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Dilepton production
Nuclear PhysicsA495(1989)423c-444~ North-Holland,Amsterdsm
423~
DILEPTON
PRODUCTION
Charles Gale Atomic Energy of Canada, Ltd. Theoretical Physics Branch Chalk River Nuclear Laboratories Chalk River, Ontario Canada KOJ - 1JO
Joseph Kapusta* School of Physics and Astronomy University of Minnesota Minneapolis, Minnesota 55455
The general features of dilepton production from nucleon-nucleon Estimates are made of and nucleus-nucleus collisions are discussed. the thermal production rates arising from incoherent nucleon-nucleon It may be possible to scattering and from two-pion annihilation. infer the pion dispersion relation in hot and dense nuclear matter by measuring the invariant mass distribution of back-to-back electrons and positrons in the center of mass frame in high energy nucleus-+ _ Comparison to the recent data on p + Be + e e X nucleus collisions. at 4.9 GeV from the Bevalac is made.
1.
INTRODUCTION One of the most impressive
the 1970's was a measurement
of pip- production
by the Columbia-Fermilab-Stony invariant
mass distribution
high energy physics
d*o/dMdy
in the data are the decays primarily
via gg fusion.
in terms of the Drell-Yan target
and projectile
The theoretical collisions
process,
for masses
states,
which
study of dilepton
with the suggestion
03759474/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
the
Quite
evident
the data can be understood from the
photon.
production
on nuclei
are formed
where a quark and an antiquark
to a virtual
out in
2 5 M zz 20
in cross section.
Away from these few peaks
annihilate
was initiated
They measured
at mid-rapidity
of cc and b6 bound
carried
from 400 GeV protons
Brook collaboration!
The data spans nine orders of magnitude
GeV.
experiments
in nucleus-nucleus
that it may serve as a
C. Gale and J. Kapusta / Dilepton production
424~
2 thermometer of quark-gluon plasma formed at ultra-relativistic energies. liowever,dileptons should be produced at lower energies as well, where no quark-gluon plasma formation is expected. The argument is simply that whenever charged objects collide they radiate real and virtual photons, the virtual photons decay into dilepton pairs. The advantage of observing electromagnetic signals from a strongly interacting many-body system is that they travel relatively unscathed from the production point to the detector. Since production rates are rapidly increasing functions of temperature and density these electromagnetic signals are good probes of the early high temperature and density stage of heavy ion collisions. A relevant analogy can be made with neutrino emission from the sun or from 8 cooling neutron star. The purpose of this work3 is to provide an exploratory study of dilepton radiation from finite temperature and density nuclear matter and to make a preliminary comparison with the recent p(4.9 GeV) + Be data of the DLS group. A rigorous study is extremely difficult due to the strongly interacting nature of nuclear matter:
for example, there are complications from two and three
particle correlations, three body collisions, identification of the relevant degrees of freedom (nucleons, baryonic resonances, mesons, collective degrees of excitation, quarks, etc.), and finite temperature and density corrections to form factors, widths, and so on.
Therefore our analysis will at times be
quite phenomenological, and our study will be based primarily on relativistic kinetic theory.
2.
GENERAL DISCUSSION OF DILEPTON PRODU~ION To begin the discussion let us first consider bremsstrahlung in hadron-
hadron collisions. For example, in the reaction np-np, photons (real and virtual) can be radiated since charged particles are accelerated. In the soft photon approximation photons are radiated only from the initial or from the 4 final charged lines. 1.
None come from the strong interaction blob.
See figure
This approximation is valid if the energy carried by the photon is less
C. Gale and J. Kapusta / Dilepton production
Fig. 1.
425c
Radiation from the external proton lines in np scattering.
than the inverse of the strong interaction collision time. The latter is usually estimated as about l-2 fm/c, so that this mechanism is dominant if E7 < r$-
100-200 MeV.
strong interaction blob.
For hard photons we must in addition look inside the One contribution is shown in figure 2.
5 represents radiation from the exchange of a charged pion. internal pion line will be important for energies E > 7; Y
It
Radiation from the - 100-200 MeV.
Of
course the separation into soft and hard photons is not sharp. The relative contribution of radiation from internal and external lines in general depends not only on the average collision time but also on other variables, such as the momentum transfer to the charged particles in the final state, on the angle of emission of the photon and on the available center of mass energy. For virtual photons, that is for dilepton production, it is possible to extrapolate the soft photon predictions from q2 - 0 to q2 - M2, M being the invariant mass of the e+e- pair. The result is6 (qP - pr + p'>
Fig. 2.
Radiation from an internal charged pien line in np scattering.
e+ A
B
e-
&
I I
1 2
A +
B
n
1
F
1 I
2
n
e+
e-
e+ A + B Fig. 3.
2
e-
1
I I
2 +
*
e
*
n
Soft radiation from nucleus-nucleus collisfons is a coherent superposition of radiation from the initial nuclei, A and B, and from the charged particles in the final state.
+ -
ee
dol E+E_ $- L-1 2 2 qo d3q * d p+d3p_ 2x q
(1)
For hard virtual photons thfs extrapolation is in general not possible since the off-shell continuation of the strong interaction blob is different for q2 -0
and q2 > 0.
Note that real photon production is of order a while dilepton
production is of order a*. In nucleus-nucleus collisions the possibilities are even more interesting. Consider a collision between nucleus A and nucleus B leading to some distribution of particles in the final state. A and B may radiate a photon before impact, or the charged particles may radiate a photon in the final state.
See figure 3.
If the photon energy is less than the inverse
427~
C. Gale and J. Kapusfa J Dil~pfon p~~~u&fion
nuclear collision time, estimated to be 10-20 fm/c, E7 < ri
- 10 - 20 MeV,
the charges will act coherently over the whole extent of the nucleus. In that case the cross section for real photons is proportional to Z2a, and for dileptons proportional to Z202. At high incident energy the cross sections peak at angles near 0' and NO0
in the CM (although they vanish at 0' and
7 180') due to the sudden deceleration along the beam axis. It is known from computer simulations that the nucleons cascade during the collision. In order to resolve the finer features of the nucleus-nucleus collision, such as the cascading, we muat look at higher energy photons. for lo-20 MaV - ri 8 cascade.
< r;=
100-200 MeV we
can see
ThUS
the individual baryons
This radiation will be incoherent, proportional to Z, but will also
increase with the number of collisions per baryon roughly as Zl/3 .
At even
higher photon energies radiation from internal NN blobs will become important, as in figure 2. The x0 lifetime is 10
-16
sec. It
decays outside the reaction zone in a
heavy ion collision. The branching ratio into 7-yis 98.85%. and into ye+=(Dalitz decay) is 1.15%. The Dalitz decay may be viewed as a background to other e+e- sources. Fortunately phase space restricts the invariant e+emass M to less than mso - 135 MeV. and 99% have
M <
9 65 MeV.
In fact, 90% of the pairs have M < 15 MeV,
Still, the Dalitz pairs can be approximately
+ subtracted out by taking the x0 spectrum to be the average of the s and Ispectra and folding in the decay distribution. Also the n meson has a decay mode ye+e-, but its branching ratio is only 0.5%, and besides the q should not be abundantly produced at the temperatures T - 40-100 MeV of interest to us here. In the nuclear many-body environment there will be a profusion of microprocesses involving nucleons, pions and deltas. If a charged particle is involved it will radiate. A detailed comparison with experiment will require a complete enumeration of all these processes and their inclusion in BUU or 10 cascade computer simulation calculations.
This will allow us to test our
428~
C. Gale and 1. Kappusta / Dile~ton production
understanding of collision processes in hot and dense nuclear matter. So far our discussion has not distinguished between real and virtual photons. In fact for measuring bremsstrahlung radiation real photons are favored over dileptons since the dilepton production rate is smaller by a factor of a.
The real advantage of dileptons over photons arises when
focussing on annihilation processes. Kinematically, annihilation of two massive particles into a single real photon is not allowed, and -y-y correlation measurements are more difficult than e'e* measurements. The most important* annihilation process is s+s- + T* * e+e-. Annihilation of hadrons always leads to a minimum energy, in this case E 2 q* - X2 2 &rnq. Thus in high temperature nuclear matter, and in high energy heavy ion collisions, we expect the rate for production of high mass e'e- pairs, M > 2m=, to be dominated by pion annihilation. That region of invariant mass should provide information on the abundance of pions in the hot and dense medium since the production rate is proportional to the square of the pion number density. We view annihilation processes to be of most interest from the point of view of dilepton production. Bremsstrahlung from cascading baryons and mesons is best studied via real photons, and from the perspective of annihilations forms a background.
3.
PROTON-NEUTRON VIRTUAL BRRMSSTRARLUNG Now we make a rough estimate of the production of dileptons by the
cascading baryons.
Consider a two-body reaction a + b + c + d where a,b,c and
d may be either a nucleon or a delta. These are the primary reactions in current cascade or BUU Monte Carlo simulations of heavy ion collisions.lo
In
the soft-photon approximation a, b, c and d may each radiate, depending on its charge. Among the three reactions pp + pp, np + np and nn * nn, the most important are neutron-proton collisions.* This is because nn + nn has no charged particles in the initial or final states, and pp -+pp is suppressed in
429~
C. Gale and J. Kapusta / Dilepton production
the dipole
limit and only makes a contribution
The suppression
of pp compared
We temporarily of view of virtual following
neglect
creation matter
the delta excitation
collisions
Finally,
which
decrease,
since a delta
other factors
being
first, order
of magnitude
baryon-baryon radiation
cascade,
cascade
so the mean temperature
decreases
than a nucleon,
Furthermore,
bremsstrahlung. less,
that to obtain
of the rate of e+e- production
a
in the
and calculate
We shall return
collisions.
the delta
it will radiate
suggest
we ignore the delta excitation
only from proton-neutron
for the
of the nuclear
the rate of (virtual)
These considerations estimate
from the point
would not change by much.
is more massive
equal.
11
so by acknowledging
is conserved,
of a delta costs energy,
would
of the nucleon
in the baryon-baryon
Baryon number
the rate of baryon-baryon
effects.
to np in fact is seen experimentally.
bremsstrahlung
reasons.
due to relativistic
the
to the delta
later. From e+e-pair
(1) we can write with
invariant
for an np collision
the cross-section
to make an
mass M as
,w
e+e-
LL
dM2
3a2
M2
In
[
2%
M
(2)
I
do
(-ej.$ 2 -dt
(T(s) -
(3)
dt
-(s-45)
where
t is the four-momentum
weighted
cross
approximation
section.
This result
for the temperatures
The np differential develops
a strong
collisions. charge
transfer
However,
exchange
force.
assumes
that
ItI < 44,
transfer
not a bad
of interest.
cross section
forward
and o(s) is the momentum
is isotropic
peak at high energy,
it also develops Approximately,
at low energy.
typical
of hadron-hadron
a strong backward the differential
It
peak due to the cross section
is
43oc
C. Gale and .I. Kapusta j Dilepton production
symmetric about Bcm - 90*, or equivalently stated, donp/dt is a symmetric function of t and u.
It then follows that
L;(s) - 2o”P&-)( el
s
-I)
.
(4)
44
For the np cross section we adopt the parametrization
18(mb.GeV)% + 10 mb .
o:;(s) s-4
(5)
3
In the independent particle approximation of kinetic theory the rate (number of reactions of the specified kind per unit time per unit volume) is 12 computed as
+-
+-
d3kl - s ---&
f,(Q
ee d3k dono s f (Z ) (2n)3 p 2 dM2
vrel'
[ (kl*k2) 2 _ q/2 Vrel
_
(6) ElE2
Here the f's are the occupation probabilities in momentum space. For our purposes we use a relativistic Boltzmann distribution and neglect the Pauli blocking in the final state:
(7)
2 I./2 , fi is the chemical potential, i refers to proton or where E - (g2 + m ) neutron and the 2 is a spin factor. It wi 1 be useful later, for comparison with the rate from n+x‘annihilation, to know the rate for producing an e+e- pair with total
C. Gale and J. Kapusta / Dilepton production
431c
momentum { - 0 in the nuclear matter rest frame. This leads to the cross section
_ a2 6n3
i(s) M4
where we have made the approximation t--
(8)
’
inp)2.
+ The rate dREpe /d3qdMI+ q-0
is then obtained by replacing &/dM2 with &/d3qdM in (6).
4.
TWOPION ANNIHILATION The most important annihilation process leading to dilepton production in
nuclear matter is2 ~+7r-+ e+e‘. This reaction proceeds through the p meson 13 which decays into a virtual photon by vector dominance. .
For quark-gluon
plasma formation at ultrarelativistic energies this is considered a background. However, in nuclear matter we view it as the process of prime interest since it can provide direct information on pion dynamics. The pion annihilation cross section is well known to be
- $
CI
$ M
(1 - 4mz/M2)1'21Fn(M)12
where Fa(M) is the form factor. We use
m4 IFn(M)12(M2 -
m
P
- 775 MeV
m' - 761 MeV P
(9)
A ralativistic Breit-Wigner would have m' - 775 MeV and PP - 155 MeV
P
parametrization (lCr>,thvugb, is a good approximation to the more 14 sophisticated Gounaris-Sakurai formula, which itself provides an excellent fit to the data.
The annihilation rate is then computed analogously to (6)
with the results
ae+“nn__
_
ete-
oETr (M)
M%
2(2?J+
cd
K (M/T)<1 -4m2/& z 1
.
WI
The pion dispersion relation refers CO the function relating the energy w(k) of a pion moving through nuclear matter to its momentum k. depends both on temperature T and baryon density n. the pion dispersion relation becomes softer
This relation
It is conjectured that
with increasing
density, that is
w(k) is reduced aC fixed k, due to the attractive p-wave interaction between I"5 pions and nucleons. In fact w(k) may even have a minimum at:some k - kg > 0 15-17 This may occur as at finite density if Che interaction is strong enough. a precursor to pion condensation, or it may occur even if pion condensation never
sets
in.
It
would
seem that
8 softening of the dispersion relation
would increase the number of pions and hence also the rate of dilepton emission. The number of real pions escaping from the reaction zone may in fact not increase at:all because these pians are not "on the mass-shell" at high density and are absorbed as the nuclear matter expands" Therefore dilepton spectrum may be tl-~e ody
the
signal we have of the pion dispersion
relation in dense nuclear matter. The most direct connection betwsen Che pion dispersion relation and the dilepton spectrum OCOIXS when the pions have equal but opposite momenta k in the
nuclear
18 matter rest frame.
Then the e+ and e- emerge back-to-back
r;;- 0) 8nd their !.nvariantmass is M * Zwfk), The total invariant ClitSS
C. Gale and J. Kapusta / Dilepton production
433c
spectrum, integrated over <, is less sensitive to the form of o(k) because there is a continuum of pion momenta which all have the same invariant mass
M.
We have computed the rate of e+e- production with ; - 0 and with arbitrary pion dispersion relation to be
dR
e’e
(12)
w&)2
such that 2u(k)-M
Notice that this rate is inversely proportional to the group velocity of the pion, and to the fourth power of k/w. The group velocity originates in the Jacobian of the transformation from momentum to energy. We use the vacuum pion form factor in this work.
On the one hand we
expect the pions to annihilate within one pion charge radius, 0.6 fm, of each other. This is a rather short-distancephenomenon in which many-body effects should not play a significant role. On the other hand the rho mass and width are expected to change with increasing temperature and density reflecting the 19 approach to chiral symmetry restoration. We leave this as an open problem. The pion dispersion relation in nuclear matter is not known.
For
purposes of illustration we take the functional form
o(k)
-
[(k-ko)2 + mi]1'2 - U .
(13)
This is suggested by the following considerations: (i) The group velocity &/dk
should not exceed the velocity of light.
(ii) At high momenta,
corresponding to short distances, many-body effects should be of secondary importance so that w -+k +
l
.* as k + 9). (iii) The energy should first
decrease as the momentum increases, corresponding to the strong p-wave attraction, reach a minimum, and then increase with increasing momentum,
434c
C. Gale and J. Kapusta / Dilepton production
I
I
I
I
5-
n/n,=
I
I
5
6
I
0
4-
1
“0
2
4
3
k/m7 Fig. 4.
following
A possible
baryons
kg, m0 and U are all temperature
dependence
which
should coma from the baryon
are primarily
altering
parametrization
is shown
in figure 4.
NUMERICAL
ESTIMATES
OF THE RATES
5.
Now we shall numerically two sections.
The simplest
The contributions annihilation n-n
0'
relation
in nuclear
matter.
(ii).
In general strongest
pion dispersion
In this figure
is apparent
the expressions
is the invariant
from the baryon-baryon
are shown separately
density
the pion dispersion
evaluate quantity
and density
cascade
since
it is the
relation.
derived
Our
in the last
mass distribution
dR/dM.
and from pion-pion
in figure 5, for T - 50 and 100 MeV and
the free space pion dispersion
that s+vr- annihilation
The
dependent.
is a rapidly
relation
increasing
is used.
function
of T.
It To
435c
I
I
I
I
-T=lOOMeV ---T = 50 MeV
n/no= ?
M (MeV) Fig. 5.
The thermal rate for producing a+e- pairs of invariant mass H at normal nuclear density and at two different+temperatures. Contributions from np collisions and from z s annihilation (the pions have their free space dispersion relation) are shown separately.
see this contribution in a heavy ion collision, clearly it is advantageous to use a high beam energy, perhaps the 1.7 to 2.1 GeV/nucleon available at the Bevalac or the lo-14 GeV/nucleon available at BNL. The best mass range is 400
C. Galeand J. Kapusta / Dilepton production
436~
< M < 850 MeV.
Note that the background from baryon-baryon cascading scales n
with density like n‘.
To predict more accurately what will actually be seen
in a heavy ion collision requires a detailed model for the dynamical evolution of the nuclear matter. A rough estimate would be to take the maximum volumeaveraged temperature and density in a hydrodynamic or cascade simulation and multiply by a typical space-time volume of 1000 fm4/c. In order to test the impact of the pion dispersion relation on dilepton emission it is best to examine the mass distribution of back-to-back e+epairs, dR/d'qdM at < = 0.
We show results in figure 6 for T - 100 MeV for a
sequence of baryon densities. The most noticeable aspect of these curves is the divergence at the threshhold for annihilation. This is a consequence of the inverse relationship between the rate and the group velocity in (12). The minimum pion energy occurs at k - k. > 0 wher,n > 0, at which point dw/dk - 0 and W(ko) < ms, with our parametrization. The threshhold mass shrinks with increasing density, as displayed in figure 4.
The rate for densities 15
n/no
< 4 and for masses M 2 800 MeV is two or more orders of magnitude greater than for noninteracting pions. The rate for s+s-
annihilation far exceeds the
rate for baryon-baryon cascading for 4 - 0 pairs and with the pion dispersion relation of figure 4.
Even if we underestimated the background due to baryon-
baryon cascading or other sources by an order of magnitude the signal from r+s- annihilation would still shine through. If the pion dispersion relation does have a dip at finite momentum it should be possible to detect it from +measurements of the e e
mass distribution at ; - 0.
This is a general
conclusion independent of the specific parametrization used in figure 4. However, we must realize that in actual heavy ion collisions the peaks will be broadened due to finite system size effects and due to the time evolution of the baryon density. 20 Recently Ko and collaborators have modeled heavy ion collisions by an expanding fireball, They integrated the dilepton emission rate
over
the
space-time history of the fireball to obtain total yields, As expected, most
C. Gale and J. Kapusta / DiEepton production
43762
n/n,=4 qI&--=O -._f
~
200
400
600
800
' ,O
~~~~V~ Fig,
6,
Comparison of the thermal rates for producing e+e* pairs of invariant m$ss M and zero total momentum coming from np collisions and from XII_ annihilation when the pions have the dispersion relation shown in figure 4.
dileptons are emitted during the early, high temperature and density epoch of the expansion. The spike in the mass spectrum due to a dip in the pion dispersion relation was broadened but stiI.1quite pronounced.
C. Galeund J. Kapusta / Dilepton production
438c
6.
COMPARISON WXTH P + BE DATA The first experimental program to study dilepton production at GeV
energies for nucleon-nucleon, nucleon-nucleus and nucleus-nucleus collisions is being carried out by the DLS group. Jim Carroll just gave a presentation of their current status. Here we compare with their published data21 for ~(4.9 GeV) + Be -te+e-X. Sfnce multiple scattering is probably a small effect we approximate this reaction by np * e'e‘X and simply divide the data by A 2/3 , We consider two contributions, nucleon-nucleon bremsstrahlung and delta production and decay, A + N e+e-.
In the absence of a calculation based on
Lagrangian field theory, which we and the Giessen group are working on, we simply add these contributions incoherently. For the nucleon-nucleon bremsstrahlung we use a phase space corrected soft virtual photon approximation.
do
_
dyd2qTdM
a2 6%'
rrls)Rz(S2) R2fs)
(14)
ME2
where R2(s) oc (1 - 44,'~)~'~ is the LIPS available to the 2-nucleon system, and s -s+M 2
2
- 2s1'2E.
22,23 TO first order in a the rate of decay of a resonance A into Befe- is
(15)
where
X-
M2
(mA -m)B
and
2
(16)
C Gale and J. Kapusta / Dilepton production
row -
Here M.P and
ML
the real photon dominance factor
a1’2(m2,m2,M2) [2K+d
16m;
are the transverse
(17)
+ %‘“‘I
and longitudinal
structure
decay mode, M,f(0) > 0 and M,_(O) - 0.
model we approximate
squared,
439c
when computing
1
I
r,(M2) = ; IF,W)
The invariant
7
these structure virtual
functions.
Following
functions
with
photon production.
For
the vector the pion
form
Thus
A"2(mz,G,M2)
.?
I-
r(A -, N-t)
cross section
for the reaction
(18)
NN + AX + e+e‘X'
is then
obviousiy
_ g tF,(M2)12
do dyd2q,dM
x
S PlbA)
o(NN+Ax)x
M
=
P2bA, z 1 P3(mA,zA,y,qT.M) B7bA) x A
A1'2(m2,<,M2) XL
x1/2
2
2
.
dmA dzA
The P's are probability
densities.
Pl is the probability
mass m A, and is taken to be proportional width.
The width
where k frame.
7r
(19)
b,,s,O)
into the aN channel
to a Lorentzian
is taken to be r
and w A are the pion recoil momentum
TC
and energy
to make a delta of with a mass-dependent = 185.5(kX/on)3
MeV,
in the delta rest
The width into the TN channel is taken to be I‘ - 1.65(? - d/m:)
MeV.
7
At the peak of the delta mass distribution, width
is Pr - 115 MeV, and the branching
mA - m. - 1232 MeV,
the delta
ratio By into the photon
channel
is
C Gale and 1. Kapusta / Dilep~o~ production
44oc
n
22
DLS data
-
hem.
---
Delta
10-I
m
a
5
1o-2
\ 1O-4 0
I
I
I
I
I
I
0.2
0.4
0.6
0.8
1 .o
1.2
M,+,Pig. 7.
0.6%.
‘,
(GeV)
The dilepton mass distribution without account taken of the experimental acceptance. The bremsstrahlung2~nd delta contributions are shown separately. The data is from DLS.
P2 is the probability density to find the delta emitted at co&A
- xA
in the NN center of mass frame, and is proportional to cosh(czA), c - 12pipf, with pi and pf the GM momenta in GeV/c. P3 is the probability density to find the e+e‘ pair emitted with rapidity yz invariant mass M and transverse momentum q given that the delta has mass m and is emitted at angle B T A. A
p3
corresponds to an isotropic distribution in the delta rest frame. Figure 7 shows our predictions for Q/dM,
i.e., the mass distributions
integrated over all accessible y and qT. Also shown is the DLS data.
The
detector sees only a restricted portion of phase space. Therefore the experimental acceptance must be folded into our predictions to make a correct comparison. This is done in figure 8 where the bremsstrahlung and delta contributions have been summed. In order of magnitude our calculations are in agreement with the data. This is important, for to search for qualitatively new phenomena in nucleus-nucleus collisions we must first have at least a rudimentary understanding of nucleon-nucleon collisions. However, two
441c
C. Gale and J. Kapusta / Dilepton production
10' l
i
-
o_
f_
2,
DLS data Brem. + Delta
:”
3-
4_ 0
0.2
0.4
0.6
0.8
1.0
1.2
Me+e- (GeV) Fig.
8.
The dilepton mass distribution from figure 7, with the DLS phase space acceptance taken into account, and summed over the bremsstrahlung and delta contributions.
discrepancies are apparent. First, in our opinion the soft virtual photon approximation is too large at large M, even with the phase space correction. Second, there is a peak around 2ms which may be a sign of two pion production and
7.
annihilation.
CONCLUSION In this work we have given an introductory account of the rate of
dilepton emission from high temperature nuclear matter. Our estimates are based
on
relativistic kinetic theory. The dominant sources of this type of
radiation seem to be two-body elastic and inelastic baryon-baryon collisions and two-pion annihilation. If the pion dispersion relation in nuclear matter has a dip in it due to the strong p-wave attraction to nucleons, then there will be a peak in the mass distribution of back-to-back electrons and positrons located at M - 2u where umin is the pion energy at the dip. min'
If
C. Gale and J. Kapusta 1 Dilepton production
442~
the pion dispersion relation in nuclear matter does not differ much from its free space form, then the dilepton mass distribution will still provide information on the pion to baryon ratio in dense, high temperature nuclear matter. In either case we can learn about pion dynamics in nuclear matter at high temperature and density, which apparently cannot be done any other way. Improvements are needed in the computation of the cross sections for dilepton production in baryon-baryon collisions. Especially needed is a relaxation of the soft-photon approximation in emission from the external baryon lines, and a calculation for the process shown in figure 2. Also missing is a concrete estimate of the cross section for the process XN + Ne+e-.
Self-consistency should play an important role so as to avoid over-
counting the available degrees of freedom in nuclear matter, especially with 17 regards to the delta versus pion-nucleon degrees of freedom. We have made a preliminary comparison with the first DLS data on p(4.9 GeV) + Be based on nucleon-nucleon bremsstrahlung and delta production and decay. When we are satisfied that we have adequate knowledge of the elementary processes then they can be used in conjunction with a dynamical calculation of the evolution of nuclear matter in a heavy ion collision (for example, the Boltzmann-Uehling-Uhlenbeckapproach) to predict absolute distributions. Soon, additional data should be available from the Dilepton Spectrometer at the Bevalac. It is amusing to recall that a decade ago one of the major hopes was to 24 discover pion condensation15 in high energy heavy ion collisions.
Although
it may not be possible to detect a condensate with dileptons either (condensation occurs in the spin-isospin sound branch, not the real pion branch) we will finally be able to study pion dynamics in dense, high temperature nuclear matter without fear that the signal will be absorbed upon expansion.
C Gale apldJ. Kapusta J Dilepton production
443c
ACKNOWLEDGMENTS J.K. would like to thank the organizers for their hospitality and for running such an interesting program. This research was supported by the U.S. Department of Energy under grant DE-FG02-87ER40328. REFERENCES 1)
D. C. Hornet al., Phys. Rev. Lett. 36 (1976) 1236; ibid. 37 (1976) 1374; D. K. Kaplan et al., Phys. Rev. Lett. 40 (1977) 435.
2)
G. Domokos and J. I. Goldman, Phys. Rev. D23 (1981) 203; G. Domokos, ibid. 28 (1983) 123; K. Kajantie and H. I. Miettenen, Z. Phys. C9 (1981) 341; ibid. 14 (1982) 357.
3)
C. Gale and J. Kapusta, Phys. Rev. C35 (1987) 2107; Nucl. Phys. A471 (1987) 35~; and to be published.
4)
J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1975).
5)
V. R. Brown and J. Franklin, Phys. Rev. C8 (1973) 1706.
6)
R. Ruckl, Phys. Lett. 64B (1976) 39.
7)
J. I. Kapusta, Phys. Rev. Cl5 (1977) 1580; M. P. Budiansky, S. P. Ahlen, G. Tarle and P. B. Price, Pbys. Rev. Lett. 49 (1982) 361; D. Vasak, W. Greiner, B. Muller, Th. Stahl and M. Uhlig, Nucl. Phys. A428 (1984) 291~.
8)
H. Nifenecker and J. P. Bondorf, Nucl. Phys. A442 (1985) 478; C. M. Ko, G. Bertsch'and J. Aichelin, Phys. Rev. C31 (1985) 2324: W. Gassing, T. Biro, U. Mosel, M. Tohyama and W. Bauer, Phys. Lett. 181B (1986) 217; W. Bauer, G. F. Bertsch, W. Cassing and U. Mosel, Phys. Rev. C34 (1986) 2127; K. Nakayama and G. Bertsch, Phys. Rev. C34 (1986) 2190. D. Neuhauser and S. E. Koonin, Nucl. Phys. A462 (1987) 163.
9)
N. P. Samios, Phys. Rev. 121 (1961) 275.
10) J. Cugnon, T. Mitzutani and J. Vandermuelen, Nucl. Phys. A3S2 (1981) 505; G. F. Bertsch, H. Kruse and S. Das Gupta, Phys. Rev. C29 (1984) 673; Y. Kitazoe, M. Gyulassy, P. Danielewicz, H. Toki, Y. Yamamura and M. Sano, Phys. Lett. 138B (1984) 34P; C. Gale, G. Bertsch and S. Das Gupta, Phys. Rev. C35 (1987) 1666. 11) J. A. Edgington and B. Rose, Nucl. Phys. 89 (1966) 523; F. P. Brady and J. C. Young, Phys. Rev. C2 (1970) 1579; ibid. 7 (1973) 1707. B.M.K. Nefkens, 0. R. Sander and D. I. Sober, Phys. Rev. Lett. 38 (1977) 876. 12) K. Kajantie, J. Kapusta, L. McLerran and A. Mekjian, Phys. Rev. D34 (1986) 2746. 3.3) R. F. Schwitters and
K.
Strauch, Ann. Rev. Nucl. Sci. 26 (1976) 89.
14)
G. J. Gounaris and J. J. Sakurai, Phys. Rev. Lett. 21 (1968) 244.
15) A. B. Migdal, Rev. Mod. Phys. 50 (1978) 107. 16)
T.E.O. Ericson and F. Myhrer, Phys. Lett. 74B (1978) 163.
17)
I. M. Mishustin, F. Myhrer and P. J. Siemens, Phys. Lett. 958 (1980) 361.
18)
This was first suggested for the quark dispersion relation in quark-gluon plasma. J. Kapusta, Phys. Lett. 136B (1984) 201.
19)
R. D. Pisarski, Phys. Lett. 1105 (1982) 155.
20)
L. H. Xia, C. M. Ko, L. Xiong and J. Q. Wu, Texas A&M preprint 1988.
21)
G. Roche et al., Phys. Rev. Lett. 61 (1988) 1069.
22) N. M. Kroll and W. Wada, Phys. Rev. 98 (1955) 1355. 23)
B. E. Lautrup and J. Smith, Phys. Rev, 03 (1971) 1122.
24)
For a more recent status report on the search for evidence of pion condensation in the pion momentum distribution see S. Nagamiya and M. Gyulassy, Adv. Nucl. Phys. 13 (1984) 201.
*
Speaker at the International Workshop on Nuclear Dynamics at Medium and High Energies, Bad Honnef, W. Germany, lo-14 October 1988.
423~
DILEPTON
PRODUCTION
Charles Gale Atomic Energy of Canada, Ltd. Theoretical Physics Branch Chalk River Nuclear Laboratories Chalk River, Ontario Canada KOJ - 1JO
Joseph Kapusta* School of Physics and Astronomy University of Minnesota Minneapolis, Minnesota 55455
The general features of dilepton production from nucleon-nucleon Estimates are made of and nucleus-nucleus collisions are discussed. the thermal production rates arising from incoherent nucleon-nucleon It may be possible to scattering and from two-pion annihilation. infer the pion dispersion relation in hot and dense nuclear matter by measuring the invariant mass distribution of back-to-back electrons and positrons in the center of mass frame in high energy nucleus-+ _ Comparison to the recent data on p + Be + e e X nucleus collisions. at 4.9 GeV from the Bevalac is made.
1.
INTRODUCTION One of the most impressive
the 1970's was a measurement
of pip- production
by the Columbia-Fermilab-Stony invariant
mass distribution
high energy physics
d*o/dMdy
in the data are the decays primarily
via gg fusion.
in terms of the Drell-Yan target
and projectile
The theoretical collisions
process,
for masses
states,
which
study of dilepton
with the suggestion
03759474/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
the
Quite
evident
the data can be understood from the
photon.
production
on nuclei
are formed
where a quark and an antiquark
to a virtual
out in
2 5 M zz 20
in cross section.
Away from these few peaks
annihilate
was initiated
They measured
at mid-rapidity
of cc and b6 bound
carried
from 400 GeV protons
Brook collaboration!
The data spans nine orders of magnitude
GeV.
experiments
in nucleus-nucleus
that it may serve as a
C. Gale and J. Kapusta / Dilepton production
424~
2 thermometer of quark-gluon plasma formed at ultra-relativistic energies. liowever,dileptons should be produced at lower energies as well, where no quark-gluon plasma formation is expected. The argument is simply that whenever charged objects collide they radiate real and virtual photons, the virtual photons decay into dilepton pairs. The advantage of observing electromagnetic signals from a strongly interacting many-body system is that they travel relatively unscathed from the production point to the detector. Since production rates are rapidly increasing functions of temperature and density these electromagnetic signals are good probes of the early high temperature and density stage of heavy ion collisions. A relevant analogy can be made with neutrino emission from the sun or from 8 cooling neutron star. The purpose of this work3 is to provide an exploratory study of dilepton radiation from finite temperature and density nuclear matter and to make a preliminary comparison with the recent p(4.9 GeV) + Be data of the DLS group. A rigorous study is extremely difficult due to the strongly interacting nature of nuclear matter:
for example, there are complications from two and three
particle correlations, three body collisions, identification of the relevant degrees of freedom (nucleons, baryonic resonances, mesons, collective degrees of excitation, quarks, etc.), and finite temperature and density corrections to form factors, widths, and so on.
Therefore our analysis will at times be
quite phenomenological, and our study will be based primarily on relativistic kinetic theory.
2.
GENERAL DISCUSSION OF DILEPTON PRODU~ION To begin the discussion let us first consider bremsstrahlung in hadron-
hadron collisions. For example, in the reaction np-np, photons (real and virtual) can be radiated since charged particles are accelerated. In the soft photon approximation photons are radiated only from the initial or from the 4 final charged lines. 1.
None come from the strong interaction blob.
See figure
This approximation is valid if the energy carried by the photon is less
C. Gale and J. Kapusta / Dilepton production
Fig. 1.
425c
Radiation from the external proton lines in np scattering.
than the inverse of the strong interaction collision time. The latter is usually estimated as about l-2 fm/c, so that this mechanism is dominant if E7 < r$-
100-200 MeV.
strong interaction blob.
For hard photons we must in addition look inside the One contribution is shown in figure 2.
5 represents radiation from the exchange of a charged pion. internal pion line will be important for energies E > 7; Y
It
Radiation from the - 100-200 MeV.
Of
course the separation into soft and hard photons is not sharp. The relative contribution of radiation from internal and external lines in general depends not only on the average collision time but also on other variables, such as the momentum transfer to the charged particles in the final state, on the angle of emission of the photon and on the available center of mass energy. For virtual photons, that is for dilepton production, it is possible to extrapolate the soft photon predictions from q2 - 0 to q2 - M2, M being the invariant mass of the e+e- pair. The result is6 (qP - pr + p'>
Fig. 2.
Radiation from an internal charged pien line in np scattering.
e+ A
B
e-
&
I I
1 2
A +
B
n
1
F
1 I
2
n
e+
e-
e+ A + B Fig. 3.
2
e-
1
I I
2 +
*
e
*
n
Soft radiation from nucleus-nucleus collisfons is a coherent superposition of radiation from the initial nuclei, A and B, and from the charged particles in the final state.
+ -
ee
dol E+E_ $- L-1 2 2 qo d3q * d p+d3p_ 2x q
(1)
For hard virtual photons thfs extrapolation is in general not possible since the off-shell continuation of the strong interaction blob is different for q2 -0
and q2 > 0.
Note that real photon production is of order a while dilepton
production is of order a*. In nucleus-nucleus collisions the possibilities are even more interesting. Consider a collision between nucleus A and nucleus B leading to some distribution of particles in the final state. A and B may radiate a photon before impact, or the charged particles may radiate a photon in the final state.
See figure 3.
If the photon energy is less than the inverse
427~
C. Gale and J. Kapusfa J Dil~pfon p~~~u&fion
nuclear collision time, estimated to be 10-20 fm/c, E7 < ri
- 10 - 20 MeV,
the charges will act coherently over the whole extent of the nucleus. In that case the cross section for real photons is proportional to Z2a, and for dileptons proportional to Z202. At high incident energy the cross sections peak at angles near 0' and NO0
in the CM (although they vanish at 0' and
7 180') due to the sudden deceleration along the beam axis. It is known from computer simulations that the nucleons cascade during the collision. In order to resolve the finer features of the nucleus-nucleus collision, such as the cascading, we muat look at higher energy photons. for lo-20 MaV - ri 8 cascade.
< r;=
100-200 MeV we
can see
ThUS
the individual baryons
This radiation will be incoherent, proportional to Z, but will also
increase with the number of collisions per baryon roughly as Zl/3 .
At even
higher photon energies radiation from internal NN blobs will become important, as in figure 2. The x0 lifetime is 10
-16
sec. It
decays outside the reaction zone in a
heavy ion collision. The branching ratio into 7-yis 98.85%. and into ye+=(Dalitz decay) is 1.15%. The Dalitz decay may be viewed as a background to other e+e- sources. Fortunately phase space restricts the invariant e+emass M to less than mso - 135 MeV. and 99% have
M <
9 65 MeV.
In fact, 90% of the pairs have M < 15 MeV,
Still, the Dalitz pairs can be approximately
+ subtracted out by taking the x0 spectrum to be the average of the s and Ispectra and folding in the decay distribution. Also the n meson has a decay mode ye+e-, but its branching ratio is only 0.5%, and besides the q should not be abundantly produced at the temperatures T - 40-100 MeV of interest to us here. In the nuclear many-body environment there will be a profusion of microprocesses involving nucleons, pions and deltas. If a charged particle is involved it will radiate. A detailed comparison with experiment will require a complete enumeration of all these processes and their inclusion in BUU or 10 cascade computer simulation calculations.
This will allow us to test our
428~
C. Gale and 1. Kappusta / Dile~ton production
understanding of collision processes in hot and dense nuclear matter. So far our discussion has not distinguished between real and virtual photons. In fact for measuring bremsstrahlung radiation real photons are favored over dileptons since the dilepton production rate is smaller by a factor of a.
The real advantage of dileptons over photons arises when
focussing on annihilation processes. Kinematically, annihilation of two massive particles into a single real photon is not allowed, and -y-y correlation measurements are more difficult than e'e* measurements. The most important* annihilation process is s+s- + T* * e+e-. Annihilation of hadrons always leads to a minimum energy, in this case E 2 q* - X2 2 &rnq. Thus in high temperature nuclear matter, and in high energy heavy ion collisions, we expect the rate for production of high mass e'e- pairs, M > 2m=, to be dominated by pion annihilation. That region of invariant mass should provide information on the abundance of pions in the hot and dense medium since the production rate is proportional to the square of the pion number density. We view annihilation processes to be of most interest from the point of view of dilepton production. Bremsstrahlung from cascading baryons and mesons is best studied via real photons, and from the perspective of annihilations forms a background.
3.
PROTON-NEUTRON VIRTUAL BRRMSSTRARLUNG Now we make a rough estimate of the production of dileptons by the
cascading baryons.
Consider a two-body reaction a + b + c + d where a,b,c and
d may be either a nucleon or a delta. These are the primary reactions in current cascade or BUU Monte Carlo simulations of heavy ion collisions.lo
In
the soft-photon approximation a, b, c and d may each radiate, depending on its charge. Among the three reactions pp + pp, np + np and nn * nn, the most important are neutron-proton collisions.* This is because nn + nn has no charged particles in the initial or final states, and pp -+pp is suppressed in
429~
C. Gale and J. Kapusta / Dilepton production
the dipole
limit and only makes a contribution
The suppression
of pp compared
We temporarily of view of virtual following
neglect
creation matter
the delta excitation
collisions
Finally,
which
decrease,
since a delta
other factors
being
first, order
of magnitude
baryon-baryon radiation
cascade,
cascade
so the mean temperature
decreases
than a nucleon,
Furthermore,
bremsstrahlung. less,
that to obtain
of the rate of e+e- production
a
in the
and calculate
We shall return
collisions.
the delta
it will radiate
suggest
we ignore the delta excitation
only from proton-neutron
for the
of the nuclear
the rate of (virtual)
These considerations estimate
from the point
would not change by much.
is more massive
equal.
11
so by acknowledging
is conserved,
of a delta costs energy,
would
of the nucleon
in the baryon-baryon
Baryon number
the rate of baryon-baryon
effects.
to np in fact is seen experimentally.
bremsstrahlung
reasons.
due to relativistic
the
to the delta
later. From e+e-pair
(1) we can write with
invariant
for an np collision
the cross-section
to make an
mass M as
,w
e+e-
LL
dM2
3a2
M2
In
[
2%
M
(2)
I
do
(-ej.$ 2 -dt
(T(s) -
(3)
dt
-(s-45)
where
t is the four-momentum
weighted
cross
approximation
section.
This result
for the temperatures
The np differential develops
a strong
collisions. charge
transfer
However,
exchange
force.
assumes
that
ItI < 44,
transfer
not a bad
of interest.
cross section
forward
and o(s) is the momentum
is isotropic
peak at high energy,
it also develops Approximately,
at low energy.
typical
of hadron-hadron
a strong backward the differential
It
peak due to the cross section
is
43oc
C. Gale and .I. Kapusta j Dilepton production
symmetric about Bcm - 90*, or equivalently stated, donp/dt is a symmetric function of t and u.
It then follows that
L;(s) - 2o”P&-)( el
s
-I)
.
(4)
44
For the np cross section we adopt the parametrization
18(mb.GeV)% + 10 mb .
o:;(s) s-4
(5)
3
In the independent particle approximation of kinetic theory the rate (number of reactions of the specified kind per unit time per unit volume) is 12 computed as
+-
+-
d3kl - s ---&
f,(Q
ee d3k dono s f (Z ) (2n)3 p 2 dM2
vrel'
[ (kl*k2) 2 _ q/2 Vrel
_
(6) ElE2
Here the f's are the occupation probabilities in momentum space. For our purposes we use a relativistic Boltzmann distribution and neglect the Pauli blocking in the final state:
(7)
2 I./2 , fi is the chemical potential, i refers to proton or where E - (g2 + m ) neutron and the 2 is a spin factor. It wi 1 be useful later, for comparison with the rate from n+x‘annihilation, to know the rate for producing an e+e- pair with total
C. Gale and J. Kapusta / Dilepton production
431c
momentum { - 0 in the nuclear matter rest frame. This leads to the cross section
_ a2 6n3
i(s) M4
where we have made the approximation t--
(8)
’
inp)2.
+ The rate dREpe /d3qdMI+ q-0
is then obtained by replacing &/dM2 with &/d3qdM in (6).
4.
TWOPION ANNIHILATION The most important annihilation process leading to dilepton production in
nuclear matter is2 ~+7r-+ e+e‘. This reaction proceeds through the p meson 13 which decays into a virtual photon by vector dominance. .
For quark-gluon
plasma formation at ultrarelativistic energies this is considered a background. However, in nuclear matter we view it as the process of prime interest since it can provide direct information on pion dynamics. The pion annihilation cross section is well known to be
- $
CI
$ M
(1 - 4mz/M2)1'21Fn(M)12
where Fa(M) is the form factor. We use
m4 IFn(M)12(M2 -
m
P
- 775 MeV
m' - 761 MeV P
(9)
A ralativistic Breit-Wigner would have m' - 775 MeV and PP - 155 MeV
P
parametrization (lCr>,thvugb, is a good approximation to the more 14 sophisticated Gounaris-Sakurai formula, which itself provides an excellent fit to the data.
The annihilation rate is then computed analogously to (6)
with the results
ae+“nn__
_
ete-
oETr (M)
M%
2(2?J+
cd
K (M/T)<1 -4m2/& z 1
.
WI
The pion dispersion relation refers CO the function relating the energy w(k) of a pion moving through nuclear matter to its momentum k. depends both on temperature T and baryon density n. the pion dispersion relation becomes softer
This relation
It is conjectured that
with increasing
density, that is
w(k) is reduced aC fixed k, due to the attractive p-wave interaction between I"5 pions and nucleons. In fact w(k) may even have a minimum at:some k - kg > 0 15-17 This may occur as at finite density if Che interaction is strong enough. a precursor to pion condensation, or it may occur even if pion condensation never
sets
in.
It
would
seem that
8 softening of the dispersion relation
would increase the number of pions and hence also the rate of dilepton emission. The number of real pions escaping from the reaction zone may in fact not increase at:all because these pians are not "on the mass-shell" at high density and are absorbed as the nuclear matter expands" Therefore dilepton spectrum may be tl-~e ody
the
signal we have of the pion dispersion
relation in dense nuclear matter. The most direct connection betwsen Che pion dispersion relation and the dilepton spectrum OCOIXS when the pions have equal but opposite momenta k in the
nuclear
18 matter rest frame.
Then the e+ and e- emerge back-to-back
r;;- 0) 8nd their !.nvariantmass is M * Zwfk), The total invariant ClitSS
C. Gale and J. Kapusta / Dilepton production
433c
spectrum, integrated over <, is less sensitive to the form of o(k) because there is a continuum of pion momenta which all have the same invariant mass
M.
We have computed the rate of e+e- production with ; - 0 and with arbitrary pion dispersion relation to be
dR
e’e
(12)
w&)2
such that 2u(k)-M
Notice that this rate is inversely proportional to the group velocity of the pion, and to the fourth power of k/w. The group velocity originates in the Jacobian of the transformation from momentum to energy. We use the vacuum pion form factor in this work.
On the one hand we
expect the pions to annihilate within one pion charge radius, 0.6 fm, of each other. This is a rather short-distancephenomenon in which many-body effects should not play a significant role. On the other hand the rho mass and width are expected to change with increasing temperature and density reflecting the 19 approach to chiral symmetry restoration. We leave this as an open problem. The pion dispersion relation in nuclear matter is not known.
For
purposes of illustration we take the functional form
o(k)
-
[(k-ko)2 + mi]1'2 - U .
(13)
This is suggested by the following considerations: (i) The group velocity &/dk
should not exceed the velocity of light.
(ii) At high momenta,
corresponding to short distances, many-body effects should be of secondary importance so that w -+k +
l
.* as k + 9). (iii) The energy should first
decrease as the momentum increases, corresponding to the strong p-wave attraction, reach a minimum, and then increase with increasing momentum,
434c
C. Gale and J. Kapusta / Dilepton production
I
I
I
I
5-
n/n,=
I
I
5
6
I
0
4-
1
“0
2
4
3
k/m7 Fig. 4.
following
A possible
baryons
kg, m0 and U are all temperature
dependence
which
should coma from the baryon
are primarily
altering
parametrization
is shown
in figure 4.
NUMERICAL
ESTIMATES
OF THE RATES
5.
Now we shall numerically two sections.
The simplest
The contributions annihilation n-n
0'
relation
in nuclear
matter.
(ii).
In general strongest
pion dispersion
In this figure
is apparent
the expressions
is the invariant
from the baryon-baryon
are shown separately
density
the pion dispersion
evaluate quantity
and density
cascade
since
it is the
relation.
derived
Our
in the last
mass distribution
dR/dM.
and from pion-pion
in figure 5, for T - 50 and 100 MeV and
the free space pion dispersion
that s+vr- annihilation
The
dependent.
is a rapidly
relation
increasing
is used.
function
of T.
It To
435c
I
I
I
I
-T=lOOMeV ---T = 50 MeV
n/no= ?
M (MeV) Fig. 5.
The thermal rate for producing a+e- pairs of invariant mass H at normal nuclear density and at two different+temperatures. Contributions from np collisions and from z s annihilation (the pions have their free space dispersion relation) are shown separately.
see this contribution in a heavy ion collision, clearly it is advantageous to use a high beam energy, perhaps the 1.7 to 2.1 GeV/nucleon available at the Bevalac or the lo-14 GeV/nucleon available at BNL. The best mass range is 400
C. Galeand J. Kapusta / Dilepton production
436~
< M < 850 MeV.
Note that the background from baryon-baryon cascading scales n
with density like n‘.
To predict more accurately what will actually be seen
in a heavy ion collision requires a detailed model for the dynamical evolution of the nuclear matter. A rough estimate would be to take the maximum volumeaveraged temperature and density in a hydrodynamic or cascade simulation and multiply by a typical space-time volume of 1000 fm4/c. In order to test the impact of the pion dispersion relation on dilepton emission it is best to examine the mass distribution of back-to-back e+epairs, dR/d'qdM at < = 0.
We show results in figure 6 for T - 100 MeV for a
sequence of baryon densities. The most noticeable aspect of these curves is the divergence at the threshhold for annihilation. This is a consequence of the inverse relationship between the rate and the group velocity in (12). The minimum pion energy occurs at k - k. > 0 wher,n > 0, at which point dw/dk - 0 and W(ko) < ms, with our parametrization. The threshhold mass shrinks with increasing density, as displayed in figure 4.
The rate for densities 15
n/no
< 4 and for masses M 2 800 MeV is two or more orders of magnitude greater than for noninteracting pions. The rate for s+s-
annihilation far exceeds the
rate for baryon-baryon cascading for 4 - 0 pairs and with the pion dispersion relation of figure 4.
Even if we underestimated the background due to baryon-
baryon cascading or other sources by an order of magnitude the signal from r+s- annihilation would still shine through. If the pion dispersion relation does have a dip at finite momentum it should be possible to detect it from +measurements of the e e
mass distribution at ; - 0.
This is a general
conclusion independent of the specific parametrization used in figure 4. However, we must realize that in actual heavy ion collisions the peaks will be broadened due to finite system size effects and due to the time evolution of the baryon density. 20 Recently Ko and collaborators have modeled heavy ion collisions by an expanding fireball, They integrated the dilepton emission rate
over
the
space-time history of the fireball to obtain total yields, As expected, most
C. Gale and J. Kapusta / DiEepton production
43762
n/n,=4 qI&--=O -._f
~
200
400
600
800
' ,O
~~~~V~ Fig,
6,
Comparison of the thermal rates for producing e+e* pairs of invariant m$ss M and zero total momentum coming from np collisions and from XII_ annihilation when the pions have the dispersion relation shown in figure 4.
dileptons are emitted during the early, high temperature and density epoch of the expansion. The spike in the mass spectrum due to a dip in the pion dispersion relation was broadened but stiI.1quite pronounced.
C. Galeund J. Kapusta / Dilepton production
438c
6.
COMPARISON WXTH P + BE DATA The first experimental program to study dilepton production at GeV
energies for nucleon-nucleon, nucleon-nucleus and nucleus-nucleus collisions is being carried out by the DLS group. Jim Carroll just gave a presentation of their current status. Here we compare with their published data21 for ~(4.9 GeV) + Be -te+e-X. Sfnce multiple scattering is probably a small effect we approximate this reaction by np * e'e‘X and simply divide the data by A 2/3 , We consider two contributions, nucleon-nucleon bremsstrahlung and delta production and decay, A + N e+e-.
In the absence of a calculation based on
Lagrangian field theory, which we and the Giessen group are working on, we simply add these contributions incoherently. For the nucleon-nucleon bremsstrahlung we use a phase space corrected soft virtual photon approximation.
do
_
dyd2qTdM
a2 6%'
rrls)Rz(S2) R2fs)
(14)
ME2
where R2(s) oc (1 - 44,'~)~'~ is the LIPS available to the 2-nucleon system, and s -s+M 2
2
- 2s1'2E.
22,23 TO first order in a the rate of decay of a resonance A into Befe- is
(15)
where
X-
M2
(mA -m)B
and
2
(16)
C Gale and J. Kapusta / Dilepton production
row -
Here M.P and
ML
the real photon dominance factor
a1’2(m2,m2,M2) [2K+d
16m;
are the transverse
(17)
+ %‘“‘I
and longitudinal
structure
decay mode, M,f(0) > 0 and M,_(O) - 0.
model we approximate
squared,
439c
when computing
1
I
r,(M2) = ; IF,W)
The invariant
7
these structure virtual
functions.
Following
functions
with
photon production.
For
the vector the pion
form
Thus
A"2(mz,G,M2)
.?
I-
r(A -, N-t)
cross section
for the reaction
(18)
NN + AX + e+e‘X'
is then
obviousiy
_ g tF,(M2)12
do dyd2q,dM
x
S PlbA)
o(NN+Ax)x
M
=
P2bA, z 1 P3(mA,zA,y,qT.M) B7bA) x A
A1'2(m2,<,M2) XL
x1/2
2
2
.
dmA dzA
The P's are probability
densities.
Pl is the probability
mass m A, and is taken to be proportional width.
The width
where k frame.
7r
(19)
b,,s,O)
into the aN channel
to a Lorentzian
is taken to be r
and w A are the pion recoil momentum
TC
and energy
to make a delta of with a mass-dependent = 185.5(kX/on)3
MeV,
in the delta rest
The width into the TN channel is taken to be I‘ - 1.65(? - d/m:)
MeV.
7
At the peak of the delta mass distribution, width
is Pr - 115 MeV, and the branching
mA - m. - 1232 MeV,
the delta
ratio By into the photon
channel
is
C Gale and 1. Kapusta / Dilep~o~ production
44oc
n
22
DLS data
-
hem.
---
Delta
10-I
m
a
5
1o-2
\ 1O-4 0
I
I
I
I
I
I
0.2
0.4
0.6
0.8
1 .o
1.2
M,+,Pig. 7.
0.6%.
‘,
(GeV)
The dilepton mass distribution without account taken of the experimental acceptance. The bremsstrahlung2~nd delta contributions are shown separately. The data is from DLS.
P2 is the probability density to find the delta emitted at co&A
- xA
in the NN center of mass frame, and is proportional to cosh(czA), c - 12pipf, with pi and pf the GM momenta in GeV/c. P3 is the probability density to find the e+e‘ pair emitted with rapidity yz invariant mass M and transverse momentum q given that the delta has mass m and is emitted at angle B T A. A
p3
corresponds to an isotropic distribution in the delta rest frame. Figure 7 shows our predictions for Q/dM,
i.e., the mass distributions
integrated over all accessible y and qT. Also shown is the DLS data.
The
detector sees only a restricted portion of phase space. Therefore the experimental acceptance must be folded into our predictions to make a correct comparison. This is done in figure 8 where the bremsstrahlung and delta contributions have been summed. In order of magnitude our calculations are in agreement with the data. This is important, for to search for qualitatively new phenomena in nucleus-nucleus collisions we must first have at least a rudimentary understanding of nucleon-nucleon collisions. However, two
441c
C. Gale and J. Kapusta / Dilepton production
10' l
i
-
o_
f_
2,
DLS data Brem. + Delta
:”
3-
4_ 0
0.2
0.4
0.6
0.8
1.0
1.2
Me+e- (GeV) Fig.
8.
The dilepton mass distribution from figure 7, with the DLS phase space acceptance taken into account, and summed over the bremsstrahlung and delta contributions.
discrepancies are apparent. First, in our opinion the soft virtual photon approximation is too large at large M, even with the phase space correction. Second, there is a peak around 2ms which may be a sign of two pion production and
7.
annihilation.
CONCLUSION In this work we have given an introductory account of the rate of
dilepton emission from high temperature nuclear matter. Our estimates are based
on
relativistic kinetic theory. The dominant sources of this type of
radiation seem to be two-body elastic and inelastic baryon-baryon collisions and two-pion annihilation. If the pion dispersion relation in nuclear matter has a dip in it due to the strong p-wave attraction to nucleons, then there will be a peak in the mass distribution of back-to-back electrons and positrons located at M - 2u where umin is the pion energy at the dip. min'
If
C. Gale and J. Kapusta 1 Dilepton production
442~
the pion dispersion relation in nuclear matter does not differ much from its free space form, then the dilepton mass distribution will still provide information on the pion to baryon ratio in dense, high temperature nuclear matter. In either case we can learn about pion dynamics in nuclear matter at high temperature and density, which apparently cannot be done any other way. Improvements are needed in the computation of the cross sections for dilepton production in baryon-baryon collisions. Especially needed is a relaxation of the soft-photon approximation in emission from the external baryon lines, and a calculation for the process shown in figure 2. Also missing is a concrete estimate of the cross section for the process XN + Ne+e-.
Self-consistency should play an important role so as to avoid over-
counting the available degrees of freedom in nuclear matter, especially with 17 regards to the delta versus pion-nucleon degrees of freedom. We have made a preliminary comparison with the first DLS data on p(4.9 GeV) + Be based on nucleon-nucleon bremsstrahlung and delta production and decay. When we are satisfied that we have adequate knowledge of the elementary processes then they can be used in conjunction with a dynamical calculation of the evolution of nuclear matter in a heavy ion collision (for example, the Boltzmann-Uehling-Uhlenbeckapproach) to predict absolute distributions. Soon, additional data should be available from the Dilepton Spectrometer at the Bevalac. It is amusing to recall that a decade ago one of the major hopes was to 24 discover pion condensation15 in high energy heavy ion collisions.
Although
it may not be possible to detect a condensate with dileptons either (condensation occurs in the spin-isospin sound branch, not the real pion branch) we will finally be able to study pion dynamics in dense, high temperature nuclear matter without fear that the signal will be absorbed upon expansion.
C Gale apldJ. Kapusta J Dilepton production
443c
ACKNOWLEDGMENTS J.K. would like to thank the organizers for their hospitality and for running such an interesting program. This research was supported by the U.S. Department of Energy under grant DE-FG02-87ER40328. REFERENCES 1)
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*
Speaker at the International Workshop on Nuclear Dynamics at Medium and High Energies, Bad Honnef, W. Germany, lo-14 October 1988.