A full calculation of lepton-pair angular characteristics is carried out for e^+e^- pairs created in pp, pn and pd collisions at intermediate energies. It is demonstrated that the proposed new observable, the dilepton decay anisotropy, quite sensitively changes for different sources and may be useful for their disentangling. The relevance of the dilepton decay anisotropy is shown in the context of a puzzling energy behavior for the ratio of the lepton yield from pd to pp reactions as observed at the BEVALAC

Bratkovskaya, E L

Cassing, W

Mosel, U

Teryaev, O V

Titov, A I

Toneev, V D

CERN Document Server

Decay Anisotropy of e
+
e
 
Sources
from pN and pd collisions

E.L. Bratkovskaya
y
, W. Cassing, U. Mosel,
O.V. Teryaev

, A.I. Titov

and V.D. Toneev

Institut fur Theoretische Physik, Universitat Giessen, D-35392 Giessen, Germany

Bogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,
141980 Dubna, Moscow region, Russia
August 28, 1995
Abstract
A full calculation of lepton-pair angular characteristics is carried out for e
+
e
 
pairs created in pp, pn and pd collisions at intermediate energies. It is demonstrated
that the proposed new observable, the dilepton decay anisotropy, quite sensitively
changes for dierent sources and may be useful for their disentangling. The rele-
vance of the dilepton decay anisotropy is shown in the context of a puzzling energy
behavior for the ratio of the lepton yield from pd to pp reactions as observed at the
BEVALAC.
Submitted to Phys. Lett. B.

Supported by BMFT and GSI Darmstadt
y
Permanent address: Bogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Re-
search, 141980 Dubna, Moscow Region, Russia
1
As is known since a long time dileptons are quite attractive probes since they provide
almost direct information on hot and dense nuclear matter during its evolution in heavy-
ion collisions at BEVALAC/SIS and SPS energies [1]. The information carried out by
leptons may tell us not only about the interaction dynamics of colliding nuclei, but also
on properties of hadrons in the nuclear environment or on a possible phase transition of
hadrons into a quark-gluon plasma.
However, there are a lot of hadronic sources for dileptons because the electromagnetic
eld couples to all charges and magnetic moments. In particular, in hadron-hadron col-
lisions, the e
+
e
 
pairs are created due to the electromagnetic decay of time-like virtual
photons which can result from the bremsstrahlung process or from the decay of bary-
onic and mesonic resonances including the direct conversion of vector mesons into virtual
photons in accordance with the vector dominance hypothesis. In the nuclear medium,
the properties of these sources may be modied and it is thus very desirable to have ex-
perimental observables which would allow to disentangle the various channels of dilepton
production.
For a decomposition of e
+
e
 
sources contributing to the dilepton invariant mass spec-
tra it seems to be natural to start from the study of elementary nucleon-nucleon collisions
and then to move successively to nucleon-deuteron and more complicated systems. A step
in this direction has been done in Ref. [2] where the pd=pp dielectron ratio was measured
in the 1 5 GeV energy range. In contrast to naive expectations, this pd=pp ratio displays
a puzzling beam-energy dependence which is still a matter of debate. There are several
interpretations of this eect: the interferences of dierent channels [3], the contribution
of the inelasticity eects [4], and of the subthreshold -meson production [5]. Even by
considering the new and more precise data on the transverse momentum distributions of
lepton pairs [2] the puzzle could not be resolved unambiguously.
Recently, we have proposed to use lepton pair angular distributions for a distinction
between dierent sources [6, 7]. Indeed, the coupling of a virtual photon to hadrons
induces a dynamical spin alignment of both the resonances and the virtual photons. One
thus can expect that the angular distribution of a lepton will be anisotropic with respect
to the direction of the dilepton (i.e. virtual photon) emission. It has been shown that due
to the spin alignment of the virtual photon and the spins of colliding or decaying hadrons,
this lepton decay anisotropy turns out to be quite sensitive to the specic production
2
channel [6, 7, 8] .
Following our previous work [6, 7] we focus in this letter on the study of the dilepton
angular anisotropy as a way to disentangle various e
+
e
 
sources in pN and pd reactions.
Alongside with an extended computation of the Dalitz decay channel for pseudoscalar
mesons, we present the rst full results for nucleon-nucleon collisions at energies of a few
GeV. By a simple extension of these calculations to the nucleon-deuteron case we will
investigate if the anisotropy eect can help in solving the pd=pp dilepton ratio puzzle.
As in [6, 7] we choose the polar  and azimuthal ' angles of the momentum
~
l
 
of
a created electron with respect to the momentum ~q of a virtual photon to characterize
the decay anisotropy, where
~
l
 
;
~
l
+
are measured in the rest frame of this virtual photon,
i.e. ~q 
~
l
 
+
~
l
+
= 0. For comparing the shape of the angular distributions for dierent
channels, the dierential cross section for dilepton production in the channel i may be
presented in the form:
S
i
(M;) 
d
i
dMd cos 
= A
i
(1 +B
i
cos
2
); (1)
where M is the invariant mass of a lepton pair (M
2
= q
2
0
  ~q
2
) and A
i
is dened by the
normalization to unity of the angular distribution of (1). The decay anisotropy coecient
for channel i, i.e. B
i
, then is given by
B
i
=
S
i
(M; = 0
o
)
S
i
(M; = 90
o
)
  1: (2)
Since the coecient B
i
is sensitive to the spin structure of the interacting hadrons, it is
in general a function of M and the masses of the hadrons involved in the reaction.
The total dierential cross section for proton{nucleon collisions can be represented as
a sum of dierential cross sections for all channels:
d
pN
dMd cos 
=
X
i=channel
d
i
dMd cos 
= A(1 +B
pN
cos
2
): (3)
For the total anisotropy coecient we then get
B
pN
=
X
i=channel
; =
d
i
dM

B
i
1 +
1
3
B
i
X
i
d
i
dM

1
1 +
1
3
B
i
; (4)
3
where the special weighting factors originate in the necessary angle-integrations. Thus, the
anisotropy coecient B
pN
is the sum of the \weighted" anisotropy coecients ()
for each channel i obtained by means of the convolution of B
i
with the corresponding
invariant mass distribution.
In a preceding study [7] the anisotropy coecients were calculated for the bremsstrah-
lung and {Dalitz decay channels within a microscopic One-Boson-Exchange (OBE)
model [3]. We will use the latter results in the present investigation without any modi-
cation. Note that we take into account the interference terms between bremsstrahlung
and {channels for proton{nucleon and proton{proton interactions. As a result, the
contributions of these two channels to (4) are reduced to a single coherent term.
For the Dalitz decay of an -meson in Ref. [6] a rst simple estimate B

= 1 had been
made appropriate for an -meson with vanishing momentum in the center-of-mass system
(cms) of the colliding hadrons. In this case the direction of a virtual photon ~q

is not
inuenced by the -meson momentum
~
P

which simplies the kinematical considerations.
In extension to [7] we now include the full dynamics,
~
P

6= 0, where the direction of the
dilepton momentum in the cms of the colliding nucleons, ~q, does not coincide with that
in the eta rest frame ~q

; as a consequence the anisotropy coecient becomes a function
of the invariant mass M and the collision energy T
lab
.
To take this functional dependence into account the {channel for the lepton dieren-
tial cross section (1) is written in the following form:
S

(M;) =
Z
d
~
P

d

d
~
P

jT (M;;
~
P

)j
2
; (5)
where the averaged transition matrix element of the {decay is given by
jT (M;;
~
P

)j
2

Z
dV
1
M
4
(

q

q
1



q

q
1

)  L

; (6)
and q; q
1
are the four momenta of a virtual and real photon, respectively. The integral is
taken over the available phase-space volume. The lepton tensor L

is dened as
L

= Tr
^
l
 


^
l
+


; (7)
where
^
l = 

l

, l
 
; l
+
are the four momenta of the leptons.
For the  production cross section d

=d
~
P

we use the phase-space oriented expression:
d

d
~
P


1
E

q
M
2
x
  4m
2
N
M
x
T
2

(P

); (8)
4
with M
2
x
= (p
a
+ p
b
  P

)
2
= s   2
p
sE

+ m
2

. Here s = (p
a
+ p
b
)
2
is the center-
of-mass energy squared and p
a
; p
b
; P

are the four-momenta of the colliding nucleons
and the {meson, respectively, while E

is the energy of the -meson in the cms of
the colliding nucleons. It is noteworthy that the particular case considered in Ref. [6]
corresponds formally to the substitution d

=d
~
P

 (
~
P

). The modication of the phase-
space formula (8) is included in the term T
2

(P

), which is the production matrix element
squared. Here we use a simple parametrization of the calculations presented in Ref. [9],
where the -production cross section in nucleon{nucleon collisions was calculated on the
basis of an eective OBE model.
Results of our numerical calculations for B

at initial energies from the  production
threshold to about 5 GeV are shown in Fig. 1. We nd that the anisotropy coecient
indeed depends on both the initial energy E and the dilepton mass M . B

approaches
1 only close to threshold or for M ! 0. This behavior is quite natural because at
the threshold energy we have j
~
P

j  0; for low mass dileptons their velocity is close to
the velocity of light and thus the dierence between the two directions discussed above
becomes negligible. If the invariant mass of a virtual photon is close to its kinematical
limit (M ! m

), the anisotropy coecient approaches a lower limit of about 0:1 0:25
for bombarding energies from 1.26 to 5 GeV.
In Fig. 2 the dilepton invariant mass distributions d=dM from pp and pn interactions
are shown at bombarding energies of 1.26 GeV and 2.1 GeV, which are the necessary
ingredients for calculating the weighted anisotropy coecients, Eq. (9). In Fig. 2 the
\" denotes the contribution of the {channel the \" labels the contribution of the {
resonance term, while \Br" denotes the bremsstrahlung channel. The dashed curves are
the sum of bremsstrahlung and {channel with interference.
Using the pN cross sections for {production and bremsstrahlung channels from
Ref. [3], we are now in the position to present the rst estimate for the weighted co-
ecients for pp and pn collisions at the energies of 1.26 and 2.1 GeV (Fig. 3).
The notation is the same as in Fig. 2. The solid line denoted by \all" represents the sum
of bremsstrahlung, {channel with interference and {decay source. At the energy of
1.26 GeV the anisotropy is determined by the {decay since its contribution is dominat-
ing at this energy (cf. Fig. 2). The large contribution of the {source at 2.1 GeV leads
to the specic structure in for M  0:3 GeV.
5
As mentioned above, a possible explanation for the experimental pd=pp dilepton ratio
is related to the contribution of leptons from {decay [5]. Whereas the energy threshold
of {production in pN interactions is about 1.26 GeV, in the proton-deuteron case the
eective threshold is lower due to the Fermi motion of the constituent nucleons in the
deuteron. Thus at projectile energies below 1:26 GeV the contribution from the {Dalitz
decay to the dilepton spectrum may be signicant for the pd case, but negligible for pp.
Such an enhancement of the -yield in pd compared to pp reactions has been also seen
experimentally by the PINOT collaboration [10]. In line with this argument we expect
larger values for the pd=pp dilepton ratio at low energy, and according to the calculations
above one can expect that subthreshold {production may manifest itself also in the
angular characteristics (anisotropy coecients) of the created dileptons.
In a rst order approximation the decay anisotropy coecient for pd collisions may
be written as the sum of B
i
for pp and pn interactions. However, this approximation is
not sucient in the energy region close to the {threshold. To take the contribution to
the dilepton spectrum from near-threshold {production into account we calculate the
coecient B
pd
as
B
pd
=
X
k=p;n
 + ; (9)
=
d
pd

dM

B
pN

1 +
1
3
B
pN

X
k=p;n
d
kN
Br;
dM

1
1 +
1
3
B
kN
Br;
+
d
pd

dM

1
1 +
1
3
B
pN

: (10)
It should be stressed that in Eqs. (9), (10) we take into account the interference between
bremsstrahlung and {channel for pp and pn separately.
Fig. 4 shows the resulting weighted anisotropy coecients for pd collisions
at 1.26 and 2.1 GeV. The cross section d
pd

=dM is taken from Ref. [5] and the Paris
deuteron wave function [11] is used for describing the internal nucleon distribution in the
deuteron. As expected, at threshold energy the total coecient for the pd
reaction diers substantially from that for pp or pn reactions due to the near threshold
{decay source. For higher energies (2.1 GeV) the shape of is similar in all
these cases. This energy-dependent eect is more pronounced in Fig. 5 where the ratio
of the anisotropy coecients for pd to pp reactions, R
B
= B
pd
=B
pp
, is represented at
6
the energies of 1.26 and 2.1 GeV. The comparison of the solid (including the {Dalitz
decay) and dashed curves (without the {channel) illustrates the relative importance of
the {decay contribution to the observable decay anisotropy of the dileptons.
Thus, summarizing, the calculated anisotropy coecients for pp, pn and pd reactions
support our suggestion in Refs. [6, 7] that the dilepton decay anisotropy may serve as an
additional observable to discriminate the dilepton sources by experimental means.
This research is partly performed in the framework of the Grants N
0
 
MP8300 and
RFE300 from the International Science Foundation and Russian Government. The work
of V.D.T was supported by Grant N
o
 
3405 from INTAS (International Association for
promotion of cooperation with scientists from the independent states of the former Soviet
Union).
References
[1] U. Mosel, Ann. Rev. Nucl. Part. Sci. 41 (1991) 29.
[2] W.K. Wilson et al., Phys. Lett. B316 (1993) 245.
[3] M. Schafer, H.C. Donges, A. Engel, and U. Mosel, Nucl. Phys. A575 (1994) 429.
[4] K. Haglin and C. Gale, Phys. Rev. C49 (1994) 401.
[5] A.I. Titov, B. Kampfer and E.L. Bratkovskaya, Phys. Rev. C51, (1995) 227.
[6] E.L. Bratkovskaya, O.V. Teryaev and V.D. Toneev, Phys. Lett. B348 (1995) 283.
[7] E.L. Bratkovskaya, M. Schafer, W. Cassing, U. Mosel, O.V. Teryaev, and
V.D. Toneev, Phys. Lett. B348 (1995) 325.
[8] J. Zhang, R. Tabti, Ch. Gale, and K. Haglin, Preprint McGill/94-56, MSUCL-961.
[9] T. Vetter, A. Engel, T. Biro, and U. Mosel, Phys. Lett. B263 (1991) 153.
[10] E. Chiavassa et al., Phys. Lett. B337 (1994) 192.
[11] M. Lacombe, B. Loiseau, J. M. Richard, and R. Vinh Mau, Phys. Rev. C21, (1980)
861.
7
Figure captions
Figure 1: The anisotropy coecient B

for the {channel at initial energies from 1.26 to
4.9 GeV.
Figure 2: The cross sections d=dM for pp and pn interactions at 1.26 and 2.1 GeV bom-
barding energies. The \" denotes the contribution of the {channel, the \" labels the
contribution of the {resonance term, while \Br" denotes the bremsstrahlung channel.
The dashed curves are the sum of bremsstrahlung and {channel with interference.
Figure 3: The weighted anisotropy coecients for pp and pn collisions at
1.26 and 2.1 GeV. The notation is the same as in Fig. 2. The solid line denoted by
\all" represents the sum of bremsstrahlung, {channel with interference and {decay
contributions.
Figure 4: The weighted anisotropy coecients for pd interactions at 1.26 and
2.1 GeV. The notation is the same as in Fig. 3.
Figure 5: The ratio of anisotropy coecients for pd to pp reactions at energies of 1.26
and 2.1 GeV. The solid lines correspond to calculations taking into account the {decay
contribution while the dashed lines do not include the {Dalitz decay.
8

Decay anisotropy of e+e- sources from pN and pd collisions

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