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PHYSICAL REVIEW C SEPTEMBER 1996VOLUME 54, NUMBER 3

0556-2813

Medium effects on binary collisions with theD resonance

T.-S. H. LeePhysics Division, Argonne National Laboratory, Argonne, Illinois 60439-4843

~Received 20 February 1996!

To facilitate the relativistic heavy-ion calculations based on transport equations, the binary collisiovolving aD resonance in either the entrance channel or the exit channel are investigated within a Hamformulation ofpNN interactions. An averaging procedure is developed to define a quasiparticleD* and toexpress the experimentally measuredNN→pNN cross section in terms of an effectiveNN→ND* crosssection. In contrast to previous works, the main feature of the present approach is that the massmomentum of the producedD* ’s are calculated dynamically from the bareD↔pN vertex interaction of themodel Hamiltonian and are constrained by the unitarity condition. The procedure is then extended to deeffective cross sections for the experimentally inaccessibleND*→NN andND*→ND* reactions. The pre-dicted cross sections are significantly different from what are commonly assumed in relativistic heacalculations. TheD potential in nuclear matter has been calculated by using a Bruckner-Hartree-Fock apmation. By including the mean-field effects on theD propagation, the effective cross sections of tNN→ND* , ND*→NN andND*→ND* reactions in nuclear matter are predicted. It is demonstrated thadensity dependence is most dramatic in the energy region close to the pion production thr@S0556-2813~96!00409-8#

PACS number~s!: 25.75.Dw, 13.75.Cs, 13.75.Gx, 24.10.Jv

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I. INTRODUCTION

Nuclear matter with a high population of nucleon resonances~D, N* ’s! is expected to exist in some astrophysicaobjects and can be created in relativistic heavy-ion collisionWith the very dedicated experimental efforts at GSI, AGSand CERN in the past decade, it appears that the properof such a nuclear system can now be investigated. Expemental evidence for creating aD-rich nuclear system duringrelativistic heavy-ion collisions has recently been reporte@1,2#. This interesting development has raised two theoreticissues. First, it is necessary to examine the extent to whthe theoretical models employed in identifying theD-richmatter are valid. In the calculations based on transport eqtions @3–9#, some assumptions were made to define the crosections of binary collisions involving aD in either the en-trance channel or the exit channel. This needs to be clarifiwithin a rigorous formulation of the scattering theory withresonance excitations. The medium effects onD propagationassumed in those calculations must also be examined frommore fundamental point of view. One possibility is to calculate theD mean field using the well-developed nuclear manbody methods@10,14#. The second theoretical issue is how tdevelop a microscopic approach to understand theD-richmatter in terms of the elementaryNN, ND, andDD colli-sions and their coupling with the pion production channelThese two theoretical issues are closely related and mustaddressed within the same theoretical framework. The heof the problem is to define precisely and to determine quatitatively all possible binary collisions in theD-rich matter.In this paper, we will address this problem by employingHamiltonian model ofpNN interactions which was previ-ously developed to describeNN scattering up to 2 GeV@11–13# and theD-nucleus dynamics in pion-nucleus reaction@14#. Our main objective is to provide the theoretical input t

54/96/54~3!/1350~10!/$10.00

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the relativistic heavy-ion calculations using transport equations @3–9#.

In Sec. II, we briefly review the coupledNN%ND scat-tering equations within thepNN Hamiltonian model devel-oped in@11–13#. We derive formulas in Sec. III to relate theexperimentally observedNN→pNN reaction to the resonat-ing ND state. An averaging procedure is then developed tdefine a quasiparticleD* and to evaluate the cross sectionsfor binary collisions with aD* in either the entrance channelor the exit channel. In Sec. IV, we recall the approach of@14#to calculate the mean-field effects onD propagation whichare then used to calculate the medium-corrected cross setions. In Sec. V, we present our predictions of the crossections forNN→ND* , ND*→NN andND*→ND* tran-sitions in free space and in nuclear medium. A summarygiven in Sec. VI.

II. HAMILTONIAN MODEL OF pNN INTERACTIONS

An important advance in intermediate-energy physics ithe development of a microscopic approach to understanthe nuclear reactions induced by pions, protons, and eletrons in terms of the interactions betweenp, N, andD de-grees of freedom. Such an approach must start with apNNmodel which can describe the following elementary processes

pN→pN, EL<300 MeV, ~1!

NN→NN

→pNN, EL<1000 MeV, ~2!

pd→pd

→NN

→pNN, EL<300 MeV. ~3!

1350 © 1996 The American Physical Society

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54 1351MEDIUM EFFECTS ON BINARY COLLISIONS WITH . . .

The developments ofpNN models have been reviewed@16#. In this work we employ thepNN model developed in@11–13#. The main feature of the constructedpNN Hamil-tonian is that theNN↔ND interaction can be directly constrained by data on theNN→ND→pNN reacton, and theND→ND interaction can also be related to thepd reactions.This allows a more realistic determination of the phenoenological parts of the interactions involving theD. For ex-ample, it was found@11–13# that the ranges of thepNN andpND vertices in theNN↔ND transition potential must beless than about 750 MeV for a monopole form if thNN→pNN reaction cross sections can be reasonablyscribed. This phenomenological procedure is not avoidain any meson-exchange model since a fundamental theoshort-rangeNN andND interactions is still not available.

In this first application of thepNN model to relativisticheavy-ion collisions, we will focus on theD resonance andneglect the weaker nonresonantpN interactions. ThepNNmodel Hamiltonian within the formulation of@11–13# canthen be written as

H5H01H int , ~4!

whereH0 is the sum of free-energy operators for theN, D,andp degrees of freedom, and

H int5(i51

2

@hpN,D~ i !1hD,pN~ i !#11

2 (i , j51

2

@VNN,NN~ i , j !

1VNN,ND~ i , j !1VND,NN~ i , j !1VND,ND~ i , j !#. ~5!

Note that the vertex interactionshpN,D and hD,pN describethe pN↔D transition. They can renormalize the bareDmass and generate aND→DN interaction due to the exchange of an ‘‘on-mass-shell’’ pion. To account for thetwo effects, the pion production channel must be includedderiving the pNN scattering equations from the modHamiltonian Eq.~5!.

The derivation of thepNN scattering equations from thabove model Hamiltonian can be found in Sec. II of@12#. Bysome straightforward derivations using Eqs.~3.27!, ~3.31!,and~3.24! of @12#, we obtain the following equations definein the coupledNN%ND space:

TNN,NN~E!5VNN,NN~E!1VNN,NN~E!GNN~E!TNN,NN~E!,~6!

TND,NN~E!5VND~2 !1~E!VND,NN@11GNN~E!TNN,NN~E!#,

~7!

TNN,ND~E!5@11TNN,NN~E!GNN~E!#VNN,NDVND~1 !~E!,

~8!

TND,ND~E!5tND,ND~E!1TND,NNVNN,NDVND~1 !~E!. ~9!

In the above equations, we have defined the propagator

GNN~E!5PNN

E2H01 i e, ~10!

GND~E!5PND

E2H02S~E!, ~11!

-

ee-bleof

einl

where theD self-energy is determined by thehpN↔D vertexinteraction in the presence of a spectator nucleon

SD~E!5(i51

2

hD,pN~ i !PpNN

E2H01 i ehpN,D~ i !. ~12!

PNN , PND , andPpNN are, respectively, the projection operators for theNN, ND, andpNN states.

The main feature of the above scattering formulationthat the effects due to theND scattering are isolated in thetmatrix tND,ND defined by

tND,ND~E!5VND,ND~E!1VND,ND~E!GND~E!tND,ND~E!,~13!

and theND scattering operators defined by

VND~1 !~E!511GND~E!tND,ND~E!, ~14!

VND~2 !1~E!511tND,ND~E!GND~E!. ~15!

The driving term of Eq.~13! contains the directND interac-tion VND,ND of the Hamiltonian Eq.~5! and the one-pion-exchangeND→DN interaction due to the vertex interactionshpN,D andhD,pN

VND,ND~E!5VND,ND~E!

1 (iÞ j51

2

hD,pN~ i !PpNN

E2H01 i ehpN,D~ j !.

~16!

Note that both the second term of Eq.~16! andSD(E) of Eq.~12! become complex at energies about the pion productithreshold. This is the consequence of the coupling with tpion production channel and is essential for a realistic dscription of thepNN processes listed in Eqs.~1!–~3!.

TheND scattering also has a contribution to the effectivNN potential in Eq.~6!. Explicitly, we have

VNN,NN~E!5VNN,NN1UNN,NN~2! ~E!, ~17!

with

UNN,NN~2! ~E!5VNN,NDGND~E!

3@11tND,ND~E!GND~E!#VND,NN . ~18!

The task is then to find an appropriate parametrizationthe model Hamiltonian Eq.~5! to best reproduce the data othe pNN processes listed in Eqs.~1!–~3!. This was firstachieved by Betz and Lee@17# using the separable parametrization. In this work we consider the meson-exchange prametrization of@11–13#. It was found that thepNN datacan be best reproduced by using~1! the vertex interactionhpN↔D determined from fitting thepN scattering phase shiftin P33 channel, and~2! the one-pion-exchange model@18# ofthe transition potentialsVNN↔ND andVND,ND with a mono-pole form factor of a cutoff parameterL5650 MeV/c, Eq.~3!, the NN interactionVNN,NN of Eq. ~17! is defined byusing a subtraction of the Paris potential@19#

VNN,NN5VParis2UNN,NN~2! ~E5Es!, ~19!

dho

y

tn

v

y

in

s

ut

he

in

1352 54T.-S. H. LEE

with Es510 MeV laboratory energy. Note that the abovdefinition of theNN interaction amounts to removing phenomenologically the two-pion-exchange with an intermeateND state from the Paris potential, in order to avoid tdouble counting of theND effect. This approach was alsdeveloped independently by Sauer and collaborators@20#.The details of the determination of the model Hamiltonianwell as comparisons with thepNN data can be found in@11–13#. Here we illustrate the validity of the model bshowing in Fig. 1 that the predictions of the model are alsogood agreement with the most recentnp polarization data@15#.

Before we proceed further, it is important to clarify thmeaning of theD degree of freedom of the consideredpNNmodel. Because of the presence of the vertex interachD↔pN , theD is certainly not a physical particle which cabe detected experimentally. To investigate theD dynamics,we need to consider theNN↔pNN andpNN→pNN reac-tions. The amplitudes of these reactions can be calculafrom the vertex interactionhD↔pN and the baryon-baryontmatrix defined in Eqs.~6!–~9! by the following equations:

TNN,pNN~E!5TNN,ND~E!GNDF(i51

2

hD,pN~ i !G , ~20!

TpNN,NN~E!5F(i51

2

hpN,D~ i !GGND~E!TND,NN~E!, ~21!

TpNN,pNN~E!5F(i51

2

hpN,D~ i !GGND~E!TND,ND~E!

3F(i51

2

hD,pN~ i !G . ~22!

III. EFFECTIVE BINARY CROSS SECTIONS WITH D*

We will only consider the transport equations of relatiistic heavy-ion collisions which are determined by the crosections of the collisions between ‘‘on-mass-shell’’ physicparticles. Within the model defined by thepNN Hamil-

FIG. 1. The polarization observablesCLL andCLS of np scat-tering predicted by thepNN model@11–13# are compared with therecent data@15#.

e-i-e

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ted

-ssal

tonian, Eqs.~4! and~5!, the transport equations can then onlbe written in terms ofp andN variables. It is however pos-sible to express the dynamics associated with the pionterms of a quasiparticleD* which has a mass equal to theinvariant mass of apN subsystem. The transport equationcan then be rewritten in terms ofN and D* . In such anapproach, the mass of theD* can range frommN1mp to thehighest energy allowed by the collision energy. To carry ocalculations, it is necessary to define theNN↔ND* andND*→ND* cross sections. In the following we will try todefine rigorously these cross sections in terms of tNN↔pNN andpNN↔pNN amplitudes defined in the pre-vious section.

To proceed, let us first consider theNN→pNN and thethree-body kinematics in the center-of-mass frame~Fig. 2!:

kW5kW152~kW21kW3!. ~23!

For a given total collision energyE and a given momentumkW , the invariant massM and the internal relative momentumqW of the pN ~23! subsystem, calledD* , are defined by thefollowing relations:

E5EN~k!1@M21k2#1/2, ~24!

M5EN~q!1Ep~q!, ~25!

whereEa(q)5(ma21q2)1/2 is the energy of a particle with

massma . Equation~24! also leads to a useful relation

M25E222EEN~k!1mN2 . ~26!

Clearly, Eqs.~25! and ~26! imply that the invariant mass oftheD* is restricted to the region (mN1mp)<M<(E2mN).By using Eqs.~24! and ~25!, the momentak andq can bewritten as

k51

2E@~E22mN

22M2!224M2mN2 #1/2, ~27!

q51

2M@~M22mN

22mp2 !224mN

2mp2 #1/2. ~28!

In terms of the kinematical variables defined above andFig. 2, the matrix element of theNN→pNN transition op-erator, Eq.~21!, can be written~suppressing spin-isospin in-dices! as

FIG. 2. Graphical representation of theNN→ND→pNNreaction.

54 1353MEDIUM EFFECTS ON BINARY COLLISIONS WITH . . .

^kW ,qW uTpNN,NN~E!ukW0&5hpN,D~qW !1

E2EN~k!2ED~k!2S~E,kW !TND,NN~kW ,kW0 ,E!. ~29!

We choose the normalizationkW8ukW &5d(kW82kW ). The scatteringt-matrix is related to theSmatrix by ~in operator form!

S~E!5122p id~E2H0!T~E!. ~30!

We next define the following partial-wave expansions:

hpN,D~qW !5yl23s23

sDmsD~ q!hpN,D~q!^sDmsDu ~31!

and

TND,NN~kW ,kW0 ,E!5 (J,MJ

(LS,LDSD

yLDSD

JMJ ~ k!TLDSD ,LSJ ~k,k0 ,E!yLS

JMJ1

~ k0!. ~32!

In the above equations, we have defined the spin-angular vector

ylsjm~ p!5 (

mlms(

ms1ms2

Ylml~ p!us1ms1

&us2ms2&^ lsmlmsu jm&^s1s2ms1

ms2usms&, ~33!

whereYlml(q) is the usual spherical harmonic function, (si ,msi

) are the spin variables for thei th particle.By using the definitions~29! and~30! and the variables defined by Eqs.~24!–~28!, we can write the total cross section of

the two-step processNN→ND→NNp ~Fig. 2! as

s tot~E!51

~2s1811!~2s2811! (spins

~2p!4

k02 Fk0EN~k0!

2 G E dkWE dqW d$E2EN~k!2A@EN~q!1Ep~q!#21k2%

3U(msD

^ms2ms3

uhpN,D~qW !umsD&

1

E2EN~k!2ED~k!2(~k,E!^msD

ms1uTND,NN~kW ,kW0 ,E!ums1

8 ms28 &U2, ~34!

where the initial momentumk0 is defined by the total energyE5EN(k0)1EN(k0). TheD self-energy is found to be

(~k,E!5E uhD,pN~q8!u2q82dq8

$@E2EN~k!#22k2%1/22EN~q8!2Ep~q8!1 id. ~35!

Substituting the expansions Eqs.~31! and ~32! into Eq. ~34! and integrating over anglesq and k, we obtain

sNNp←NNtot ~E!5 (

J,LS,LDSD

dsLDSD ,LSJ ~E!, ~36!

with

dsLDSD ,LSJ ~E!5

4p

k02 S J1

1

2D E0kmaxdkuNND1/2~k!TLDSD ,LS

J ~k,k0 ,E!rNN1/2~E!u2, ~37!

wherekmax can be calculated from Eq.~27! by settingM5mp1mN , and

rNN~E!5pk0EN~k0!

2, ~38!

NND~k!5pk2qEN~q!Ep~q!EM~k!

M2 U hD↔pN~q!

E2EN~k!2ED~k!2(~k,E!U2. ~39!

To have a better understanding of Eq.~37!, we use Eq.~26! to change the integration variable

dk52MEN~k!

kEdM. ~40!

Equation~37! can then be written as an integral over the invariant massM of thepN subsystem

1354 54T.-S. H. LEE

dsLDSD ,LSJ 5

4p

k02 S J1

1

2D Emp1mN

E2mNdMuGND~M !1/2TLDSD ,LS

J ~k,k0 ,E!rNN1/2~E!u2, ~41!

where

GND~M !5pFkEN~k!EM~k!

E GFqEN~q!Ep~q!

M GU hD,pN~q!

E2EN~k!2ED~k!2(~k,E!U2. ~42!

u

y

iou

e

e

’c.

ed

hefg

Note that in evaluating the above integral, we need toEqs.~27! and~28! to expressk andq in terms of the invari-ant massM and the total energyE. By Eqs.~27! and ~35!,the self-energyS(k,E) is also a function of the total energE and the invariant massM .

Equation~41! is an exact expression. The integrand cotains the dependence on the invariant massM of the D* ,dsLDSD ,LS

J /dM, which is needed for a detailed transpor

equation calculation. For pratical applications and makcontact with the current calculations of relativistic heavy-icollisions, it is interesting to develop an averaging procedsuch that the cross section for a given total energyE can bewritten in terms of only one averagedD* . This is our nexttask.

We use Eqs.~27! and ~42! to define an average relativmomentumk of theND* two-body system:

k5

EmN1mp

E2mNGND~M !kdM

EmN1mp

E2mNGND~M !dM

. ~43!

The corresponding average mass ofD* is then calculatedfrom Eq. ~26!:

M25E222EEN~ k!1mN2 . ~44!

Assuming that the transitiont matrix in the integrand of Eq.~41! can be calculated at the average momentumk, we thenobtain the following ‘‘factorized’’ form:

sND*←NNtot

5 (J,LDSD ,LS

dsLDSD ,LSJ , ~45!

with

dsLDSD ,LSJ 5

4p

k02 S J1

1

2D urND1/2~E!

3TLDSD ,LSJ ~ k,k0 ,E!rNN

1/2~E!u2. ~46!

se

n-

t-

ngnre

Here we have introduced a phase-space factor for theND*two-body state

rND~E!5EmN1mp

E2mNdMGND~M !. ~47!

Equation~46! has the form similar to that introduced in thliterature@4,5,23#. Here we have explicitly shown how it canbe derived from the Hamiltonian formulation@11–13# of theD excitation by using the ‘‘factorization approximation.’The accuracy of this approximation will be examined in SeV. We emphasize here that theGND(M ), defined in Eq.~42!,is the consequence of a unitary formulation of thepNN pro-cesses listed in Eqs.~1!–~3!. Equation~47! is significantlydifferent from the parametrizations of theND phase spaceintroduced in@4,5#.

We note that Eq.~46! has the usual form of the binarycollisions involving only stable particles. For example, thexact expression for theNN→NN process can be obtaineby replacingrND(E) in Eq. ~46! by rNN(E).

By using theND* representation of thepNN state, it isstraightforward to carry out similar derivations to express tpNN→NN and pNN→pNN cross sections in terms oND*→NN andND*→ND* cross sections. If the averaginprocedure defined in Eqs.~43! and ~47! is also applied, weobtain the following expressions:

sNN←ND*tot

5 (J,LS,LDSD

dsLS,LDSD

J ~48!

with

dsLS,LDSD

J 54p

k2urNN

1/2~E!TLS,LDSD

J ~k0 ,k,E!rND1/2~E!u2,

~49!

for theND*→NN reaction, and

sND*←ND*tot

5 (J,LD8SD8 ,LDSD

dsLD8SD8 ,LDSD

J, ~50!

with

dsLD8SD8 ,LDSD

J54p

k2S J1

1

2D urND1/2~E!TL

D8SD8 ,LDSD

J~ k,k,E!rND

1/2~E!u2, ~51!

for theND*→ND* reaction.We can further extend the above formula to calculate the differential cross sections involving aD* resonance in either the

entrance channel or the exit channel. Specifically, we can define a general scattering amplitude

g

54 1355MEDIUM EFFECTS ON BINARY COLLISIONS WITH . . .

Tab~kWa ,kWb ,E!5 (JMJ ,LaSa ,LbSb

yLaSa

JMJ ~ ka!yLbSb

JMJ1

~ kb!ra1/2~E!TLaSa ,LbSb

J ~ ka ,kb ,E!rb1/2~E!, ~52!

wherea,b can beNN orND* states, andukWau5 k,k0 for a5ND* ,NN. The differential cross sections are then of the followinform

dsab~E!

dV516p2

kb2

1

~2s1b11!~2s2b

11! (spins

u^m1am2a

uTab~kWa ,kWb ,E!um1bm2b

&u2. ~53!

n-eaac

rin

Th

-ithurricodctoti

ae

ro

ial-ruc-

icle-

It is easy to show that the total cross sections Eqs.~45!, ~48!,and ~50! can be obtained by integrating Eq.~53! over thescattering angleV.

IV. MEDIUM EFFECTS ON THE D PROPAGATION

The medium effects on theD propagation have been ivestigated in the study of intermediate-energy nuclear rtions. Within the isobar-hole model of pion-nucleus retions, theD-nuclear potentials have been determined@24#phoneomenologically from fitting the pion-nucleus scattedata. An experimentally based attempt was made in@25# byconsidering inclusive electron scattering from nuclei.theoretical approaches for calculating theD mean field weredeveloped in@14# and also in@26,27#. In this work the approach developed in@14# is used, since it is consistent wthepNN formulation outlined in the previous sections. Fthermore, it had obtained a good description of the empiD-nucleus potentials determined in the isobar-hole m@24#. We will first outline the main steps of the approadeveloped in@14#. The resultingD potential is then usedcalculate the medium effects on the scattering cross secdefined in the previous section.

In the one-hole-line approximation, the propagator ofDwith momentumpW D and energyeD in nuclear matter can bwritten as

GD~pW D ,eD!51

eD2ED~pW D!2S~1!~pW D ,eD!2S~2!~pW D ,eD!,

~54!

with

S~1!~pW D ,eD!5E QpN~k,pD!uhpN↔D~k!u2k2dk

AeD22pD

22EN~k!2Ep~k!1 i e,

~55!

whereQpN(k,pD) is an angle-average Pauli operator@14# forthe pN state. The medium effect on theD propagation iscalculated from theND G matrix;

S~2!~eD ,pW D!

5Ep<pF

dpW ^pW DpW uGND,ND$W5@eD1eN~pW !#%upW DpW &,

~56!

wherepF is the Fermi momentum of nuclear matter.

c--

g

e

-alelh

ons

It is convenient to calculate theG matrix in partial-waverepresentation. For symmetric nuclear matter~the total isos-pin and the total angular momentum are both equal to ze!,we find that

S~2!~pW D ,eD!51

~2sD11!~2tD11!

3 (LDSDJT

~2J11!~2T11!

4p

3Ep<pF

dpWGLDSD ,LDSD

JT

3@q,q,v~eD ,pW D ,pW !,P#, ~57!

whereq is the relative momentum,v is the collision energyin theND center-of-mass frame, andP is the total momen-tum. In the nonrelativistic baryon approximation, we have

P5upW D1pW u,

q5UmD0pW 2mNpW D

mD01mN

U,v~eD ,pW D ,pW !5eD1eN~pW !2mD

02mN2~pW D1pW !2

2~mD01mN!

,

wheremD051280 MeV is the bare mass of theD determined

from fitting thepN phase shifts inP33 channel.The equations for theG matrix are identical to Eqs.~6!–

~9! except that the propagators are modified. In the partwave representation, each equation has the following stture:

Ga,dJT ~q8,q,v,P!5Va,d

JT ~q8,q,v!1(gE q92dq9

3Va,gJT ~q8,q9,v!Qg~q9,P,pF!

v2Wg~q9,P!1 i e

3Gg,dJT ~q9,q,v,P!, ~58!

where the Greek subindices denote collectively the partchannels,NN or ND, and the orbital and spin quantum numbers. In the propagator of Eq.~58!, Qg(q9,P,pF) is theangle-average Pauli operator@14# and

Wg~q,P!5eN~ p!1eN~ p! ~59!

ve

lh

o

t

t

.a-

no-een

t

er-tri-

l

li-it

-

1356 54T.-S. H. LEE

for g5NN channel, and

Wg~q!5eN~ pN!1eD~ pD! ~60!

for g5ND channel. The single-particle energies in the aboequations are calculated by using the angle-average mom

p5Aq21 14 P2,

and

pN5AS mNP

mD01mN

D 21q2,

pD5AS mDP

mD01mN

D 21q2.

The nucleon single-particle energyeN(p) is taken from avariational many-body calculation by Wiringa@21#. Thesingle-particle energy for theD is determined by the follow-ing self-consistency condition:

Re@eD~pW D!#5mD01

pD2

2mD0 1Re$S~1!@pW D ,eD~D!#%

1Re@UD~pW D!#, ~61!

where we have defined theD potential

UD~pW D!5S~2!@pW D ,eD~pW D!#. ~62!

The medium effects on the cross sections of binary cosions defined in Sec. III can be included by replacing tscatteringt matrix in Eqs.~46!, ~49!, ~51!, and~52! by theGmatrix. The collision energy is calculated from real partsthe single-particle energieseN(p) of @21# andeD~pD! definedby the self-consistent condition, Eq.~61!. Accordingly, thesame mean-field effects are also included in calculatingphase-space factorsrNN andrND defined by Eqs.~38!, ~42!,and ~47!.

V. RESULTS AND DISCUSSIONS

We have applied the formula developed in Sec. IIIinvestigate the cross sections of binary collisions involvingD* excitation in either the entrance channel or the exit cha

FIG. 3. The calculatedNN→pNN ~solid curve! andNN→ND* ~dotted curve! total cross sections are compared. Thdata are from the compilation of@22#.

enta

li-e

f

he

oan-

nel. In Fig. 3, we show that theNN→ND* cross sections~dashed curve! calculated by using the factorized form Eqs~45!–~47! can reproduce reasonably well the exact calcultion of NN→pNN using Eqs.~36!–~39!. This suggests thatthe factorized forms defined by Eqs.~45!–~53! are sufficientfor investigating the main features of relativistic heavy-iocollisions. The calculated total cross sections for all prcesses involvingD* are compared in Fig. 4. They have quitdifferent energy dependencies. The difference betweNN→ND* andND*→NN is due to the flux factors 1/k0

2 ofEq. ~46! and 1/k2 of Eq. ~49!. This can be understood fromFig. 5 in which the calculatedk for the ND* channel iscompared with the momentumk0 of the NN channel. Thecalculated mass of theD* is also displayed there. We see thaas the collision energy increases the producedD* becomesheavier and moves faster.

The predicted differential cross sections at several engies are compared in Fig. 6. We see that the angular disbutions for NN→ND* are quite different from that ofND*→NN. This is due to the fact that the important partiawaves inNN andND are quite different. For example, in theJ52 channel the outgoing particles are in1D2 for the

e

FIG. 4. The predicted total cross sections for the binary colsions involving aD* in either the entrance channel or the exchannel are compared.

FIG. 5. The massM* and momentumk of the D* in theNN→ND* reaction.EL is the incident nucleon energy in the laboratory frame.

hehe

of

ema-

54 1357MEDIUM EFFECTS ON BINARY COLLISIONS WITH . . .

ND*→NN, but are in5S2 for theNN→ND* . Clearly such adynamical difference in differential cross sections cannoteasily obtained by using detailed balance@3–5# to relateND*→NN to the data of theNN→NNp reaction. We fur-ther notice that the predictedND*→ND* differential crosssections are forward peaked. This is also quite different frothe usual assumption thatND*→ND* is identical toNN→NN apart from some isospin factors. The resultshown in Figs. 4 and 6 are the consequences of the consered meson-exchange mechanisms.

To get some ideas about the medium effects on collisicross sections, we assume that the mean-field effects onD propagation are contained in the energy shiftS~1! of Eq.~55! and the potentialUD defined in Eq.~62!. As seen in Eq.~55!, the only medium effect included in the calculation otheD self-energyS~1! is the Pauli blocking of the intermedi-ate nucleon states. Consequently, its effect is mainly in tlow-momentum region and depends strongly on the nucledensity. This is clearly seen in Fig. 7. The most noticeabresult is that the imaginary part ofS~1! in the low-momentumregion is drastically reduced as the density increases. Thisof course due to the decrease of the available phase spacetheD→pN decay as the Fermi momentum in Pauli operatincreases. In a more ambitious approach, the mean fieldpion propagation must also be included in the calculationS~1!. This, however, requires a much more difficult numericatask in a self-consistent approach and is beyond the scopethis work.

The calculatedUD at several densities are displayed iFig. 8. Both the real parts and imaginary parts depestrongly on the density and the momentum. Qualitativelour results indicate that theD moves almost freely in nuclearmedium at high momentum, but is slowed down consideably and is easily annihilated at low momentum. It will beinteresting to explore the consequence of the predictedUD indetermining the pion yields in relativistic heavy-ion colli-sions.

The predictedUD(p) is used to evaluate theD single-particle energy, Eq.~61!, and theG-matrix elements whichare the inputs to the calculations of the cross sections

FIG. 6. The predicted angular distributions evaluated at energ~in terms of the corresponding nucleon laboratory energie!EL5400 MeV ~dotted curves!, 700 MeV ~solid curves!, and 900MeV ~dashed curves!.

be

m

sid-

onthe

f

heonle

isfor

orofoflof

nndy,

r-

in

medium, as explained in Sec. IV. We have found that tmedium effects are most dramatic in the region where tcollision energies are close to thepNN threshold in freespace. This can be illustrated in a calculation where aD*with a mass of 1236 MeV propagates under the influencethe mean fieldUD and collides with a nucleon at rest innuclear matter. The predicted dependence of theND*→NNandND*→ND* total cross sections on the density and thmomentum is displayed in Fig. 9. There are two mediueffects. The first one is the mean-field effect on the propag

iess

FIG. 7. The calculatedD self-energy(~1!~pD! at several densi-ties. The densityr is in units ofr50.16 nucleon fm23.

FIG. 8. The calculatedD potentialUD(pD) at several densities.The densityr is in units ofr050.16 nucleon fm23.

ta

hn-

toe

nf

io

re

d

tp

onu

he

ess

yd

lyts

is-

l-

f

-

e

--

r d

1358 54T.-S. H. LEE

tors of theG matrix in Eq.~58!. This effect is found to be novery large in the momentum region considered. The mmedium effect is due to the change of the collision energby the D potentialUD in the entrance channels. Since tpotential energy~real part! becomes more attractive as desity increases~Fig. 8!, the collision energy for a given momentum of the incidentD is shifted to the lower energyregion closer to the pion production threshold wherecross sections vary rapidly as shown in Fig. 4. We therefsee in Fig. 9 that theND* collision cross sections in thlow-momentum region depend strongly on the density, aare greatly suppressed at high densities. A similar situatioalso found forNN→ND* , as illustrated in the lower half oFig. 10. The corresponding medium effects on theNN→NNare less dramatic since theNN elastic cross sections do novary so rapidly in the energy region near pion productthreshold.

VI. SUMMARY

The binary collisions involving aD resonance in eithethe entrance channel or the exit channel have been invgated within a Hamiltonian formulation ofpNN interactions@11–13#. An averaging procedure has been developed tofine the experimentally measuredNN→pNN cross sectionin terms of an effectiveNN→ND* cross section. In contrasto previous works@4,5#, the main feature of the present aproach is that the mass and the momentum of the produD* at each collision energy are calculated dynamically frthe bareD↔pN vertex interaction of the model Hamiltoniaand are constrained by the unitarity condition. The proced

FIG. 9. The momentum dependence of the calculated total csections ofND*→NN andND*→ND* collisions in nuclear mat-ter. The densityr is in units ofr050.16 nucleon fm23.

n

iniese-

here

ndis

tn

sti-

e-

-cedm

re

is then extended to define the effective cross sections for texperimentally inaccessibleND*→NN andND*→ND* re-actions. The calculations have been performed to predict thenergy dependencies and angular distributions of these crosections.

TheD mean field in nuclear matter has been calculated busing the Bruckner-Hartree-Fock approximation developein @14#. It is found that theD moves almost freely at highmomenta, but is slowed down considerably and get easiabsorbed at low momenta. By including the medium effecon theD propagation, the dependence of the effective crossections of theND* collisions on the density has been inves-tigated. It is demonstrated that the density dependencemost dramatic in the energy region close to the pion production threshold.

Our results can be used in the relativistic heavy-ion caculations using transport equations@3–9#. To make furtherprogress in investigatingD-rich matter, it is necessary to in-vestigate theD*D* collisions and the effects due to thehigher massN* resonances. This requires the extension othepNN model Hamiltonian defined by Eqs.~4! and ~5! toinclude the twop channels and the development of a scattering theory withppNN unitary condition. Such a Hamil-tonian model can be accurately constructed only when thdynamics of our present understanding of theN* excitationsin pN andgN reactions is improved quantitatively. The pro-posed pion facility at GSI@2# and theN* program at CEBAFare essential for making progress in this direction.

This work was supported by the U.S. Department of Energy, Nuclear Physics Division, under Contract W-31-109ENG-38.

oss FIG. 10. The momentum dependence of the calculateNN→NN andNN→ND* collisions in nuclear matter. The densityr is in unit of r050.16 nucleon fm23.

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