonstrate that the explicit inclusion of the propagation of negative-energy states does not affect the qualitative results of relativistic Brueckner-Hartree-Fock calculations. In the calculation of nucleon propagators for the 2hlp and 2plh terms we set _Z& = 0 and disregard the momentum dependence of xi?r, and _Z& for which we take their values at the Fermi momentum kF. This approximation relies on the weak momentum dependence of the Hartree-Fock self-energies [ k* = k, and the only effective quantity that depends on the nucleon momentum is EC, which from now on will be labeled with a subindex. It is also useful to split up the
fermion-propagator into a particle and a hole term and consider explicitly all terms of Fig. 1 by taking the following prescriptions for the nucleon propagators in the diagrams: k’+A4*
Particle line :
iG, (k) = ip 2E*
Hole line :
iG,,(k)
where the g-functions gp(k) = These
= &$+g,Jk), :
@
g-functions
are formally applying
of Eqs. (13) and (14), iZx;ah(kov k) = i
(14)
for particle and hole states are defined as
k,* - E;t + iv ’
tors. Therefore,
(13)
&(k)’
O( E; - E;;)
g/I(k) =
k,* - Ei -iq’
similar
to the non-relativistic
the standard
Feynman
rules together
we obtain a general expression
J
d4q
d4p*
HF nucleon propagawith the prescriptions
for the 2plh contribution
--MMX;ab(k,p,q)gp(P+q)g/~(P)gp(k-q)~ c2Tj4 (TT>4
(15)
while the formula for the 2hlp state is identical to the above except for the exchange of the p and h subscripts in the g-functions. The subscript X stands for D (direct) and E (exchange) diagrams, while a and b stand for the different meson types. The function MXiab( k, p, q) contains, basically, the Dirac structure of the interaction and, for direct terms, it has the general form
A. Trasobares ef al./Nuclear Physics A 640 (1998) 471-489
416
-h%
MD;odk Pt 4) =
8E*E*
rb(‘l)‘b(q)Tr[rb(‘l)(f
E*
d* + M*)ro(q)
+
P P+9 k-q
x(Id* +M*)l(r The constant
AI is the isospin degeneracy
the meson-nucleon
coupling
equivalent
cut-off
constants.
vertices are denoted by r
mass of _4 = 1500 MeV, has been attached
to modifying
(16)
and comes from the loop trace, g, and gb are The meson-nucleon
by A. A form factor of the type F(q)
and meson propagators a typical
-d* +M*)A,(q)~o(q).
the meson propagators
= n*/(n*
- q*), with
to each vertex. This is
in the following
way:
4(s>~*W
A,(q) -
(17)
Note as well the minus sign that comes from the fermion loop. For exchange terms one can write
ME;adk
s2,gt
Pv q> =
8EeEq
rb(k-q-P)Ab(k-q-p)(ti*+d*+M*)
Ee
P
pfq
k-q
xTa(q)(p*
+M*)rb(k-q
-p)(r
-d*
+M*)&(q)r&q). (18)
The imaginary
part of the self-energy
@(k,p,q)
(2a)4
- p;)B(k;
x@po*
- Q&PO*
(19)
(2a)4
the factor 0 has the following =e(E;
as
d4q d4p* ---Mx;ab(k,p,q)@(k,p,q),
WX;ab(kO, k) = 4r3 where, for 2plh,
is obtained
form:
- qo - E;)b’(p; + 40 - $+,vw,*
+ qo -E;)
(20)
- qo - E;;_,)
and, for 2hlp,
Wk, p, 4) = -e(p; - E;>e
- E;)&P;
k,*+ qo)w;
+ qo -
E;+qP(k;
- P; - 90) - qo - E;;-,I.
(21)
One can now use the step functions to define the boundaries for the integrations over dqo and dpo. Working in spherical coordinates we can automatically perform one of the axial integrations (say over the axial q-angle). By rewriting the first S-function in terms of the 3-momentum variable, the dp integration can be easily performed. After that we obtain, for the 2plh diagrams, k;-E;
1 WX;ab(kO>
k)
=
-
2(2a)4
J 0
/dp,‘p/&pd(co~e dqo J dqq*d(co%)
xMx;odwq)E;~(pg*
E:
Pa
+ qo -
E;+q)W,* -
qo
-E;_,),
(22)
A. Trasobares et al./Nuclear
where p takes the value p = dv, p0 = max(Eg
Physics A 640 (1998) 471-489
417
and
- qo,M*).
(23)
For the 2hlp contribution one obtains a similar expression up to an overall minus sign. The polar integration over the angle ey between q and the external momentum k (taken along the z-axis) 6(k,” - qo -
is readily performed
E;-,,
by using the delta function
6 COSekq-
2k;qo
-
q2 +
M*2 _
k*2
2kq
which, in order to keep 1cos 6&l < 1, imposes the following constraints
to the q variable:
q- 64<4+, where
The integration over the polar angle 8, can also be performed very easily by referring the two angular variables of the momentum p (qpp,0,) to a reference frame in which q acts as the z-axis and using the remaining a-function
%a(
&Pi +40-E;+,)
= pq
c0se,,
-
+q2> .
2P,*c?o
2Pq
The requirement that the absolute value of the cosine must be less than one puts restrictions to the values of p;, similar to those obtained for q, but more complicated to be implemented analytically due to the two different sign possibilities for q2, For this reason, we have taken care of the restrictions over p; numerically through an explicit step function inside the integral. After all these considerations of the self-energy.
WX;adkov
we can write our final expression
for the imaginary
part
For the 2plh state we have
k) = x~x;ub(k~rk,PO*,qo,q,~~),
(24)
and for the 2hlp state
Wx:rrdkork)
=
-’ 2( 2r)4k k;-E;
q-
A. Trasobares et al. /Nuclear Physics A 440 (1998) 471-489
478
where it is understood that fi does not contain the energies in the denominators of Eqs. ( 16) and ( 18) because they have canceled out with the energy factors that appeared on rewriting
the S-functions.
3. Results and discussion In the first part of this section we will concentrate
the discussion
of results on the
example of nuclear matter at a density close to the saturation density, in particular we will consider a Fermi momentum k~ of 1.4 fm-‘. We are going to discuss the contribution of the diagrams of second order in the NN interaction to the various components of the imaginary part in the nucleon self-energy. The NN interaction is described in terms of the exchange of an effective scalar, u, meson and an effective vector, w, meson. The masses of these mesons are fixed to the masses of the corresponding mesons in a realistic meson exchange model of the NN interaction (m, = 550 MeV, m, = 782.6 MeV), while the effective coupling constants have been adjusted [ 81 such that a Dirac-Hartree-Fock calculation of nuclear matter at kF = 1.4 fm-’ using these constants would reproduce results of a Dirac-Brueckner-Hartree-Fock (DBHF) calculation employing Bonn A NN potential [ 31. As described in more detail in Ref. [8], the coupling constants have been adjusted such that the scalar part of the self-energy for a nucleon with momentum k = kF, as well as the total binding energy, are reproduced. This DiracHartree-Fock parameterization of the DBHF results yields, at this density, coupling constants of g@ = 8.531 and g, = 9.539 for the (T and the w meson, respectively (see Table 1 of Ref. [ 81). It is worth noting that these effective coupling constants are smaller than the corresponding coupling constants of the bare NN potential, which for the Bonn A potential are gl = 10.22 and g,” = 31. This reduction of the bare coupling constants to the effective ones can be interpreted as describing the effects of short range correlations
in the wave function
of the two interacting
nucleons.
This would explain
that the quenching is larger for the w meson than for the (+ meson: The mass of the w is larger, which corresponds to a shorter range for the interaction which makes it more sensitive
to short range correlations.
Calculating
the 2hlp and 2plh contributions
to the self-energy as terms of second order in this effective meson exchange interaction implies that we account for the effects of the strong scalar and vector meson exchange terms in a realistic NN interaction (modified by NN correlations). An important piece missing in the present approach are the effects due to one-pion exchange. The single-particle Green function G of Eq. ( 11) for the propagation of the intermediate states is also defined employing the results of this DBHF calculation. This means that we consider a HF self-energy defined in terms of Xi?ir = -374.9
MeV,
$?uF = -289.8
MeV.
As a first example we show in Fig. 2 the imaginary
(26) part of the self-energy
calculated
A. Trasobares et al. /Nuclear Physics A 640 (1998) 471-189
479
t~~‘,~~‘,“‘,“~,“‘,“‘,“‘l -400
-200
0
200
400
600
800
1000
E[MeV]
Fig. 2. Imaginary part of the self-energy calculated for a fixed momentum k = kF = 1.4 fm-’ as a function of the energy variable, which is normalized such that it is zero at the Fermi energy. Results are displayed for the scalar part ( W’, solid line), the time-like vector part ( W”, dashed line) and the space-like vector component ( W”, dashed dotted line).
for nucleons with a momentum Note that the energy variable
fixed to k = kF = 1.4 fm-’ as a function of the energy. in this figure is normalized such that an energy zero
corresponds to the Fermi energy l,Q= EHF( kF) as defined in ( 10). Results are displayed for the scalar component, Im,Y = W’, the time-like vector component, Imp = W”, and the space-like vector component, kIm 3’ = W”. All these components are negative at energies below the Fermi energy and positive for those above. They are much larger at positive energies, reflecting the fact that the phase space of 2plh states with this momentum k is considerably larger than the corresponding phase space of 2hlp states. The scalar and time-like vector part are of similar size and exhibit the same sign. Using our notation this means that these contributions cancel each other to a large extent in calculating the expectation value for a Dirac spinor u representing a nucleon, i.e. a solution
of the Dirac equation
fi(k)ImZ(ka,k)u(k)
at positive energy:
= -W”(ko,k)
+ EW(ko,k) 4
+ g,
(27) k
where ii= kW”(ko,k)
(28)
and Ei, M*, k* are defined in Eqs. (6)-( 8). The Dirac spinors u are normalized such that .t, = 1. As the absolute value of Wc’ is always larger than the absolute value of WS at the same energy, this expectation value, which is roughly proportional to WS - Ws, shall be positive at energies below the Fermi energy and negative above, as is the case for the imaginary part of the self-energy calculated within a non-relativistic frame. The space-component of the vector part, k, is significantly smaller than the other terms.
480
A. Trasobares et al. /Nuclear
lo-
,/
_
/
/ ,-
\
Physics A 640 (1998) 471-489
\
scalar part - - vector part - - vectorpart (space) -
\
kF=1.4
fm-l
\
-2o-
\ I \ /
”
Zhlp
Contribution-
.,O~,...,.,.,,,,..,...,,,,~,,....,..,,,.... -400
-300
-200
0
-100
100
200
300
400
E[MeV]
Fig. 3. Real part of the 2hlp contribution to the self-energy for a fixed momentum as a function of energy. For further details see Fig. 2.
This difference, however, is not as large as one finds, e.g., in the real part of the selfenergy calculated within the Dirac-Hartree-Fock approach. Similar results are obtained for other momenta and nuclear densities. From the 2hlp contribution to the imaginary part, i.e. the one at energies below the Fermi energy E; one can determine the corresponding contribution to the real part by applying
the dispersion
relation 0
L%4&,(0,k)
=ReSZ&,(W,k)
= E
I
dw’
Irn ~&p(m’, w-uf
k, ’
-cm
where the P is used to indicate s, 0 and U, referring the following
the principal
value integral and the index a represents
to the scalar and vector components
the energy variables
w are redefined
of 2’. Note that here and in
such that w = 0 corresponds
to the
Fermi energy ko = EF. Typical examples for the real part of the 2hlp contribution
to the
self-energy are displayed in Fig. 3. The energy dependence of these terms can easily be understood from the energy dependence of the imaginary part, shown in Fig. 2, and the dispersion relation of Eq. (29). All three contributions are negative for energies between -100 MeV and the Fermi energy, which are typical energies for quasihole states, as well as for positive energies, i.e. particle states with energies above the Fermi energy 6~. As the absolute value of the time-like vector component Re S_$&, (dashed line) is consistently larger than the corresponding scalar component (solid line), we obtain a repulsive contribution to the quasiparticle energy, arising from this 2hlp term from energies starting around - 100 MeV below the Fermi energy up to infinity. This repulsive contribution is largest for quasihole states with energies below EF and will decrease for energies above EF with increasing energy. It should be noted that the space-like vector component exhibits a sizable contribution.
A. Trasobares et al./Nuclear
Physics A 640 (1998) 471-489
481
500
400
7
300
” 3
200
scalar part
-
-
vector
-
- - vector
part part (space)
100
E[MeV]
Fig. 4. Real part of the 2plh contribution For further details see Fig. 2.
to the self-energy
for a fixed momentum
as a function
of energy.
The real part of the 2plh contribution to the nucleon self-energy can be calculated from the corresponding imaginary part by a dispersion relation rather similar to Eq. (29), co
Re8ZjPn,(W,
k) = -f
I
dw’
Im-Z&h(J, k) w-co’
.
(30)
0
We already discussed above that the 2plh states lead to larger contributions to the imaginary part than the 2hlp terms. Therefore it is clear that the 2plh contributions to the real part of the self-energy are significantly larger than those originating from 2hlp terms. This can be seen in Fig. 4, which employs a scale which is about a factor 10 larger than the one used in Fig. 3 to visualize the 2hlp contributions. In the energy range of interest all three terms are positive. This means that the 2plh contribution to the scalar part of the self-energy .Z” tends to compensate the negative Hartree-Fock contribution to this term originating mainly from the exchange of two correlated pions, which is parameterized in the realistic OBE potentials by means of the u meson
exchange.
A similar
situation
also arises in the case of the time-like
vector
component of the nucleon self-energy 9: the negative Hartree-Fock contribution to Zc’, which is mainly due to the exchange of the w meson, is compensated to some extent by the 2plh terms. This means that the 2plh terms tend to reduce the Hartree-Fock contributions to the various terms in the self-energy while the 2hlp corrections yield contributions to Z” and _Ze with the same sign as the Hartree-Fock terms. Similar results are obtained for other momenta k. Since, however, our Hartree-Fock approximation to the self-energy has already been extracted from a DBHF calculation, we are not allowed to simply add the real part of 2plh contribution to the self-energy to the corresponding DBHF results. This would lead to a double counting of these 2plh terms. Instead we use a subtracted dispersion
A. Trasobares et al. /Nuclear
482
Physics A 440 (1998) 471-489
relation defined by -Re6~$,lh(~~~(k)
---~F,kj
(31)
to the various terms LYin the self-energy
as defined in (30).
8V2npn,(~,k) =ReSx&,(@,k) using the real contributions This subtraction
ensures
that the real part of the self-energy
remains
identical
DBHF result, if only the 2plh term is considered. With these definitions a quasiparticle self-energy, which is real and energy dependent, to be V$(&
k) = -Z&(k)
and we can determine
+ ~V&h(&
a quasiparticle
k) + ~V&(~,
k)
to the
we now define
(32)
spinor
(33)
with
q,,=E&,-l$,(,,,k)
-E,7.
(34)
This means that this spinor is an eigenstate
of the Dirac equation
[(l+V$,)~~k+y”(M+V~)-V&r,=~qpr+, which uses the self-energy VW of (32) calculated the quasiparticle energy defined by e,(k)
(35) at the energy w which corresponds
= E& - i@cow,k).
to
(3’5)
If for the moment we ignore the 2hlp contribution, SK&,, to the quasiparticle self-energy in (32)) the subtraction defined in (3 1) ensures that the SV&, terms vanish on-shell and the quasiparticle
energy coincides
with the HF energy, thus avoiding
double counting.
However, when the contribution of the 2hlp terms is taken into account, the selfconsistent definition of the energy variable wqp gives rise to a non-vanishing correction due to the energy dependence of the 2plh terms. Therefore the resulting contribution of the calculated correction to the real part of the 2plh term originates from the dispersive correction of the self-energy which is induced from the 2hlp term and therefore goes beyond the conventional Brueckner Hartree Fock approach. Results for the quasiparticle energy are displayed in Fig. 5 for various momenta k. It can bee seen that the inclusion of the 2hlp terms yields a significant increase in the quasiparticle energy eqp as compared to the corresponding HF result. This increase is as large as 17.5 MeV for momenta close to zero, reduces to values around 5.3 MeV for
A. Trasobares et al. /Nuclear
-
-
-
Physics A 640 (1998) 471-489
483
HartreemFock Quasiparticle
/ ---I
II
0.0
8 5’ 1 c
1.0 k
/
”
I 2.0
kF
Fig. 5. Results for the quasiparticle energy, calculated according to eq. (36), are compared energies calculated in the HF approximation for the self-energy (dashed line).
to the corresponding
k = kF and becomes negligibly small around k = 2kp. This repulsive effect terms in the quasiparticle energy has been discussed already in context with a large effect for the quasiparticle energy may indicate that the 2hlp terms have some effect in the calculation of the total energy. For the calculation
of the 2hlp Fig. 3. Such may as well of the total
energy, however, it is not sufficient to evaluate the quasiparticle energy, but one would need the whole spectral distribution of hole strength for states with momenta below and above the Fermi momentum [ 191. It is one aim of our study to explore the effects of the Dirac structure of the nucleon self-energy, calculated beyond the mean field approximation, on the complex optical potential for nucleon-nucleus scattering. The Dirac equation reads now [(l+~~)cu.k+y’(M+~,)-Z0,]u,=~~,~~
(37)
where the full complex self-energy is used. Real C9p solutions of Eq. (37) involve complex values of k. We take as approximate solutions of Eq. (37) the values lqp determined from Eq. (36). It is convenient to rewrite Eq. (37) into a form which only contains an effective scalar V” and vector potential V” [20] [(Y. k + y’(A4 + V”) - V”] uqp = eqpuqp, where
(38)
A. Trasobares et al. /Nuclear Physics A 640 (1998) 471-489
484
Fig. 6. The real parts of the bare scalar and vector components (Re Z&, and Re 2&), represented by the dashed line, are compared to the real parts of Vs and V”, renormalized according to Eq. (38).
The self-energy
terms _Z& have been calculated
in the quasiparticle
approach
of (32)
using the self-consistent relation between the three-momentum and the energy as defined in (34). This Dirac equation can be transformed into a Schrodinger-type equation for the large component of the Dirac spinor, leading Schrijdinger equivalent potential of the form
u = V”
- p+
&- [(lq2
- (vo>2].
to a complex
and energy-dependent
(39)
Results for the real part of the renormalized components Vs and V” are displayed in Fig. 6 as a function of the energy variable wqp, i.e. the quasiparticle energy Q, normalized such that the Fermi energy occurs at zero. For a comparison we also show the unrenormalized components V$. The difference between the solid and dashed line is a measure for the importance of the space-like vector component Z’ of the self-energy. We find that this space-like components yield a slight reduction of the absolute values for RealV, which is of the order of 3%. Larger effects only occur at negative energies, below the Fermi energy, where the 2hlp contribution gets more important. This figure also demonstrates that the energy-dependence of the various Dirac components in the real part of the self-energy remains weak, again with the exception of energies below EF. Also the deviations from the Hartree-Fock result, -375 MeV and -290 MeV for the scalar and vector parts, respectively, are not very pronounced. This can also be seen from the left part of Fig. 7, exhibiting the real part of the Schrodinger equivalent potential U defined in (39). The results derived from the quasiparticle approximation (solid line) including the effects of the energy dependence in the 2pl h and 2hlp terms, exhibit a dependence on the quasiparticle energy, which is very similar to the one derived from the HF approximation. Therefore one may conclude that the
A. Trasobares et al./Nuclear
Physics A 640 (1998) 471-489
-80 ~....,....,....,....,....~....,....,....,....,.,..~-3o -50 0 50 100 150
-50
0
Energy[MeV]
50
100
485
150
Energy[MeV]
Fig. 7. Energy dependence of the real part (left) and imaginary part (right) of the Schrirdinger equivalent potential. The result for the real part is compared to the corresponding prediction obtained in the HF approximation. The dashed curve in the right part of the figure exhibits the expectation value of W calculated according to Eq. ( 40).
energy-dependence in the depth of central, Woods-Saxon type, optical potentials, used to describe nucleon-nucleus scattering is mainly due to the relativistic structure of the underlying Hartree-Fock self-energy to be used in a Dirac equation. Dispersive effects due to the consideration of 2plh and 2hlp terms also lead to an energy dependence, which, however, is much smaller. These dispersive corrections tend to make the potential slightly more attractive at higher energies. It should be mentioned, however, that an energy dependence of the central Schrodinger potential similar to the one obtained here within the relativistic scheme can also be obtained within a non-relativistic Hartree-Fock due to large non-localities in the Hartree-Fock potential [ 211. An analysis rather similar to the one just outlined for the real components of the self-energy can also be performed for the imaginary parts. Results for the renormalized Dirac components W” = Imag V” (a = s and 0) are presented in Fig. 8 and compared to the bare scalar and vector imaginary
components.
beginning
of this section,
part of the bare components
energies
below the Fermi
positive energies
(2plh).
are much more important
the imaginary energy
(2hlp),
The renormalizing for the imaginary
zero around
As we discussed
already
at the
are negative
for
w = 0 ( EF) and positive
for
effects of the space-like
vector components
part than for the real part. This can be
deduced from the differences between W” and W”: the bare components are as large as twice the renormalized quantities. The imaginary part of the Schradinger optical potential [ Eq. (39) ] is shown on the right of Fig. 7. An alternative way of evaluating this imaginary part would be to calculate the expectation value of the Dirac operator ii,
[W” - row” + y . kWU] uqp
(40)
using the quasiparticle Dirac spinors of (33). These expectation values lead to the dashed line on the right part of Fig. 7. We see that the results for the imaginary part of
486
A. Trasobares et al. /Nuclear
_:,,, -50
Physics A 640 (1998) 471-489
,,,,
,,,, 0
,,,, 100
50
,,,, 150
200
Energy[MeV] Fig. 8. Bare and renormalized scalar and vector components of the imaginary part of the quasiparticle self-energy. For further details see Fig. 6.
the Schrodinger potential are rather insensitive on the way of calculation. These results for the imaginary part of the Schrodinger potential may be compared with the volume contribution to the absorption part of the nucleon-nucleus optical potential. As an example we consider the fit to the elastic scattering data of protons on “aCa of Tornow et al. [ 22,231. They obtain a volume term of -7.8 MeV at an energy of 80 MeV for the incoming proton. For the same energy we get -8.3 MeV which is in fair agreement, keeping in mind that we performed our calculation in nuclear matter. Also the energy dependence of the imaginary part displayed in Fig. 7 is similar to the empirical fit. In particular, we obtain an almost symmetric behavior around the Fermi energy. In the calculation of the imaginary components tion 2 and consequently also in the corresponding mined
the direct and exchange
explore the importance results obtained with (solid
contributions
discussed in Secwe always deter-
(see Eqs. ( 16) and ( 18)).
In order to
of the 2plh and 2hlp exchange terms we compare in Fig. 9 the
for the real and imaginary
lines)
of the self-energy real components,
and without
(dashed
part of the Schrodinger
lines)
inclusion
equivalent
of the exchange
potential
terms in the
2pl h and 2hlp parts of the self-energy. One finds that the effects of the exchange terms on the real part are rather small typically around 1 MeV However, the effects are significantly larger for the imaginary part where the difference gets as large as 30% of the total result. This can be understood from the fact that the real part of the self-energy is dominated by the Hartree-Fock contribution, which is identical in these two approaches. If one ignores the exchange terms in calculating the 2plh and 2hlp terms, one may consider it more consistent to ignore the exchange term also in the leading contribution and replace the Hartree-Fock approximation by the Hartree approach. In Ref. [ 81 effective meson-nucleon coupling constants were determined to reproduce the DBHF results within a Dirac-Hartree model. The resulting coupling constants are a bit larger than
A. Trasobares et al. /Nuclear
Physics A 640 (1998) 471-489
487
-30 -50
0
50
100
150
-50
0
Energy[MeV]
50
100
150
Energy[MeV]
Fig. 9. Real and imaginary parts of the SchrGdinger equivalent potential, calculated using the full model (solid lines), ignoring exchange diagrams in the evaluation of the terms of second order (dashed line), and using the Hartree approximation also for the lowest order term (dashed-dotted line).
those derived from the Dirac-Hartree-Fock analysis. If we use these Hartree coupling constants and ignore the effects of exchange terms in the leading term as well as in the 2plh and 2hlp terms, one arrives at a Schrodinger equivalent potential displayed by the dash-dotted lines in Fig. 9. The differences to the solid lines get now quite pronounced for the real part. This can be traced back to the fact that in the Dirac-Hartree
approxi-
mation _Z” and 9 are independent on the momentum or energy of the state and Z” is identical to zero. The results for the imaginary part obtained in this approach, however, are rather close to those evaluated with inclusion of the exchange contributions. It is also interesting to investigate the density dependence of the self-energy in order to extend these calculations to finite nuclei. As an example we present some results obtained for nuclear matter with a Fermi momentum kF = 1.2 fm-’ in Fig. 10. The results obtained at these various densities, either for the self-energy keeping track of the Dirac structure, or using the Schrodinger equivalent potentials derived from these components,
may then be used in a local density
nucleon-nucleus
scattering.
Such investigations
approximation
for a prediction
of
are in progress.
4. Conclusions The relativistic structure of the self-energy for a nucleon in nuclear matter is investigated by including all irreducible terms of first and second order in the residual interaction. For the NN interaction a parameterization of the Dirac-Brueckner-HartreeFock (DBHF) G-matrix in terms of the exchange of effective scalar and vector mesons has been used. The 2plh and 2hlp contributions to the imaginary part of the self-energy are evaluated keeping track of all direct and exchange terms. The corresponding 2plh and 2hlp contributions to the real part are derived from these imaginary components by means of dispersion relations. A subtracted dispersion relation must be used for the
488
A. Trasobares et al. /Nuclear
Physics A 640 (1998) 471-489
Fig. 10. Real and imaginary parts of the Schriidinger equivalent potential calculated for nuclear matter with a Fermi momentum of kF = 1.4 fm-’ (solid line) are compared to those obtained for k,c = 1.2 fm-t.
2plh term to avoid double-counting on which these studies are based. The inclusion of 2hlp diagrams
with the G-matrix in the evaluation
underlying
the DBHF approach
of the real part of the self-energy
yields a non-negligible modification of the scalar and vector components in particular for states with momenta below the Fermi momentum. Also the value of the quasiparticle energy is increased by a value as large as 17.5 MeV for momenta close to zero and to values around 5.3 MeV for k = kF. Such a large effect for the quasiparticle energy may indicate that the 2hlp terms should have some effect in the calculation of the total energy. The calculated self-energy can also be transformed into a Schrijdinger equivalent optical potential, to be used in the study of nucleon-nucleus scattering. Exchange diagrams are non-negligible in the evaiuation of the imaginary part. The energy or momentum dependence of the central component of the real potential, however, is dominated by the effects of the Dirac-Hartree-Fock contribution. The 2plh and 2hlp terms give rise to a more attractive SEP at positive energies and introduce which is very weak.
an additional
energy dependence
Acknowledgements This project has been supported
by the Spanish research grant DGICYT, PB95-1249, and by the EC contract CHRX-CT93-0323. One of us (H.M.) is pleased to acknowledge the warm hospitality at the Facultat de Fisica, Universitat de Barcelona, and the support from the program of visiting Professor at this university.
References [ 11 B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. [2] M.R. Anastasia, L.S. Celenza, W.S. Pong and CM. Shakin, Phys. Rep. 100 (1983) 327.
A. Trasobares et al. /Nuclear [3] [4] [S] [6] [7] [S] [9]
[lo] [ 111 [ 121 1131 ]14] [ 151 [ 161 [ 171 ] 181 [ 191 [20] [2l] [ 221 [23]
Physics A 640 (1998) 471-489
489
R. Machleidt, Adv. Nucl. Phys. 19 (1989) 189. R. Brockmann and R. Machleidt, Phys. Rev. C 42 ( 1990) 1965. B. ter Haar and R. Malfliet, Phys. Rep. 149 (1987) 207. F. &ester, S. Cohen, B.D. Day and C.M. Vincent, Phys. Rev. C 1 (1970) 769. K. W. Schmid, H. Miither and R. Machleidt, Nucl. Phys. A 530 (1991) 14. R. Fritz and H. Mtither, Phys. Rev. C 49 (1994) 633. F. Boersma and R. Malfliet, Phys. Rev. C 49 (1994) 1495. H. Mtither and L.D. Skouras, Nucl. Phys. A 581 (1995) 247. M.F. Jiang, T.T.S. Kuo and R. Machleidt, Phys. Rev. C 43 (1991) 1469. M.F. Jiang, T.T.S. Kuo and H. Mtither, Phys. Rev. C 38 (1988) 240. C.J. Horowitz, Nucl. Phys. A 412 (1984) 228. S.A. Chin, Ann. Phys. 108 (1977) 301. M. H. Steffani, M. Betz and Th.A.J. Maris, Nucl. Phys. A 493 (1989) 493. Y.L. Han, Q.B. Shen, Y.Z. Zhuo and T.M. Geng, Nucl. Phys. A 569 (1994) 732. H. Elsenhans and H. Miither, Nucl. Phys. A 515 (1990) 715. F. de Jong and H. Lenske, nucl-th/9709012. W.H. Dickhoff and H. Mtither, Rep. Prog. Phys. 11 (1992) 1947. T. Mitsumori, N. Noda, K. Koide, H. Kouno, A. Hasegawa and M. Nakano, Prog. Theor. Phys. 96 (1996) 179. M. Kleinmann, R. Fritz, A. Ramos and H. Mtither, Nucl. Phys. A 579 (1994) 85. W. Tornow, Z.P. Chen and J.P. Delaroche, Phys. Rev. C 42 (1990) 693. J.P. Jeukenne and C. Mahaux, Nucl. Phys. A 394 (1983) 445.
On the Dirac structure of the nucleon self-energy in nuclear matter A. Trasobares”,
A. Polls a, A. Ramosa, H. Miitherb
a Departament d’Estructura i Constituents de la Matkia Universitat de Barcelona, E-08028 Barcelona, Spain b Institut fir Theoretische Physik, Vniversitiit Tiibingen, D-72076 Tiibingen, Germany
Received 10 November 1997; revised 22 June 1998; accepted 14 August 1998
Abstract The relativistic structure of the self-energy of a nucleon in nuclear matter is investigated including the imaginary and real components which arise from the terms of first and second order in the NN interaction. A parameterized form of the Brueckner G-matrix is used for the NN interaction. The effects of the terms beyond the DBHF approximation on quasiparticle energies and the optical potential for nucleon-nucleus scattering are discussed. @ 1998 Elsevier Science B.V.
1. Introduction During
the last few years the attempts
clear systems
from a realistic
to derive the ground-state
nucleon-nucleon
(NN)
interaction,
properties
of nu-
have been promoted
very much by the understanding that relativistic effects may be non-negligible in such investigations. These ideas were originally developed within the various versions of the phenomenological Walecka model [ 11. The Dirac structure of a strong repulsive component originating from the exchange of an attractive component of medium range described in terms leads to a self-energy of the nucleon in the nuclear medium
the NN interaction with the w vector meson and of a scalar meson ((T) which contains a large
scalar component 2” and a large time-like vector component 2?. These two components compensate each other to a large extent if one calculates the single-particle energy. This leads to the well-known fact that the binding energy of nuclei are very small as compared to the rest mass M of the nucleon, a fact which has often been used to argue that relativistic effects should be small in the many-body problem of nuclear physics. The individual components of the self-energy 27 are of the order of the nucleon rest mass 0375-9474/98/$ - seefront matter @ 1998 Elsevier Science B.V. All rights reserved. PIISO375-9474(98)00471-O
A. Trasobares et al. /Nuclear
412
(typically
one third of M) . Therefore
medium
is modified
nucleon.
This medium
the NN interaction
Physics A 640 (1998) 471-489
the structure of the Dirac spinors in the nuclear
to quite some extent as compared dependence
in the medium,
These more or less empirical
to the Dirac spinor for a free
of the Dirac spinors, which affects the evaluation leads to a saturation
mechanism
models received support from calculations
from a realistic One-Boson-Exchange
model of the NN interaction
realistic NN interaction means that the parameters contained to describe the experimental data of free NN scattering.
of
for nuclear matter. which start
[ 2,3]. In this context
in these models are adjusted Using such a realistic NN
interaction in a many-body calculation of the Dirac-Brueckner-Hartree-Fock (DBHF) type, results were obtained for the saturation point of nuclear matter which were in quite a good agreement with the empirical saturation point [4,5]. Such a DBHF calculation accounts for the effect of correlations on the level of the BHF approximation, i.e. the lowest order in the hole line expansion, and allows for the relativistic effects which we just outlined. This success of the relativistic features contained in the DBHF approach could be a solution of an old problem: the so-called Coester band phenomenon [6], which stands for the fact that many-body calculations based on various realistic models of the NN interaction lead to predictions for the saturation point of nuclear matter which either fail to yield enough binding or predict a saturation density twice as large as the empirical value. Such a Coester band can also be observed in non-relativistic studies of finite nuclei [ 71. Therefore it was quite obvious that attempts have been made to include the relativistic features also in DBHF calculations of finite nuclei. Indeed the inclusion of relativistic effects gave a substantial improvement in the calculated binding energies and radii of finite nuclei [ 8,9]. However, there remains still a discrepancy between the DBHF results and the experimental values. On the other hand it was observed that an extension of the non-relativistic BHF approach to a definition of the nucleon self-energy, which accounts in a symmetric way not only for the particle-particle ladders included in the Brueckner G-matrix but also for the corresponding hole-hole scattering term, may lead to a larger binding energy and a larger radius than obtained in the BHF approach [ lo]. A similar feature, moving the saturation point away from the Coester band by including hole-hole scattering terms, has also been observed in particle-particle hole-hole RPA calculation of nuclear matter [ 11,121. Taking a very optimistic point of view one may argue that the combination of relativistic effects and hole-hole scattering terms may lead to an improved microscopic understanding of ground state properties of nuclear systems. In the work presented here we would like to investigate the effects of particle-particle and hole-hole scattering terms on the relativistic structure of the nucleon self-energy. For that purpose we consider the nucleon self-energy defined with all terms of first and second order in the NN interaction (see Fig. 1). For the NN interaction we use a parameterization of a G-matrix, evaluated within the DBHF approach, in terms of effective (+ and w mesons [ 81. Note that this parameterization does not include the pion exchange explicitly, therefore we may miss part of the second-order tensor contribution, which is included only in an average way. Since this effective interaction has been
A. Trasobares et al./Nuclear
Physics A 640 (1998) 471-489
413
Fig. 1. Graphical representation of the Hartree (a), Fock (b), 2plh direct (c), 2plh exchange (d), 2hlp direct (e) and 2hlp exchange (f) contributions to the self-energy of the nucleon. The momenta labeling the various contractions in (c) correspond to the nomenclature used in Eq. (15). constructed from a DBHF G-matrix we have to use a subtracted dispersion relation to avoid a double counting of the 2 particle-l hole contributions to the real part of the self-energy. In contrast to earlier investigations [ 13-161 we evaluate the imaginary contributions to the nucleon self-energy allowing for all possible combinations of momentum k and energy k”. Therefore we can use dispersion relations to evaluate also the real part of the second-order diagrams. We investigate in detail the effect of these higher order diagrams on the Dirac structure of the self-energy. Furthermore we derive from this relativistic self-energy
an optical potential to be used in a Schradinger
equation for nucleon-nucleus
scattering. The technical details for the calculation of the imaginary components in the self-energy are presented in Section 2. The dispersion relations used to evaluate the corresponding real components are given in Section 3. That section also contains a detailed discussion of the results. The main conclusions
2. Imaginary
are summarized
in the final section.
part of the nucleon self-energy
The physical system well as rotational
of
or nuclear it
by definition, to
In general,
as be
written ’ as [l] kxY(k),
(1)
’ Notation: within formulae we shall use reman type (p,q, .) for 4-vectors, boldface type (p. q, 3-vectors and normal italic type (p,q, .) for the norm of 3-vectors.
.) for
Z(k) = -Z’(k) -rap(k)
+y.
A. Trasobares et al. /Nuclear Physics A 640 (1998) 471-489
474
where the functions
2:’ (r = s, 0, u) can be projected
out by taking the appropriate
traces: (2) Ae = $Tr[yaP],
(3) (4)
As mentioned Hartree-Fock
in the Introduction,
approximation
in this work we want to go beyond
to the self-energy,
visualized
by the diagrams
the usual a and b of
Fig. 1, and pay special attention to the 2 hole-l particle (2hlp) and 2 particle-l hole (2plh) diagrams displayed in Figs. lc-f. Therefore, the different components of the self-energy
are complex functions
and, in general, we will write them as
2:’ = V’ + iW’.
(5)
Previous works have focused on the imaginary part of the self-energy at the on-shell energy of the propagating nucleon. In Ref. [ 131 the 2plh and 2hlp direct contributions where studied, while the work of Ref. [ 151 considered also the Fock-exchange terms. This study, however, was restricted to the on-shell 2hlp contributions. In the present work we extend these calculations and evaluate the direct and exchange terms for both 2hlp and 2plh contributions to the nucleon self-energy considering offshell effects, i.e. investigating the self-energy .Z(ke, k) for all combinations of energy ka and momentum k. In a first step we calculate the imaginary part of the self-energy, as described below, from which we later obtain the real part by means of a dispersion relation. The nucleon-nucleon interaction is derived from a G-matrix evaluated within the Dirac-Brueckner-Hartree-Fock approach. This G-matrix is parameterized in terms of an exchange of effective g and w mesons [ 8,171. The starting point of our discussion will be the Hartree-Fock (HF) approach. Therefore, and for the sake of simplifying the notation, we will define the following effective quantities in terms of the Hartree-Fock components
of the self-energy:
M*(k)
= M+ Z&(k),
(6)
k* = k(1 + a&(k)),
(7)
E*(k)
= di&%,
(8)
k*-(k;,k*)=(ko+ZO,,(k),k*), W(k)
= E*(k) - p&(k).
The nucleon G(k)
=
HF propagator
(9) (10)
is then
‘*2E*(k) +M*(k)g(k) ’
(11)
where
B(E*(k)-E;)
+
g(k)=k; - E*(k) +iq
1
e(E; -E*(k))
with EF = dv.
The first two terms on the right-hand
refer to the propagation
of solutions
of the Dirac equation
positive energies. These two terms distinguish i.e. states with energies
(12) side of this equation in nuclear
between the propagation
matter
with
of particle states,
above the Fermi energy, and hole states in the Fermi sea. The
last term in Eq. ( 12) refers to the propagation of solutions of the Dirac equation negative energies. In calculating the 2plh and 2hlp contributions to the nucleon
with self-
energy, this propagation of negative energy solutions will be ignored. This is done to be consistent with the propagators used in solving the NN scattering equations, where the terms are ignored as well [ 31. Recent studies of de Jong and Lenske [ 181
corresponding
demonstrate that the explicit inclusion of the propagation of negative-energy states does not affect the qualitative results of relativistic Brueckner-Hartree-Fock calculations. In the calculation of nucleon propagators for the 2hlp and 2plh terms we set _Z& = 0 and disregard the momentum dependence of xi?r, and _Z& for which we take their values at the Fermi momentum kF. This approximation relies on the weak momentum dependence of the Hartree-Fock self-energies [ k* = k, and the only effective quantity that depends on the nucleon momentum is EC, which from now on will be labeled with a subindex. It is also useful to split up the
fermion-propagator into a particle and a hole term and consider explicitly all terms of Fig. 1 by taking the following prescriptions for the nucleon propagators in the diagrams: k’+A4*
Particle line :
iG, (k) = ip 2E*
Hole line :
iG,,(k)
where the g-functions gp(k) = These
= &$+g,Jk), :
@
g-functions
are formally applying
of Eqs. (13) and (14), iZx;ah(kov k) = i
(14)
for particle and hole states are defined as
k,* - E;t + iv ’
tors. Therefore,
(13)
&(k)’
O( E; - E;;)
g/I(k) =
k,* - Ei -iq’
similar
to the non-relativistic
the standard
Feynman
rules together
we obtain a general expression
J
d4q
d4p*
HF nucleon propagawith the prescriptions
for the 2plh contribution
--MMX;ab(k,p,q)gp(P+q)g/~(P)gp(k-q)~ c2Tj4 (TT>4
(15)
while the formula for the 2hlp state is identical to the above except for the exchange of the p and h subscripts in the g-functions. The subscript X stands for D (direct) and E (exchange) diagrams, while a and b stand for the different meson types. The function MXiab( k, p, q) contains, basically, the Dirac structure of the interaction and, for direct terms, it has the general form
A. Trasobares ef al./Nuclear Physics A 640 (1998) 471-489
416
-h%
MD;odk Pt 4) =
8E*E*
rb(‘l)‘b(q)Tr[rb(‘l)(f
E*
d* + M*)ro(q)
+
P P+9 k-q
x(Id* +M*)l(r The constant
AI is the isospin degeneracy
the meson-nucleon
coupling
equivalent
cut-off
constants.
vertices are denoted by r
mass of _4 = 1500 MeV, has been attached
to modifying
(16)
and comes from the loop trace, g, and gb are The meson-nucleon
by A. A form factor of the type F(q)
and meson propagators a typical
-d* +M*)A,(q)~o(q).
the meson propagators
= n*/(n*
- q*), with
to each vertex. This is
in the following
way:
4(s>~*W
A,(q) -
(17)
Note as well the minus sign that comes from the fermion loop. For exchange terms one can write
ME;adk
s2,gt
Pv q> =
8EeEq
rb(k-q-P)Ab(k-q-p)(ti*+d*+M*)
Ee
P
pfq
k-q
xTa(q)(p*
+M*)rb(k-q
-p)(r
-d*
+M*)&(q)r&q). (18)
The imaginary
part of the self-energy
@(k,p,q)
(2a)4
- p;)B(k;
x@po*
- Q&PO*
(19)
(2a)4
the factor 0 has the following =e(E;
as
d4q d4p* ---Mx;ab(k,p,q)@(k,p,q),
WX;ab(kO, k) = 4r3 where, for 2plh,
is obtained
form:
- qo - E;)b’(p; + 40 - $+,vw,*
+ qo -E;)
(20)
- qo - E;;_,)
and, for 2hlp,
Wk, p, 4) = -e(p; - E;>e
- E;)&P;
k,*+ qo)w;
+ qo -
E;+qP(k;
- P; - 90) - qo - E;;-,I.
(21)
One can now use the step functions to define the boundaries for the integrations over dqo and dpo. Working in spherical coordinates we can automatically perform one of the axial integrations (say over the axial q-angle). By rewriting the first S-function in terms of the 3-momentum variable, the dp integration can be easily performed. After that we obtain, for the 2plh diagrams, k;-E;
1 WX;ab(kO>
k)
=
-
2(2a)4
J 0
/dp,‘p/&pd(co~e dqo J dqq*d(co%)
xMx;odwq)E;~(pg*
E:
Pa
+ qo -
E;+q)W,* -
qo
-E;_,),
(22)
A. Trasobares et al./Nuclear
where p takes the value p = dv, p0 = max(Eg
Physics A 640 (1998) 471-489
417
and
- qo,M*).
(23)
For the 2hlp contribution one obtains a similar expression up to an overall minus sign. The polar integration over the angle ey between q and the external momentum k (taken along the z-axis) 6(k,” - qo -
is readily performed
E;-,,
by using the delta function
6 COSekq-
2k;qo
-
q2 +
M*2 _
k*2
2kq
which, in order to keep 1cos 6&l < 1, imposes the following constraints
to the q variable:
q- 64<4+, where
The integration over the polar angle 8, can also be performed very easily by referring the two angular variables of the momentum p (qpp,0,) to a reference frame in which q acts as the z-axis and using the remaining a-function
%a(
&Pi +40-E;+,)
= pq
c0se,,
-
+q2> .
2P,*c?o
2Pq
The requirement that the absolute value of the cosine must be less than one puts restrictions to the values of p;, similar to those obtained for q, but more complicated to be implemented analytically due to the two different sign possibilities for q2, For this reason, we have taken care of the restrictions over p; numerically through an explicit step function inside the integral. After all these considerations of the self-energy.
WX;adkov
we can write our final expression
for the imaginary
part
For the 2plh state we have
k) = x~x;ub(k~rk,PO*,qo,q,~~),
(24)
and for the 2hlp state
Wx:rrdkork)
=
-’ 2( 2r)4k k;-E;
q-
A. Trasobares et al. /Nuclear Physics A 440 (1998) 471-489
478
where it is understood that fi does not contain the energies in the denominators of Eqs. ( 16) and ( 18) because they have canceled out with the energy factors that appeared on rewriting
the S-functions.
3. Results and discussion In the first part of this section we will concentrate
the discussion
of results on the
example of nuclear matter at a density close to the saturation density, in particular we will consider a Fermi momentum k~ of 1.4 fm-‘. We are going to discuss the contribution of the diagrams of second order in the NN interaction to the various components of the imaginary part in the nucleon self-energy. The NN interaction is described in terms of the exchange of an effective scalar, u, meson and an effective vector, w, meson. The masses of these mesons are fixed to the masses of the corresponding mesons in a realistic meson exchange model of the NN interaction (m, = 550 MeV, m, = 782.6 MeV), while the effective coupling constants have been adjusted [ 81 such that a Dirac-Hartree-Fock calculation of nuclear matter at kF = 1.4 fm-’ using these constants would reproduce results of a Dirac-Brueckner-Hartree-Fock (DBHF) calculation employing Bonn A NN potential [ 31. As described in more detail in Ref. [8], the coupling constants have been adjusted such that the scalar part of the self-energy for a nucleon with momentum k = kF, as well as the total binding energy, are reproduced. This DiracHartree-Fock parameterization of the DBHF results yields, at this density, coupling constants of g@ = 8.531 and g, = 9.539 for the (T and the w meson, respectively (see Table 1 of Ref. [ 81). It is worth noting that these effective coupling constants are smaller than the corresponding coupling constants of the bare NN potential, which for the Bonn A potential are gl = 10.22 and g,” = 31. This reduction of the bare coupling constants to the effective ones can be interpreted as describing the effects of short range correlations
in the wave function
of the two interacting
nucleons.
This would explain
that the quenching is larger for the w meson than for the (+ meson: The mass of the w is larger, which corresponds to a shorter range for the interaction which makes it more sensitive
to short range correlations.
Calculating
the 2hlp and 2plh contributions
to the self-energy as terms of second order in this effective meson exchange interaction implies that we account for the effects of the strong scalar and vector meson exchange terms in a realistic NN interaction (modified by NN correlations). An important piece missing in the present approach are the effects due to one-pion exchange. The single-particle Green function G of Eq. ( 11) for the propagation of the intermediate states is also defined employing the results of this DBHF calculation. This means that we consider a HF self-energy defined in terms of Xi?ir = -374.9
MeV,
$?uF = -289.8
MeV.
As a first example we show in Fig. 2 the imaginary
(26) part of the self-energy
calculated
A. Trasobares et al. /Nuclear Physics A 640 (1998) 471-189
479
t~~‘,~~‘,“‘,“~,“‘,“‘,“‘l -400
-200
0
200
400
600
800
1000
E[MeV]
Fig. 2. Imaginary part of the self-energy calculated for a fixed momentum k = kF = 1.4 fm-’ as a function of the energy variable, which is normalized such that it is zero at the Fermi energy. Results are displayed for the scalar part ( W’, solid line), the time-like vector part ( W”, dashed line) and the space-like vector component ( W”, dashed dotted line).
for nucleons with a momentum Note that the energy variable
fixed to k = kF = 1.4 fm-’ as a function of the energy. in this figure is normalized such that an energy zero
corresponds to the Fermi energy l,Q= EHF( kF) as defined in ( 10). Results are displayed for the scalar component, Im,Y = W’, the time-like vector component, Imp = W”, and the space-like vector component, kIm 3’ = W”. All these components are negative at energies below the Fermi energy and positive for those above. They are much larger at positive energies, reflecting the fact that the phase space of 2plh states with this momentum k is considerably larger than the corresponding phase space of 2hlp states. The scalar and time-like vector part are of similar size and exhibit the same sign. Using our notation this means that these contributions cancel each other to a large extent in calculating the expectation value for a Dirac spinor u representing a nucleon, i.e. a solution
of the Dirac equation
fi(k)ImZ(ka,k)u(k)
at positive energy:
= -W”(ko,k)
+ EW(ko,k) 4
+ g,
(27) k
where ii= kW”(ko,k)
(28)
and Ei, M*, k* are defined in Eqs. (6)-( 8). The Dirac spinors u are normalized such that .t, = 1. As the absolute value of Wc’ is always larger than the absolute value of WS at the same energy, this expectation value, which is roughly proportional to WS - Ws, shall be positive at energies below the Fermi energy and negative above, as is the case for the imaginary part of the self-energy calculated within a non-relativistic frame. The space-component of the vector part, k, is significantly smaller than the other terms.
480
A. Trasobares et al. /Nuclear
lo-
,/
_
/
/ ,-
\
Physics A 640 (1998) 471-489
\
scalar part - - vector part - - vectorpart (space) -
\
kF=1.4
fm-l
\
-2o-
\ I \ /
”
Zhlp
Contribution-
.,O~,...,.,.,,,,..,...,,,,~,,....,..,,,.... -400
-300
-200
0
-100
100
200
300
400
E[MeV]
Fig. 3. Real part of the 2hlp contribution to the self-energy for a fixed momentum as a function of energy. For further details see Fig. 2.
This difference, however, is not as large as one finds, e.g., in the real part of the selfenergy calculated within the Dirac-Hartree-Fock approach. Similar results are obtained for other momenta and nuclear densities. From the 2hlp contribution to the imaginary part, i.e. the one at energies below the Fermi energy E; one can determine the corresponding contribution to the real part by applying
the dispersion
relation 0
L%4&,(0,k)
=ReSZ&,(W,k)
= E
I
dw’
Irn ~&p(m’, w-uf
k, ’
-cm
where the P is used to indicate s, 0 and U, referring the following
the principal
value integral and the index a represents
to the scalar and vector components
the energy variables
w are redefined
of 2’. Note that here and in
such that w = 0 corresponds
to the
Fermi energy ko = EF. Typical examples for the real part of the 2hlp contribution
to the
self-energy are displayed in Fig. 3. The energy dependence of these terms can easily be understood from the energy dependence of the imaginary part, shown in Fig. 2, and the dispersion relation of Eq. (29). All three contributions are negative for energies between -100 MeV and the Fermi energy, which are typical energies for quasihole states, as well as for positive energies, i.e. particle states with energies above the Fermi energy 6~. As the absolute value of the time-like vector component Re S_$&, (dashed line) is consistently larger than the corresponding scalar component (solid line), we obtain a repulsive contribution to the quasiparticle energy, arising from this 2hlp term from energies starting around - 100 MeV below the Fermi energy up to infinity. This repulsive contribution is largest for quasihole states with energies below EF and will decrease for energies above EF with increasing energy. It should be noted that the space-like vector component exhibits a sizable contribution.
A. Trasobares et al./Nuclear
Physics A 640 (1998) 471-489
481
500
400
7
300
” 3
200
scalar part
-
-
vector
-
- - vector
part part (space)
100
E[MeV]
Fig. 4. Real part of the 2plh contribution For further details see Fig. 2.
to the self-energy
for a fixed momentum
as a function
of energy.
The real part of the 2plh contribution to the nucleon self-energy can be calculated from the corresponding imaginary part by a dispersion relation rather similar to Eq. (29), co
Re8ZjPn,(W,
k) = -f
I
dw’
Im-Z&h(J, k) w-co’
.
(30)
0
We already discussed above that the 2plh states lead to larger contributions to the imaginary part than the 2hlp terms. Therefore it is clear that the 2plh contributions to the real part of the self-energy are significantly larger than those originating from 2hlp terms. This can be seen in Fig. 4, which employs a scale which is about a factor 10 larger than the one used in Fig. 3 to visualize the 2hlp contributions. In the energy range of interest all three terms are positive. This means that the 2plh contribution to the scalar part of the self-energy .Z” tends to compensate the negative Hartree-Fock contribution to this term originating mainly from the exchange of two correlated pions, which is parameterized in the realistic OBE potentials by means of the u meson
exchange.
A similar
situation
also arises in the case of the time-like
vector
component of the nucleon self-energy 9: the negative Hartree-Fock contribution to Zc’, which is mainly due to the exchange of the w meson, is compensated to some extent by the 2plh terms. This means that the 2plh terms tend to reduce the Hartree-Fock contributions to the various terms in the self-energy while the 2hlp corrections yield contributions to Z” and _Ze with the same sign as the Hartree-Fock terms. Similar results are obtained for other momenta k. Since, however, our Hartree-Fock approximation to the self-energy has already been extracted from a DBHF calculation, we are not allowed to simply add the real part of 2plh contribution to the self-energy to the corresponding DBHF results. This would lead to a double counting of these 2plh terms. Instead we use a subtracted dispersion
A. Trasobares et al. /Nuclear
482
Physics A 440 (1998) 471-489
relation defined by -Re6~$,lh(~~~(k)
---~F,kj
(31)
to the various terms LYin the self-energy
as defined in (30).
8V2npn,(~,k) =ReSx&,(@,k) using the real contributions This subtraction
ensures
that the real part of the self-energy
remains
identical
DBHF result, if only the 2plh term is considered. With these definitions a quasiparticle self-energy, which is real and energy dependent, to be V$(&
k) = -Z&(k)
and we can determine
+ ~V&h(&
a quasiparticle
k) + ~V&(~,
k)
to the
we now define
(32)
spinor
(33)
with
q,,=E&,-l$,(,,,k)
-E,7.
(34)
This means that this spinor is an eigenstate
of the Dirac equation
[(l+V$,)~~k+y”(M+V~)-V&r,=~qpr+, which uses the self-energy VW of (32) calculated the quasiparticle energy defined by e,(k)
(35) at the energy w which corresponds
= E& - i@cow,k).
to
(3’5)
If for the moment we ignore the 2hlp contribution, SK&,, to the quasiparticle self-energy in (32)) the subtraction defined in (3 1) ensures that the SV&, terms vanish on-shell and the quasiparticle
energy coincides
with the HF energy, thus avoiding
double counting.
However, when the contribution of the 2hlp terms is taken into account, the selfconsistent definition of the energy variable wqp gives rise to a non-vanishing correction due to the energy dependence of the 2plh terms. Therefore the resulting contribution of the calculated correction to the real part of the 2plh term originates from the dispersive correction of the self-energy which is induced from the 2hlp term and therefore goes beyond the conventional Brueckner Hartree Fock approach. Results for the quasiparticle energy are displayed in Fig. 5 for various momenta k. It can bee seen that the inclusion of the 2hlp terms yields a significant increase in the quasiparticle energy eqp as compared to the corresponding HF result. This increase is as large as 17.5 MeV for momenta close to zero, reduces to values around 5.3 MeV for
A. Trasobares et al. /Nuclear
-
-
-
Physics A 640 (1998) 471-489
483
HartreemFock Quasiparticle
/ ---I
II
0.0
8 5’ 1 c
1.0 k
/
”
I 2.0
kF
Fig. 5. Results for the quasiparticle energy, calculated according to eq. (36), are compared energies calculated in the HF approximation for the self-energy (dashed line).
to the corresponding
k = kF and becomes negligibly small around k = 2kp. This repulsive effect terms in the quasiparticle energy has been discussed already in context with a large effect for the quasiparticle energy may indicate that the 2hlp terms have some effect in the calculation of the total energy. For the calculation
of the 2hlp Fig. 3. Such may as well of the total
energy, however, it is not sufficient to evaluate the quasiparticle energy, but one would need the whole spectral distribution of hole strength for states with momenta below and above the Fermi momentum [ 191. It is one aim of our study to explore the effects of the Dirac structure of the nucleon self-energy, calculated beyond the mean field approximation, on the complex optical potential for nucleon-nucleus scattering. The Dirac equation reads now [(l+~~)cu.k+y’(M+~,)-Z0,]u,=~~,~~
(37)
where the full complex self-energy is used. Real C9p solutions of Eq. (37) involve complex values of k. We take as approximate solutions of Eq. (37) the values lqp determined from Eq. (36). It is convenient to rewrite Eq. (37) into a form which only contains an effective scalar V” and vector potential V” [20] [(Y. k + y’(A4 + V”) - V”] uqp = eqpuqp, where
(38)
A. Trasobares et al. /Nuclear Physics A 640 (1998) 471-489
484
Fig. 6. The real parts of the bare scalar and vector components (Re Z&, and Re 2&), represented by the dashed line, are compared to the real parts of Vs and V”, renormalized according to Eq. (38).
The self-energy
terms _Z& have been calculated
in the quasiparticle
approach
of (32)
using the self-consistent relation between the three-momentum and the energy as defined in (34). This Dirac equation can be transformed into a Schrodinger-type equation for the large component of the Dirac spinor, leading Schrijdinger equivalent potential of the form
u = V”
- p+
&- [(lq2
- (vo>2].
to a complex
and energy-dependent
(39)
Results for the real part of the renormalized components Vs and V” are displayed in Fig. 6 as a function of the energy variable wqp, i.e. the quasiparticle energy Q, normalized such that the Fermi energy occurs at zero. For a comparison we also show the unrenormalized components V$. The difference between the solid and dashed line is a measure for the importance of the space-like vector component Z’ of the self-energy. We find that this space-like components yield a slight reduction of the absolute values for RealV, which is of the order of 3%. Larger effects only occur at negative energies, below the Fermi energy, where the 2hlp contribution gets more important. This figure also demonstrates that the energy-dependence of the various Dirac components in the real part of the self-energy remains weak, again with the exception of energies below EF. Also the deviations from the Hartree-Fock result, -375 MeV and -290 MeV for the scalar and vector parts, respectively, are not very pronounced. This can also be seen from the left part of Fig. 7, exhibiting the real part of the Schrodinger equivalent potential U defined in (39). The results derived from the quasiparticle approximation (solid line) including the effects of the energy dependence in the 2pl h and 2hlp terms, exhibit a dependence on the quasiparticle energy, which is very similar to the one derived from the HF approximation. Therefore one may conclude that the
A. Trasobares et al./Nuclear
Physics A 640 (1998) 471-489
-80 ~....,....,....,....,....~....,....,....,....,.,..~-3o -50 0 50 100 150
-50
0
Energy[MeV]
50
100
485
150
Energy[MeV]
Fig. 7. Energy dependence of the real part (left) and imaginary part (right) of the Schrirdinger equivalent potential. The result for the real part is compared to the corresponding prediction obtained in the HF approximation. The dashed curve in the right part of the figure exhibits the expectation value of W calculated according to Eq. ( 40).
energy-dependence in the depth of central, Woods-Saxon type, optical potentials, used to describe nucleon-nucleus scattering is mainly due to the relativistic structure of the underlying Hartree-Fock self-energy to be used in a Dirac equation. Dispersive effects due to the consideration of 2plh and 2hlp terms also lead to an energy dependence, which, however, is much smaller. These dispersive corrections tend to make the potential slightly more attractive at higher energies. It should be mentioned, however, that an energy dependence of the central Schrodinger potential similar to the one obtained here within the relativistic scheme can also be obtained within a non-relativistic Hartree-Fock due to large non-localities in the Hartree-Fock potential [ 211. An analysis rather similar to the one just outlined for the real components of the self-energy can also be performed for the imaginary parts. Results for the renormalized Dirac components W” = Imag V” (a = s and 0) are presented in Fig. 8 and compared to the bare scalar and vector imaginary
components.
beginning
of this section,
part of the bare components
energies
below the Fermi
positive energies
(2plh).
are much more important
the imaginary energy
(2hlp),
The renormalizing for the imaginary
zero around
As we discussed
already
at the
are negative
for
w = 0 ( EF) and positive
for
effects of the space-like
vector components
part than for the real part. This can be
deduced from the differences between W” and W”: the bare components are as large as twice the renormalized quantities. The imaginary part of the Schradinger optical potential [ Eq. (39) ] is shown on the right of Fig. 7. An alternative way of evaluating this imaginary part would be to calculate the expectation value of the Dirac operator ii,
[W” - row” + y . kWU] uqp
(40)
using the quasiparticle Dirac spinors of (33). These expectation values lead to the dashed line on the right part of Fig. 7. We see that the results for the imaginary part of
486
A. Trasobares et al. /Nuclear
_:,,, -50
Physics A 640 (1998) 471-489
,,,,
,,,, 0
,,,, 100
50
,,,, 150
200
Energy[MeV] Fig. 8. Bare and renormalized scalar and vector components of the imaginary part of the quasiparticle self-energy. For further details see Fig. 6.
the Schrodinger potential are rather insensitive on the way of calculation. These results for the imaginary part of the Schrodinger potential may be compared with the volume contribution to the absorption part of the nucleon-nucleus optical potential. As an example we consider the fit to the elastic scattering data of protons on “aCa of Tornow et al. [ 22,231. They obtain a volume term of -7.8 MeV at an energy of 80 MeV for the incoming proton. For the same energy we get -8.3 MeV which is in fair agreement, keeping in mind that we performed our calculation in nuclear matter. Also the energy dependence of the imaginary part displayed in Fig. 7 is similar to the empirical fit. In particular, we obtain an almost symmetric behavior around the Fermi energy. In the calculation of the imaginary components tion 2 and consequently also in the corresponding mined
the direct and exchange
explore the importance results obtained with (solid
contributions
discussed in Secwe always deter-
(see Eqs. ( 16) and ( 18)).
In order to
of the 2plh and 2hlp exchange terms we compare in Fig. 9 the
for the real and imaginary
lines)
of the self-energy real components,
and without
(dashed
part of the Schrodinger
lines)
inclusion
equivalent
of the exchange
potential
terms in the
2pl h and 2hlp parts of the self-energy. One finds that the effects of the exchange terms on the real part are rather small typically around 1 MeV However, the effects are significantly larger for the imaginary part where the difference gets as large as 30% of the total result. This can be understood from the fact that the real part of the self-energy is dominated by the Hartree-Fock contribution, which is identical in these two approaches. If one ignores the exchange terms in calculating the 2plh and 2hlp terms, one may consider it more consistent to ignore the exchange term also in the leading contribution and replace the Hartree-Fock approximation by the Hartree approach. In Ref. [ 81 effective meson-nucleon coupling constants were determined to reproduce the DBHF results within a Dirac-Hartree model. The resulting coupling constants are a bit larger than
A. Trasobares et al. /Nuclear
Physics A 640 (1998) 471-489
487
-30 -50
0
50
100
150
-50
0
Energy[MeV]
50
100
150
Energy[MeV]
Fig. 9. Real and imaginary parts of the SchrGdinger equivalent potential, calculated using the full model (solid lines), ignoring exchange diagrams in the evaluation of the terms of second order (dashed line), and using the Hartree approximation also for the lowest order term (dashed-dotted line).
those derived from the Dirac-Hartree-Fock analysis. If we use these Hartree coupling constants and ignore the effects of exchange terms in the leading term as well as in the 2plh and 2hlp terms, one arrives at a Schrodinger equivalent potential displayed by the dash-dotted lines in Fig. 9. The differences to the solid lines get now quite pronounced for the real part. This can be traced back to the fact that in the Dirac-Hartree
approxi-
mation _Z” and 9 are independent on the momentum or energy of the state and Z” is identical to zero. The results for the imaginary part obtained in this approach, however, are rather close to those evaluated with inclusion of the exchange contributions. It is also interesting to investigate the density dependence of the self-energy in order to extend these calculations to finite nuclei. As an example we present some results obtained for nuclear matter with a Fermi momentum kF = 1.2 fm-’ in Fig. 10. The results obtained at these various densities, either for the self-energy keeping track of the Dirac structure, or using the Schrodinger equivalent potentials derived from these components,
may then be used in a local density
nucleon-nucleus
scattering.
Such investigations
approximation
for a prediction
of
are in progress.
4. Conclusions The relativistic structure of the self-energy for a nucleon in nuclear matter is investigated by including all irreducible terms of first and second order in the residual interaction. For the NN interaction a parameterization of the Dirac-Brueckner-HartreeFock (DBHF) G-matrix in terms of the exchange of effective scalar and vector mesons has been used. The 2plh and 2hlp contributions to the imaginary part of the self-energy are evaluated keeping track of all direct and exchange terms. The corresponding 2plh and 2hlp contributions to the real part are derived from these imaginary components by means of dispersion relations. A subtracted dispersion relation must be used for the
488
A. Trasobares et al. /Nuclear
Physics A 640 (1998) 471-489
Fig. 10. Real and imaginary parts of the Schriidinger equivalent potential calculated for nuclear matter with a Fermi momentum of kF = 1.4 fm-’ (solid line) are compared to those obtained for k,c = 1.2 fm-t.
2plh term to avoid double-counting on which these studies are based. The inclusion of 2hlp diagrams
with the G-matrix in the evaluation
underlying
the DBHF approach
of the real part of the self-energy
yields a non-negligible modification of the scalar and vector components in particular for states with momenta below the Fermi momentum. Also the value of the quasiparticle energy is increased by a value as large as 17.5 MeV for momenta close to zero and to values around 5.3 MeV for k = kF. Such a large effect for the quasiparticle energy may indicate that the 2hlp terms should have some effect in the calculation of the total energy. The calculated self-energy can also be transformed into a Schrijdinger equivalent optical potential, to be used in the study of nucleon-nucleus scattering. Exchange diagrams are non-negligible in the evaiuation of the imaginary part. The energy or momentum dependence of the central component of the real potential, however, is dominated by the effects of the Dirac-Hartree-Fock contribution. The 2plh and 2hlp terms give rise to a more attractive SEP at positive energies and introduce which is very weak.
an additional
energy dependence
Acknowledgements This project has been supported
by the Spanish research grant DGICYT, PB95-1249, and by the EC contract CHRX-CT93-0323. One of us (H.M.) is pleased to acknowledge the warm hospitality at the Facultat de Fisica, Universitat de Barcelona, and the support from the program of visiting Professor at this university.
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