Dielectron production from nucleus-nucleus collisions

Nuclear Physics A512 (1990) 772-786 North-Holland

DIELECTRON

PRODUCTION

FROM

NUCLEUS-NUCLEUS

COLLISIONS L. XIONG, Z.G. WU, C.M. KO and J.Q. WU ~ Cyclotron Institute and Physics Department, Texas A&M University, College Station, Texas 77843, USA

Received 17 November 1989 (Revised 30 January 1990) Abstract: Dielectron pairs can be produced in nuclear reactions from many processes such as the

proton-neutron bremsstrahlung, the delta decay, the annihilation of pions on nucleons, and the pion-pion annihilation. Based on the transport model, we calculate the dielectron-production cross section from nucleus-nucleus collisions and compare it to the available data.

I. Introduction

O n e o f the m a j o r p r o b l e m s in h e a v y - i o n collisions o f Bevalac energies is to u n d e r s t a n d the d y n a m i c s o f p i o n s in a d e n s e n u c l e a r matter. A c c o r d i n g to m u l t i p l e c o l l i s i o n m o d e l s , p i o n s d o not m a t e r i a l i z e until a late stage o f the collision w h e n the n u c l e a r d e n s i t y is quite low '). It is not k n o w n w h e t h e r this is a valid a s p e c t o f the m o d e l , a l t h o u g h these m o d e l s d e s c r i b e the e x p e r i m e n t a l d a t a r e a s o n a b l y well. Also, the m o d i f i c a t i o n o f the p i o n p r o p e r t i e s in n u c l e a r m a t t e r d u e to the strong p i o n - n u c l e o n i n t e r a c t i o n is not i n c l u d e d in these m o d e l s . Recently, Bertsch et al. 2) have s h o w n that the p i o n collectivity in d e n s e n u c l e a r m a t t e r increases a p p r e c i a b l y the n u c l e o n - n u c l e o n i n e l a s t i c cross section in the m e d i u m . A s i m i l a r b u t s m a l l e r effect has also b e e n o b t a i n e d b y W u a n d K o 3) for the k a o n p r o d u c t i o n cross section in dense n u c l e a r matter. It is c e r t a i n l y i m p o r t a n t in the future to c o n t i n u a l l y d e v e l o p m o d e l s t h a t w o u l d i n c l u d e these effects. But b e f o r e such a realistic m o d e l is a v a i l a b l e , it is o f equal i m p o r t a n c e to conceive e x p e r i m e n t s that w o u l d d e t e r m i n e directly the p i o n d y n a m i c s in h e a v y - i o n collisions. In this respect, m e a s u r e m e n t s o f b o t h p i o n m u l t i p l i c i t y a n d energy s p e c t r a are likely to p r o v i d e o n l y limited k n o w l e d g e a b o u t p i o n d y n a m i c s as t h e y m o s t p r o b a b l y reflect the final l o w - d e n s i t y stage o f the collision. It is t h e r e f o r e o f interest to find o b s e r v a b l e s w h i c h p r o b e the p i o n d y n a m i c s in d e n s e n u c l e a r m a t t e r t h a t exists in the e a r l y stage o f the collision. O n e u n i q u e o b s e r v a b l e is the p r o d u c t i o n o f dielectrons. Since d i e l e c t r o n s will not u n d e r g o further i n t e r a c t i o n s with the h a d r o n i c m a t t e r o n c e t h e y are created, they have an i n c r e a s e d sensitivity to early stages o f the collision. * Present address: Teleco Oilfield Services Inc., 105 Pondview Drive, Meriden, CT 06450, USA. 0375-9474/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)

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773

Experiments have already begun at the Bevalac to study dielectron production in both proton-nucleus and nucleus-nucleus reactions 4.5). Preliminary data on the dielectron invariant mass spectra are available and have stimulated much theoretical work 6-,3). In this paper, we shall first discuss in sect. 2 the various elementary processes that contribute to the production of dielectron pairs. In sect. 3, dielectron production from nucleus-nucleus collisions is studied in the Vlasov-Uehling-Uhlenbeck model. The results will be compared to the available data. Finally, conclusions are given in sect. 4.

2. Elementary processes for dielectron production 2.1. PROTON-NEUTRON BREMSSTRAHLUNG

Dielectron production from proton-neutron interaction is more important than that from p r o t o n - p r o t o n interaction because of the cancellation of the electromagnetic currents in the latter case. The Feynman diagrams for this process are shown in fig. 1 where the dashed lines denote all allowed mesons. Three diagrams are not shown explicitly. Two are the exchange diagrams of (a) and (b) and are obtained by interchanging the initial proton and neutron. The other corresponds to dielectron production at the left vertex of diagram (c). The first microscopic calculation of these diagrams has been carried out by Sch~ifer et al. 6). Similar attempts are being pursued by Haglin et al. 7). For simplicity, we shall use in the present work the n

p

n

P

p

n

e

P

((]) p

n

(b) n

Pe-

e+n

....... < i i n

p

(c)

n

P

(d)

Fig. 1. Feynman diagrams for dielectron production from proton-neutron bremsstrahlung.

774

L. Xiong et al. / Dielectron production

soft-photon approximation with the phase-space correction 8). In this approximation, the cross section for producing a dielectron pair of invariant mass M and momentum p is given by do'e;eot2 ~(s) R2(v~2) d3p d M ~67r 3 M E 3 R2(v/'s) "

(1)

In the above, a is the fine-structure constant and O(s) is the momentum-transfer weighted proton-neutron elastic cross section at a center-of-mass energy x/~ [ref. 9)]. The function R2(x/~) is the Lorentz invariant two-body phase-space integral of the final two nucleons of energy x/-~. The energy of the two nucleons after emitting a dielectron with energy E is denoted by s2. In ref. 7), the soft-photon approximation with the phase-space correction gives a smaller contribution than that of the explicit diagrammatical calculations. But the approximation of using only the pion-exchange in ref. 7) may not be adequate. In ref. 6), comparisons have been made between the soft-photon approximation without the phase-space correction with the explicit evaluation of the bremsstrahlung diagrams including the exchange of other mesons besides the pion. Although the former is seen to lead to a larger contribution than that from the latter, the inclusion of the phase-space correction to the soft-photon approximation, however, improves the agreement between the two ~3).

2.2. DELTA DECAY The delta particle is produced from the nucleon-nucleon inelastic scattering. Its contribution to dielectron pair production can be obtained from the diagrams (a) and (b) of fig. 1 with the delta inserted between the strong-interaction vertex and the photon vertex. Since the final state of the delta decay diagrams is indistinguishable from the proton-neutron bremsstrahlung diagrams, all these diagrams should be summed coherently. Because of the finite width of the delta particle, the amplitude with the delta in the intermediate state has a time delay with respect to that without the delta. The interference between the two processes is thus negligible for dielectron pairs with energies greater than the width of the delta, i.e. about 115 MeV. The corresponding dielectron invariant mass above which the interference effect is insignificant will be even smaller. As this is the region of dielectron invariant masses we are interested in, we shall therefore treat separately the proton-neutron bremsstrahlung with and without the intermediate delta. The latter case can further be considered as a two-step process in which a delta is first produced in the nucleon-nucleon inelastic scattering and then decays into the dielectron pair. The partial width of delta decaying into a dielectron pair with invariant mass M can then be evaluated according to the diagram in fig. 2. We take the yNA transition vertex from ref. ~4), i.e. F ¢ . = G , ( q 2 ) r ~ . + G~(q 2) F ~2 . + G3(q2)F3~.

(2)

L. Xiong et al. / Dielectron production

775

P_

Fig. 2. Feynman diagram for delta decay into the dielectron. where

F ~ = e(q~%, - g ~ , q " Y)Ys, F ~ , = e[q~(PN+Pa), --~g~q" (pN +pa)]'Y5, IF3~, = e( q~q~, - g ~ q 2 ) y s .

(3)

In the above, the nucleon and delta masses are denoted by m N and ma, respectively; their four m o m e n t a are PN and pj while that of the virtual photon is q; the Dirac matrices are denoted by "y and g ~ is the metric tensor; e is the proton charge; and the form factor at the vertex is given by Gi(q2). Eq. (2) contains contributions from magnetic dipole, electric quadrupole, and C o u l o m b quadrupole interactions. Keeping only the dominant magnetic dipole term, the y N A vertex can be written as M --3et~(ma + mN)GM(q 2) F ¢ , = 2mN[(ma + mN) 2 -- M 2] '

(4)

where 1

3

(5)

~. = - m ~ G . + r~. + ~r~.,

and GM(q 2) is the magnetic dipole form factor. The square of the invariant amplitude, Mri, for delta decay into the dielectron can be expressed as the product of the leptonic part ~:"" and the hadronic part W "", i.e. IM,-d2 = W . . s~'~. The leptonic part has the form v_~_

~:~'"=4[p~p_

p +v p _p. - ~ g1

~tv

2

(p++p_) ]

(6)

where p+ and p_ are, respectively, the four-momenta of the positron and the electron. Averaging over the angle of the electron m o m e n t u m , k, in the dielectron center-ofmass, we have 1

4~

f I~~" d ~ = 2 ( 4 k 2 - q 2 ) g ~ + 8 ( ~ k 2 + m ~ ) q " q " / q 2 ~ q Z ( _ g ~ , . + q~ q,,/ q2) .

(7)

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776

The second step follows when the small electron mass is neglected compared to q2, which is just the square of the dielectron invariant mass. The hadronic part can be written as Wl~ v =

M+ M Tr [P ot fl (pa)yoFo. To(T" PN + mN)F~..],

(8)

where P~t~(pa) is the projection operator of delta and has the form e"°(Pa)

=

(Y" Pa + ma)(g'~O 2p~p~3m~ Y"T°3+pxy-~3ma-p°ay~]"

(9)

After some lengthy algebras, we obtain the following expression for the partial width of delta decaying into a dielectron pair with invariant mass M, dFe+e- a 2 2,2 d M - 12~-IOM(q )

T(M, ma),

Iql

(10)

where [ 2mE

3ma

T(M, ma)=Lma+mN (ma+mN)2

l= M2j (T,+T2+Ts),

(11)

with

8[

TI=~ 2 ( p a ' q ) ( p N ' q ) - - 4 16 [(Pa" q)2 + - 3- [_ m~

m, ( P a ' q ) 2 + ( p a ' p N + m a m N ) M 2 ] ma

] M 2_(Pa'pN+2mamN)'

T2=3-~~ (p,. pN-m~mN){2(p,,. PN)(Pa" q)(P~ " q)+ m2a( pN • q)2 -mam~ L - ~

]}

M2

,

T3 = 16mN [(pa_~_q)2 ] 3m---~-[ m2a M2 (Pa'pN--mamN)+~(PN'q)2 32mN 8m~ 3ma (Pa" q)(PN" q)+~5-(Pa'3ma q)2 16mN 8M 2 3m 3 (Pa" PN)(Pa " q)2+ 3--~a [(Pa" PN) 2- m2am2N].

(12)

The magnetic dipole form factor GM at the yNA vertex has been extracted in ref. ~4) for space-like photon and has a value of ~3 for real photon. For time-like photon as in delta decay into the dielectron pair, no experimental information is known. To account for the structure of the vertex, we use therefore the following form, GM(q 2) = 3A21/(A~+ q2), where q is the photon three-momentum in the delta

L. Xiong et aL / Dielectron production

777

rest frame. To reproduce the experimental branching ratio of 0.6% for delta of mass ma = 1232 MeV decaying into a photon requires a cutoff parameter A, = 0.725 GeV. The probability that a delta decays into a dielectron pair with invariant mass M is thus given by the ratio of this partial width to the delta total width which is given by ~5) 0.47q~

F(q,~) [l+0.6(q,~/m~.)2]m2,

(13)

where rn= is the pion mass and q= is the value of its m o m e n t u m in the delta rest frame. We note that only za+ and a ° can decay into the dielectron due to charge conservation. Our treatment of the delta contribution to dielectron production is different from that of ref. 8) in which not only a different form factor but also the soft-photon approximation has been used. In the soft-photon approximation, the partial width of delta decaying into a dielectron is obtained from the ratio of its width F~ of decaying into a real photo n to its total width corrected by the phase-space factors. Leaving out the form factor, we find that for a delta of given mass the dielectron yield obtained with the soft-photon approximation (dashed curves) is larger than that from the exact calculations (solid curves) as shown in fig. 3 for four different

1

°-34]

10-41

1.132 GeV

1.232 GeV

I0-' I 0-"

10-~

1.332 GeV ",

10-'

10"

4

kl (GeV)

0

0.1 012 013 0~4

M

(GeV)

0

Fig. 3. Partial width of delta decay into a dielectron pair with invariant mass M for delta masses of 1.132, 1.232, 1.332 and 1.432 GeV. The solid curves are from the microscopic calculation of the diagram in fig. 2 while the dashed curves are from the soft-photon approximation.

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778

delta masses of 1.132, 1.232, 1.332 and 1.432GeV. In the calculation with the soft-photon approximation, we have used the exactly calculated Fr and the total width in eq. (13) instead of the forms used in ref. 8).

2.3. PION ANNIHILATION ON NUCLEON

Although the interaction of a pion with nucleon is most likely to form a delta again, it is possible that the total center-of-mass energy of the pion and nucleon system is such that the delta-formation cross section is negligible. In this case, we need to take into account the contribution to dielectron production from the pion-nucleon interaction. Energetically, we expect that dielectron production from the pion-nucleon bremsstrahlung is less important than that from pion annihilation on nucleon and is thus neglected in the present work. The contribution from pion annihilation on nucleon is calculated according to the diagrams in fig. 4. The diagram (d) which contains a four-particle vertex is a result of gauge invariance as we have used the pseudovector coupling for the pion-nucleon interaction. In diagrams (b) and (c), we do not allow a delta in the intermediate state as it has been treated explicitly via delta decay. Due to isospin conservation, we only have diagrams (a), (b), and (d) for 7r+n interaction; diagrams (b) and (c) for 7r°p interaction; and diagrams (a), (c), and (d) for 7r-p interaction. The isospin-averaged cross section

(a)

(b)

.-jo, (c)

(d)

Fig. 4. Feynman diagrams for dielectron production from pion annihilation on nucleon.

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779

for pion-nucleon annihilation into a dielectron pair with invariant mass M is thus e+e-

dtr,N dM

e+e e+e 1 ( 2 d ° ' r r + n 4- dOrTr°p 4- 2

6\

dM

dM

e+e -

dtr;p~ dM /

.

(14)

As in delta decay, the square of the invariant amplitude can be written as the product of the leptonic and the hadronic parts. Evaluating the leptonic part using eq. (7), we then have d O ' ~e+e N-

dM

g~a21qlM 3

1

36~r x / s ( s - ( m N + m ~ ) 2 ) ( s - ( m N - m ~ )

xlF((v,-vO~)l ~

d(cosO) 2

2)

-g'"4-

p,,v= 1,4

× [ 2 T r ( T , . T ~ .+) + T r ( T 2 , . T 2 ~+) + 2 T r (T3~. T3~)] + ,

(15)

where T j ~ --- T(a) T(b) 4/z 4- --,u,

T~ )

T2,~ -----Y (b) 4- Y (c) Z3p~ ~--"z~a) --[--T ~ ) 4 - T(~d) '

(16)

with T(a)= _F~(q2)(k +P~),ff(Pf)Ys(Y" k)u(Pi)/[q2( k 2 - m2)] kL T~ )=

-ff( Pr ) %, ( Y " P, + mN) yS( y " p~)u( pO /[ q 2( p~ - m ~ ) ] ,

T(d)= - a ( p f ) r d T - p,~)(7 • p2+ mN)%,u(pi)/[q2(p~-- m ~ ) ] , T ~d)= a ( p f ) y s y , u(pi)/q 2 .

(17)

In the above, the four-momentum of the nucleon before and after interaction is denoted by pi and pf, respectively, while q is that of the virtual photon. The angle between the initial and final momenta of the nucleon is given by 0. We denote the nucleon spinor by u(p). The pion-nucleon coupling constant g= is taken to have a value of g~/4~- = 14.4. To take into account the complicated structure of the strong-interaction vertex and to preserve also gauge invariance, an overall form factor of monopole type as that used in studying photopion production ~6) has been introduced, i.e. F ( ( p i - p f ) 2 ) = A2/(A2+ 2 2 (p~-pr)2), where A2 is the cutoff parameter with a value ~0.6 GeV. We have also included the pion electromagnetic form factor at the yercr vertex of diagram (a). It is dominated by the rho meson and has the form 9) 4

m, IF~(q)] 2 - ( q 2 rap) t2 2 --52 2, rFlpCp

(18)

with mp = 775 MeV, m~ = 761 MeV and Fp = 118 MeV. Another way of preserving gauge invariance in the presence of the strong-interaction form factor is to introduce

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780

additional diagrams in which the virtual photon is emitted from a heavy meson of mass A2 as in ref. 7). It can be easily shown that the latter method is not applicable to the present case as the additional diagrams will simply cancel the effect from the strong-interaction form factor. It is complicated to carry out the traces in eq. (15). Instead, we evaluate them numerically. In fig. 5, we show the dielectron invariant mass spectra for different center-of-mass energies of the pion-nucleon system. The solid, long-dashed, short-dashed and dotted curves are obtained, respectively, for the total pion-nucleon center-of-mass energy of 1.33, 1.58, 1.83 and 2.08 GeV. We note that its contribution decreases with the energy. The probability of dielectron production from pion-nucleon annihilation is given by the ratio of eq. (15) to the total pion-nucleon nonresonant cross section which is about 30 mb [ref. 17)]. 2.4. P I O N - P I O N A N N I H I L A T I O N

A dielectron pair can be produced in p i o n - p i o n annihilation via the diagram in fig. 6. For two pions with a total center-of-mass energy M, the cross section is given by 9) ~+~_ 4"n" O~2 { _ 4 m 2 " ~ 1/2 ~r~= - 3 M2 \1 M e ] IF=(M)[ 2.

(19)

10.

1:

(_9

I0-'! °O~ '%.. %

ci.

--~

-o 10 -2

....

.,,

\ I

b

I

k,

.-'1 10 -3

1%,, I '

~s <

<

I 10-~).0

0:2

04

M

o:~ (o v)

I I

o28

~o

Fig. 5. Dielectron invariant mass spectra from pion annihilation on nucleon. The solid, long-dashed, short-dashed and dotted curves are obtained, respectively, for the total pion-nucleon center-of-mass energy of 1.33, 1.58, 1.83 and 2.08 GeV.

L. Xiong et al. / Dielectron production 7]" ~.,

781 +

TI.+ s" Fig. 6. Feynman diagram for pion-pion annihilation into the dielectron.

Due to the rho meson dominance of the pion electromagnetic form factor, only ~r+~r- can annihilate into dielectrons as a result of isospin conservation. The probability of dielectron production from pion-pion annihilation is given by the ratio of eq. (19) to the total pion-pion cross section, which we use the parametrization of Bertsch et al. is).

3. Dielectron production from nucleus-nucleus collisions For collisions between heavy nuclei, the compression of nuclear matter leads to a substantial change of the nuclear mean-field potential as a result of its density dependence. One of the major goals of heavy-ion collisions is to extract this density dependence so that the nuclear equation of state at high densities can be determined. The model which extends the cascade model to include the mean-field is called the Vlasov-Uehling-Uhlenbeck (VUU) model ~9). In this model, the nucleon phasespace distribution function f(r, p) satisfies the equation

Of~/Ot+v'V,fl-VrU'Vpf,=f

d3p2 f dg2cr(~)[v~-v2' × [f',f~(1 - f 0 ( 1 - f 2 )

-flf2(1

-F)(1 -f~)] •

(20)

In the above, the mean field potential U is related to the density (denoted by p) derivative of the single-particle potential energy V via U = d(pV)/dp. Based on the Skyrme interaction, V can be expressed as

V(p) = a(p/ po) + b(p / po) ~ ,

(21)

where po denotes the normal nuclear matter density. The parameter a determines the stiffness of the nuclear equation of state. For a = 2, the nuclear compressibility is 380 MeV and is normally called a stiff equation of state. In this case, one has a = - 6 2 MeV and b = 23.5 MeV from fitting the nuclear matter binding energy at saturation density. A soft equation of state with nuclear compressibility of 200 MeV is obtained if c~ =7. Then one has a = - 1 7 8 MeV and b = 140 MeV. The right-hand side of the above equation is the collision term with tr(g2) being the nucleon-nucleon differential cross section which is normally taken to be that in the free space. It also includes the Pauli-blocking effect of the final phase space as shown by the factors in the square bracket. The VUU model has been generalized to include a momentum dependent potential 20) and also to the relativistic case using the Walecka model 21,22).

782

L. Xiong et al. / Dielectron production

The V U U model is solved with the test particle method 23) by carrying out an ensemble of parallel cascade-type simulations. Particles can be shown to move in the mean-field potential following the classical equations of motion until it collides with another particle. This occurs when the distance between them is less than x/-~/7r with tr being the interaction cross section of the two particles. After the collision, the direction of the particle changes in a statistical way according to the angular distribution. Collisions are allowed only among particles in the same simulation but the nuclear mean field is computed from the local density using eq. (21), which is, however, calculated with all nucleons in the ensemble. The probability that a collision occurs also depends on the available phase space for the two particles after the collision. A delta is produced in the collision with a probability given by its production cross section divided by the nucleon-nucleon total cross section. The mass of the produced delta is determined by the following probability distribution is)

¼F2(q~) P(m,a) = (ma - mo)2+¼F2(q~) '

(22)

where the centroid is mo = 1232 MeV and the width F(q=) is given by eq. (13). The decay of the delta is treated statistically based on its half-life, which is the inverse of its width. After the decay of the delta, a pion is produced isotropically in its rest frame. The motion of the pion then follows that of a free particle as in the cascade model ~) until it makes a collision with another particle. The collision of the pion with a nucleon leads mainly to the formation of the delta. However, a nonresonant elastic cross section of about 30 mb [ref. ~7)] is used for describing their collision if the total energy of the pion-nucleon system is such that the delta-formation cross section is less than this value. Similarly, the cross section for pion-delta interaction is taken to be 30 mb. For the pion-pion elastic scattering, we use the cross section of ref. 18). We have used the VUU model with a stiff equation of state to calculate dielectron production in the collision of two Ca nuclei at an incident energy of 1.05GeV/nucleon. Dielectron production is determined perturbatively as in photon 24) and kaon 25) production. Its production probability at each impact parameter is calculated by adding contributions from the proton-neutron bremsstrahlung, the delta decay, and the pion-nucleon interaction. The total dielectron-production cross section is then obtained by integrating the production probability over the impact parameter. Because of the small pion density in each simulation, the probability for the occurrence of a p i o n - p i o n interaction is small. To calculate the probability of pion-pion annihilation into dielectron, we thus allow pions from different simulations to interact as well. The true dielectron production probability is then obtained from dividing the total probability by the square of the number of simulations in the ensemble rather than simply the number of simulations as in photon and kaon production. This method is valid since we treat pion-pion annihilation perturbatively.

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783

The results are shown in fig. 7 for the dielectron invariant mass spectrum. In order to compare to the experimental data 4.5) shown by the solid squares, the theoretical results have been corrected for the detector acceptance filter which requires that we also determine the m o m e n t u m distribution of the dielectron pairs. This reduces substantially the dielectron yield at small invariant masses. The total contribution given by the solid curve agrees fairly with the data. We have also shown in the figure the contributions from all four elementary processes. It is seen that the p i o n - p i o n annihilation (dashed-dotted curve) dominates the dielectron spectrum at large invariant masses while both p r o t o n - n e u t r o n bremsstrahlung (long-dashed curve) and delta decay (short-dashed curve) are important at smaller invariant masses. The contribution from the p i o n - n u c l e o n interaction (dotted curve) is, however, unimportant as all pions are produced from delta decays and are likely to form deltas again when they interact with nucleons. A dip occurs in the dielectron spectrum from the p i o n - n u c l e o n interaction and can be understood as follows. We have already shown in fig. 5 that the dielectron production cross section decreases with the energy of the p i o n - n u c l e o n system. On the other hand, the probability that the p i o n - n u c l e o n system does not form a delta resonance has a minimum at the delta mass. Weighting the dielectron production probability by the probability of finding the nonresonant pions leads to the appearance of a dip in the dielectron invariant mass spectrum. In fig. 8, the time dependence of the total number of dielectron pairs (solid curve) from the central collision corresponding to an impact parameter b = 0.65 fm is shown. Also shown in the figure is the time dependence of the central density of

Ca+Ca 1.05 GeV/A pn .........

B

aN

lOrD

'

~. I 0 -~

~

',, •

0-3

b "~ 10-4 0.0

0.2

0.4

0.6

0.8

1.0

M (GeV) Fig. 7. Dielectron invariant mass spectra for Ca + Ca collisions at 1.05 GeV/nucleon. The experimental data of refs. 4,5) are given by the solid squares while the theoretical total yield is given by the solid curve. The contributions from p r o t o n - n e u t r o n bremsstrahlung, delta decay, pion-nucleon interaction and p i o n - p i o n annihilation are given by long-dashed, short-dashed, dotted and dashed-dotted curves, respectively.

L. Xiong et al. / Dielectron production

784

3..

,30. Ca+Ca 1.05 GeV/A

b = 0.65 fm 20. 0 I

..

oO

. o

,.Q

.o...o.o,. .....

Q,

10. z

o&

10.

15.

2b.

25.

,30. 0.

t ( fm/o ) Fig. 8. Time dependence of the central density and the n u m b e r of dielectron pairs, deltas and pions for the central collision of two Ca nuclei at 1.05 GeV/nucleon.

the colliding system, and the delta (dashed curve) and pion (dotted curve) numbers. It is clearly seen that dielectron pairs are mainly produced in the compression stage of the reaction when the density is more than twice the normal nuclear matter density. It is worth noting that there are more deltas than pions in this stage of the collision. Our results on dielectron invariant mass spectrum are similar to that of ref. 13) in which the contributions to dielectron production from the proton-neutron bremsstrahlung and the pion-pion annihilation have also been calculated with the VUU model.

4. Conclusions

In nucleus-nucleus collisions, dielectron pairs can be produced from protonneutron bremsstrahlung, delta decay, pion-nucleon interactions and pion-pion annihilations. These elementary processes have been evaluated. For proton-neutron bremsstrahlung, the soft-photon approximation with the phase-space correction has been used. The branching ratio of delta decay into the dielectron has been calculated in the magnetic dipole approximation. For dielectron production from both pionnucleon interaction and pion-pion annihilation, we have evaluated the Born diagrams exactly. To describe the dynamics of nucleus-nucleus collisions, we have used the VlasovUehling-Uhlenbeck model. The calculated dielectron invariant mass spectrum for C a + C a at 1 GeV agrees fairly with the data. It has been found that pion-pion annihilations dominate at large dielectron invariant masses while both protonneutron bremsstrahlungs and delta decays are important at smaller dielectron invariant masses. At this energy, the contribution from the pion-nucleon interaction

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is not important. Most dielectrons are produced in the initial stage of the collision when the density is high. It is thus expected that they will carry information on the property of high-density nuclear matter. In the above calculations, pions are treated as free particles. As pointed out in the introduction, the property of pion in nuclear matter is expected to be modified due to the strong p i o n - n u c l e o n p-wave interaction. Gale and Kapusta 9) have demonstrated with a simple form of pion e n e r g y - m o m e n t u m relation that the yield of dielectron with invariant mass around twice the pion mass is substantially enhanced if the softening of pion spectrum in dense nuclear matter is taken into account. Xia et al. lo) have confirmed this effect with a more realistic pion dispersion relation obtained in the delta-hole model. They have also included the expansion dynamics of the nuclear system in a simple hydrodynamical model and have found that the above effect remains present. Both these earlier studies ignored the imaginary part of the pion self-energy, and therefore obtained singular cross sections. The effect of the imaginary part of the pion self-energy on dielectron production has been studied by Ko et al. 11). They have found that this indeed leads to nonsingular dielectron yield. This is unfortunately not clear in C a + C a data due to the large experimental uncertainties. It is thus important in the future to include the medium effect in the VUU model so that we will be able to explore the possibility of studying the property of pion in dense nuclear matter when more accurate data are available. The authors are grateful to Linhua Xia for helpful discussions. This work was supported in part by the National Science Foundation under Grant No. PHY8608149 and No. PHY-8907986, and the Robert A. Welch Foundation under Grant No. A-1110.

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