An investigation of nuclear medium effects in the Dirac-Brueckner optical potential

Volume 253, number 1,2

PHYSICS LETTERS B

3 January 1991

An investigation of nuclear medium effects in the Dirac-Brueckner optical potential Yoshinori Miyama and Toru Suzuki

Research Centerfor Nuclear Physics, Osaka University, lbaraki, Osaka 567. Japan Received 19 September 1990

The effects of the nuclear medium on relativistic optical potentials in nuclear matter are studied within the framework of the Dirac-Brueckner method. The medium effects are classified into Pauli blocking, dispersive, and Dirac spinor modification effects, where the last is shown to be most sizable and persistent up to high energies. Various meson exchange contributions to the spinor modification effect are investigated. It is shown that pionic contributions to the optical potentials are suppressed because of medium effects.

For some time the so-called Dirac- (or relativistic-) approach has been successfully applied to a variety of nuclear phenomena, for instance, the nuclear saturation problem and elastic scattering of intermediate energy nucleons from nuclei. To draw a definite conclusion about the success o f this approach, however, it seems necessary to estimate various corrections not taken into account in a lowest-order theory such as the Hartree-Fock or impulse approximations. One such correction in nuclear reactions known to be important in non-relativistic theories is the effect of the nuclear m e d i u m on the nucleon-nucleon interaction, which is neglected in the relativistic impulse approximation. This means that the nucleon-nucleon scattering matrix (t-matrix) should be modified in nuclear matter because of the presence of other nucleons. The effects of the nuclear m e d i u m have been considered within a ladder approximation (Brueckner Gmatrix) for relativistic single-particle potentials in nuclear matter [1,2] and nucleon-nucleus optical potentials in finite nuclei [ 3,4 ]. It turned out that the Dirac scalar and vector potentials are reduced considerably when the m e d i u m effects were taken into account by solving the Bethe-Salpeter equation in nuclear matter [2,4 ]. This leads in particular to a reduction of the spin-orbit part o f the nuclear optical potential, which thus becomes much weaker than the standard phenomenological values obtained from a

fit to elastic scattering data [4 ]. It is the purpose o f the present letter to investigate the origin of the nuclear medium effects and to clarify the role of various meson exchanges. The calculation procedure follows that of refs. [ 5 ] and [ 4 ]: We solve the Bethe-Salpeter (BS) equation using the one-boson exchange kernel and the singleparticle Green's function in nuclear matter. The latter is defined in terms of the nucleon self-energy (optical potential), which in turn is calculated from the nucleon-nucleon scattering amplitude obtained from the BS equation in nuclear matter. The whole procedure thus imposes a self-consistency requirement on the Dirac spinor, or the effective mass. The BS equation for the nucleon-nucleon scattering amplitude F in nuclear matter is given by

F=K-i

f KGGI-'daq'

where K is a two-body interaction kernel gle-particle Green's function. Actually three-dimensionally reduced version T h o m p s o n equation. Once F i s obtained self-energy is given as

( 1) and G a sinwe solve a using the the nucleon

S ( p ) = - i f [ (GF)dir- (GF)exch] d 4 k ' =,~S(p) _yoSO(p ) + ~ . ~ ( p ) .

0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )

(2)

23

Volume 253, number 1,2

PHYSICSLETTERSB

In the calculation of ( 1 ) and (2) we ignore the contribution coming from negative-energy states. The positive-energy part of the single-particle Green's function is given by M*

G+ (p) = E,(p---~A+ (p; M*) x{

O(IPl-kv)

+

0(kF--IPl)

~

(3)

Here

pU*=pU+Zu, M * = M + Z s, E*(p) = ~ / M * E + p : , and A+ (p; M*) denotes the projection operator to positive-energy solutions of the Dirac equation in the presence of potentials. Eq. ( 1 ) is solved using matrix inversion with partial waves up to the total angular momentum equal to ten. The equation is transformed into the center of mass (CM) frame and is solved iteratively until the self-consistency for effective mass is obtained. In the course of evaluating the integral ( 1 ) we ignore the small quantity ~ and take the potentials to be independent of momentum, the values of which are fixed at the Fermi momentum. We adopt the one-boson exchange kernel of the Bonn group [6 ], who utilized the Thompson equation to obtain the parameters for the free nucleon-nucleon interaction. Using the parameters of ref. [ 6 ] with n, p, o, co, 11and 8 mesons, we obtain the self-consistent value 591.8 MeV for the effective mass M* at normal nuclear matter density p=0.164 fm -3 (kF=265 MeV/c). To investigate in detail the effects of the nuclear medium on the nucleon optical potentials, we first classify them according to the way they appear in the single-particle Green's function (3) as follows: [P] the Pauli blocking effect, which is represented by the 0-function in G÷; [D] the potential dependence of the energy denominator, that is p ° * - E * ; [M ] the Dirac-spinor modification effect owing to the scalar potential. This effect is contained in the M*dependence of the factor in front of the parentheses ineq. (3). The first two effects appear in non-relativistic theories as well, while the last one is of genuine relativ24

3 January 1991

istic origin. We discuss only the scalar potential Z s and the time component X ° of the vector potential, since the space component • of the vector potential is rather small, coming only from the exchange matrix elements. The magnitudes of the different medium effects can be extracted by performing the following calculations: ( 1 ) The full calculation with all types of medium effects [ P + D + M ] . (2) A calculation that neglects the spinor modification effect, by using free spinors in the evaluation of the interaction matrix elements. As we are interested in the decomposition of medium effects, we use the same effective mass in the energy denominator as that used in the full calculation, and do not impose self-consistency within this calculation [ P + D ]. (3) A calculation with only the Pauli blocking effect, that is using free spinors and neglecting the potentials in the energy denominator [P]. (4) A calculation with no medium effects included. IN] The calculation [N] may be denoted as the impulse approximation, but it is different from the socalled relativistic impulse approximation (RIA), because in the RIA the optical potentials are calculated from the free t-matrix using medium-modified Dirac spinors. The difference between the RIA and our calculation [N] is small, however. We notice in addition that the calculation [ P + D ] should be comparable to the standard non-relativistic Brueckner calculation, in the sense that the Pauli and dispersive effects are included. The results are shown in fig. 1. As mentioned earlier, the effect of the nuclear medium drastically reduces the potential strengths. This reduction amounts to 30% and remains large up to high energies. From the figure one can extract the size of each medium effect, and find the following characteristic features: (i) Naturally the Pauli blocking effect [P] is important close to the Fermi surface, but is not necessarily a dominant medium effect for the real part of the self-energies. (ii) The dispersive effect (the difference of [P] and [P + D] ) is relatively small, and is almost negligible for vector potentials. (iii) The effect of the spinor modification [M] is large and weakly dependent on the incident momentum. Thus it is the dominant medium effect at high

Volume 253, number 1,2

PHYSICS LETTERS B Scalar

200

,

-

100

-- -

3 January 1991

Self-Energy

,

Scalar i

200

,

,m~

--_. _--_ _-_____________. _ _

i

i

(a)

(a)

(P+D)

Self-Energy

i

, mE

i

s

100 0

0 -100

>

% -20o

-~

%

-100 -200

-300

-300

-400

-400

~~-=" f R e TS

-500

R e~ I

-600 300

I

400

,

.

A >

.

400

i

300

"..__...~_..~

.

.

I

-600

700

300

800

.

.

.

.a~ o

(b) 100

i

i

(b)

800

i

,m~S

0

200

-100

100

-200

o

-300

. :.._ :__::

-100

I

700

(P+D+M)

Self-Energy

i

200

,

I

500 600 P(MeV/c)

Scalar ,

.

I

400

Self-Energy

,

.

-500

s

I

500 600 P(MeV/c)

Vector 500

I

-

--

~

--

. . . .

_ =.1 ~

~

~

~

~--

-'-

='-

--'- --'-

---- --'-

---- --'-

~ - ' -- -

Re~ s

-400

-200

~

-500

-300

I 300

400

I 500

I 600 P(MeV/c)

I 700

I

-600 800

Fig. l. (a) Momentum dependence of the real and imaginary parts of the scalar potential for [ P + D + M ] (full line), [ P + D ] (shortdashed line), [P] (dot-dashed line) and [N] (long-dashed line). (b) Same as (a) but for the vector potential.

energies. The rather small values of the optical-potential strength in finite nuclei obtained in ref. [ 4 ] are mainly the result of this effect. In the rest of the letter we investigate the origin of the spinor modification effect, the main part of the medium effects, focusing on the role played by various meson exchanges. For this purpose we compare the scalar potential with full medium effects ( [ P + D + M ] ) with that for Pauli and dispersive effects only ( [ P + D ] ). Analysis of the vector potential shows that a qualitatively similar conclusion is obtained. Among the six mesons included in the calculation, contributions coming from o, (o and x mesons would be the most important. This in fact comes out from the calculation. In fig. 2 we show the calculation us-

3O0

400

I

I

500 600 P(MeV/c)

I

700

800

Fig. 2. Momentum dependence of the real and imaginary parts of the scalar 2: s potential in nuclear matter. (a) The spinor modification effect is not included ( [ P + D ] ). (b) Full medium effects ( [P + D + M] ) are included, that is M* = 591.8 MeV is used. Solid lines show the total potential. Calculations using only the o and co mesons (short-dashed lines), and those omitting pions (long-dashed lines) are also shown.

ing all six kinds of mesons (solid lines) together with the one using only the ~ and c0 mesons (short-dashed lines) and the one without pions (long-dashed lines). As the latter two calculations almost coincide we conclude that the net effect of the p, ~ and ~ mesons is quite small, except possibly when the p mesons come in together with the pions. Comparing (a) and (b) of fig. 2, we find that nearly half ( ~ 50 MeV reduction) of the spinor modification effect arises from the o and co mesons, while the other half comes from the meson exchange process having at least one pion exchange. At this stage it will be reasonable to separate the 25

Volume 253, number 1,2

PHYSICS LETTERSB

lowest-order term of ( 1 ), which is referred to as the Hartree-Fock (or more precisely the static mean field) term, as follows:

Xn=X~F+AX=

(#=S, 0).

(4)

The Pauli and dispersive effects contribute only to AS, while the spinor modification effect [M] is present in both the SHF and AS. Fig. 3 shows the decomposition (4) for the real part of the scalar potential

with full medium effects (fig. 3a) and with only the Pauli and dispersive effects (fig. 3b). Let us first study the Hartree-Fock term, the main part of which comes from the direct a-meson exchange diagram. The small energy dependence of 2;HF arises from the Fock term; it slowly decreases as the incident momentum increases because of the larger exchange momentum. One can see that the spinor modification effect on the HF term is quite small. This may easily be understood by noting that essentially SSHF ~

ReZ s (P+D) 200

(a)

.

.

.

.

100

.........................

;-

o

I

-100

¢

-200

;ito--;i .........

• ~ - - . . . . . . . . . . . . . . . . . . . . . . . . . . A~lot t~I;[n-x]

-300 -400 -500

Ztot I

-600 300

400

l

I

I

500 600 P{MeV/c)

700

800

Re£ s (P+D+M) 200

t~ b

•- ,

i

=

=

i .

. . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

. . . .

100

;-

o ¢D X v -100

n-

AZto t .........................................

ar[~-x]

-200

'~lot -300 £H-F

-400 -500 I

-600 300

400

I

I

500 600 P(MeV/c)

I 700

800

Fig. 3. Decompositionof the real scalar potential into HartreeFock (~rNv)and higher-order (AXto,) contributions. Dominant contributions to the latter are also shown (AX(oo~) and AS(g-x), see text). (a) The spinor modification is not included ([P+D]). (b) Full medium effects are included ( IP+D+M] ). Solid lines show the total scalar potential, longdashed lines show the Hartree-Fock contributions. The total higher-ordercontributions are shownby short-dashed lines. Those coming from o and to, and those from pion plus other mesonsare also shown by triple-dashed lines and dash-dotted lines, respectively. 26

3 January 1991

V'pS

,

,~O F ~ r.p

( g ~

Vdirect "t-

gexch ) ,

and that Ps/Pis only weakly dependent on the effective mass, that is about 0.94 (for M * = 592 MeV) and 0.98 ( M * = M = 9 3 9 MeV). We now consider the higher-order contributions AS corresponding to the second term of eq. (1). The momentum dependence of this term is rather small as seen from the figures. Its magnitude (sign) is, however, quite sensitive to the spinor modification effect: the attractive potential obtained in fig. 3a turns into a weak repulsive one because of this effect. The results are in accord with the behavior of the imaginary part of the potential shown in fig. 2. As is well known [7] the dispersive part of the real potential is related to the imaginary part of the potential through a dispersion relation. With only the Pauli and dispersive effects (fig. 2a), the imaginary part of the scalar potential is peaked at low momenta (hence at low energies), thus giving a negative contribution to the dispersive integral. Inclusion of the spinor modification effect drastically reduces the imaginary part and removes the peak at low momenta. The dispersive integral now receives a contribution from high energies and becomes weakly positive. To study the mesonic origin of the behavior of AS we include in fig. 3 the contribution coming solely from a and m mesons ( ~ S [ a o ) ] ) and the one from pion plus other mesons A S ( n - x ) (x denotes any combination of mesons including pion). We first note that the o-m contribution is repulsive while the n - x term is attractive. This may also be understood from the imaginary part in fig. 2. For instance, the low-momentum peak of Im X in fig. 2a comes mainly from the process involving pion exchanges, giving a strong attraction in the real part. As shown in fig. 3b, the inclusion of spinor modification results in the reduction of the attractive pion contribution and the en-

Volume 253, number 1,2

PHYSICS LETTERS B

hancement of the repulsive a-to part. Thus one may conclude that the attraction coming from pions becomes rather ineffective because of this medium effect. On the other hand, the effect on the o-to term is more complicated. One may roughly say that a strong attractive potential owing the o meson alone is reduced and overcome by the contributions involving the co meson. It should be noted, however, that there are strong cancellations among-processes involving strong attractive (o) and repulsive (to) interactions. In this letter we have studied the nuclear medium effects on the Dirac potentials by solving the threedimensional version of the Bethe-Salpeter equation in nuclear matter. The calculated results show the importance of medium effects even at high incident energies. The most important one arises from the modification of the Dirac spinor. It was found that the exchange of pions (and other mesons) becomes rather ineffective because of this effect, giving rise finally to a much weaker potential compared with, for example, the impulse approximation. Although the above analysis has been done for the scalar potential, a similar conclusion is obtained for the vector poten-

3 January 199"1

tial, where the roles of the attractive and repulsive contributions are interchanged. It remains to see if this result may not be altered by, for example, the higher-order contributions not taken into account in the present calculation, or better approximation schemes than that adopted here. In view of the cancellations among large terms observed in the calculation, it may also be important to study the diagrammatic convergence based on the multiple-scattering formalism.

References [ 1] L.S. Celenza and C.M. Shakin, Relativistic nuclear physics (World Scientific, Singapore, 1986 ); M.R. Anastasio et al., Phys. Rep. 100 (1983) 327. [ 2 ] B. ter Haar and R. Malfliet, Phys. Rep. 149 ( 1987 ) 207. [3] D.P. Murdock and C.J. Horowitz, Phys. Rev. C 35 (1987) 1442. [4] Y. Miyama, Phys. Lett. B 215 (1988) 602; in preparation. [ 5 ] C.J. Horowitz and B.D. Serot, Nucl. Phys. A 464 (1987) 613. [6] R. Machleidt, Relativistic dynamics and quark nuclear physics, eds. M. Johnson and A. Picklesimer (Wiley, New York, 1986) p. 71. [7] C. Mahaux et al., Phys. Rep. 120 (1985) I.

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