Analysis of low energy pion spectra
Analysis of Low Energy Pion Spectra
Suk Choi and Kang Seog Lee∗
Department of Physics, Chonnam National University, Gwangju 500-757, Korea
(Dated: Apr. 11, 2007)
The transverse mass spectra and the rapidity distributions of π+ and π− in Au-Au collisions at
2, 4, 6, and 8 GeV·A by E895 collaboration are fitted using an elliptically expanding fireball model
with the contribution from the resonance decays and the final state Coulomb interaction. The
ratio of the total number of produced π− and π+ is used to fit the data. The resulting freeze-out
temperature is rather low(Tf < 60 MeV) with large transverse flow and thus resonance contribution
is very small. The difference in the shape of mt spectra of the oppositely charged pions are found
to be due to the Coulomb interaction of the pions.
PACS numbers: 24.10.Pa,25.75.-q
Pion production just above the threshold energy is
quite different from that at very high energies such as
RHIC energy since the ratio of π− to π+ at very high en-
ergies is one which is not the case at low energies. At just
above the threshold energy, pions are produced through
the production of ∆ resonances and counting all the pos-
sible channels of ∆ decay the difference in the compo-
sition of isospins in the colliding nuclei appears as the
difference in the numbers of the two oppositely charged
pions[1, 2, 3], whereas at high energies many channels
producing pions are open and small asymmetry in the
initial isospin does not matter.
Other features of the pion spectra at low energies
are[1, 2, 3, 4, 5, 6, 7]: (1)The transverse momentum
spectra both of the π− and π+ seem to have two temper-
atures. Usually the low temperature component in the
low momentum region is attributed to the pions decayed
from resonances, especially the delta resonance, while the
higher temperature component in the mid-momentum re-
gion is the thermal ones. (2) Transverse momentum spec-
tra of π− and π+ at very small momentum are different
in the sense that the π+ spectra is convex in its shape
while the π− spectra does not show this behavoir. This
difference in low momentum region is due to the Coulomb
effect. The hadronic matter formed during the collision
has charge which comes from the initially colliding two
nuclei and thus the thermal pions escaping from the sys-
tem experience the Coulomb interaction. The Coulomb
interaction of π− may bend the spectrum in the low mo-
mentum region upward and thus it is hard to disentan-
gle the contribution from the delta resonance and the
Coulomb interaction in the low momentum region. (3)
Width of the rapidity spectra of π− and π+ are much
wider than those from the isotropically expanding ther-
mal model. The wide width may either come from partly
transparent nature of the collision dynamics or the ellip-
soidal expansion geometry. In order to fit large rapidity
width using expanding fireball model one usually needs
large longitudinal expansion velocity.
∗kslee@chonnam.ac.kr
Even though all those features mentioned above are not
new, calculations with all those features put in together
comparing each contributions in detail is hard to find.
There are claims that the properties of ∆ resonance are
modified inside the hadronic matter formed even at this
low energy[5, 6]. In order to draw any conclusion, one
should have a model which can explain all of the features
above mentioned. Lacking one or two features in the
calculations, the result may not be conclusive.
In this paper, we analyze the pion spectra in Au+Au
collisions at 2, 4, 6, and 8 A·GeV measured by the E895
collaboration[4] using the expanding fireball[8, 9, 10]
with the resonance contribution[10] and the final state
Coulomb interaction[1, 11, 12, 13]. The geometry of
the expansion used is ellipsoidal[8] and can be varied to
sphere and cylinder, by taking the transverse size R as a
function of the longitudinal coordinate, z.
At just above the threshold energy, pions are produced
through the production of ∆ resonances and their sub-
sequent decays. The ratio of π− and π+ is given from
the initial isospin conservation as π
5Z2+NZ
∼ 1.94
for Au+Au collisions[1, 2, 3] . Hence it is expected that
at low beam energies near 2 GeV·A in Au+Au collisions,
the ratio is near 1.94 and then as the beam energy is in-
creased the ratio will decrease eventually to one. In the
present calculation, the ratio of normalization constants
for π− and π+, R is taken as a fit parameter in order to
investigate the beam energy dependence of R.
We assume that once the pions are produced they
rescatter among themselves and thermalize before they
decouple from the system. Hence we assume thermaliza-
tion of the pionic matter. However, the total number of
negatively and positively charged pions are not in chem-
ical equilibrium and the ratio is governed by the isospin
asymmetry of the initially colliding nuclei. We keep the
ratio as a fitting parameter and want to compare with the
expected value of 1.94 near threshold of the pion produc-
tion.
After the formation of a pionic fireball, it expands and
cools down until freeze-out when the particle production
is described from the formalism of Cooper-Frye[14]. For
the equilibrium distribution function we use the Lorentz-
boosted Boltzmann distribution function, where uµ is the
http://arxiv.org/abs/0704.1529v1
TABLE I: Fitted values for each parameters.
Ebeam V ηm ρ0 T Pc π
−/π+ χ2 /n
(GeV) (×105) GeV GeV/c
2 1.41 1.12 0.88 46 25 1.96 1.3
4 0.93 1.32 0.92 57 24 1.95 2.9
6 1.44 1.50 1.11 54 18 1.40 2.4
8 1.62 1.58 1.12 55 15 1.38 1.8
expansion velocity the space-time of the system.
(2π)3
pµdσµ(x)f(x, p) (1)
where
f(x, p) = exp(−
pνuν(x)− µ
) (2)
For the geometry of the expanding fireball[8], we as-
sume azimuthal symmetry and further assume boost-
invariance collective dynamics along the longitudinal
direction[15]. In this case it is convenient to use as the
coordinates the longitudinal proper time τ =
t2 − z2,
space-time rapidity η = tanh−1(z/t) and the trans-
verse coordinate r⊥. Then the 4-velocity of expansion
can be expressed as uµ(x) = γ(1, v⊥(x)er , vz(x)) where
vz(τ, r⊥, η) = tanh η, which is the result of the longitudi-
nal boost-invariance. In the transverse direction we take
a linear flow rapidity profile, tanh−1 v⊥ = ρ(η)(r⊥/R0)
where R0 is the transverse radius at midrapidity. Here
one takes ρ(η) = ρ0
1− (η2/η2max) for the elliptic ge-
ometry and a constant value of ρ, i.e. ρ(η) = ρ0 for the
cylindrical geometry. As is the same for the SPS energy
by Dobbler et. al., the elliptic case fits the pion spectra
a little better.
As pions escape from the system at freeze-out, they ex-
perience the Coulomb interaction with the charge of the
system which are mainly due to the initial protons in the
colliding nuclei. The Coulomb effect on the particle spec-
tra are studied in detail in refs.[11, 12, 13] for the static
and dynamical cases. Here due to the low beam energy
we restrict ourselves only to the static case. The system
is expanding rapidly in the longitudinal direction and the
change in the longitudinal momentum is negligible. Only
the transverse momentum of the charged particles will be
shifted by an average amount[11]
pc = ∆p⊥ ∼ 2e2
dN ch
, (3)
where Rf is the transverse radius of the system at freeze-
out. Due to the lack of detailed knowledge, pc is taken
as a fit parameter in the present calculation. And in this
way the beam energy dependence of pc can be studied.
Since the transverse momentum of the escaping ther-
mal pions are shifted by the amount pc, i.e. pt = pt,0±pc.
the invariant cross section can be written in terms of the
unshifted momentum (pt,0, y0).
-1.5 -1 -0.5  0  0.5  1  1.5
E895, Au+Au at 2A·GeV
π- fit
π+ fit
FIG. 1: Rapidity distribution of π− and π+ in Au+Au col-
lision at 2 A·GeV by E895 collaboration[4].
). (4)
where (E d
)0 is the unshifted invariant cross section.
Integrating over the rapidity y one gets the equation for
the transverse momentum spectra,
ptdpt
pt,0dpt,0dy
. (5)
and the rapidity spectra is obtained by integrating over
the transverse mass.
mt,0dmt,0(
pt,0dpt,0
. (6)
Finally one has to add the contribution from resonance
decay[10] to both the transverse spectrum and rapidity
distribution. Here we assume that the resonances decay
far outside the system and the Coulomb interaction of
the pions decayed from the resonances is neglected.
The fitted values for the parameters are tabulated in
the Tab. 1 and the results of the fitting are shown in
Figs.1-2 for 2 A·GeV. The fitted value for the ratio R =
π−/π+ is close to 1.94 at 2 and 4 A·GeV as expected and
decreases to 1.38 at 8 A·GeV, which eventually becomes 1
at higher energies such as RHIC energies. In other words,
at this very low energy pion isospin is not in chemical
equilibrium.
 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1
mt - m0
E895, Au+Au at 2A·GeV
 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1
mt - m0
E895, Au+Au at 2A·GeV
FIG. 2: π− transverse mass spectrum for each rapidity
bin(∆y = 1.0) measured by E895 collaboration[4]in Au+Au
collisions at 2 A·GeV. Data on the top line is for the ra-
pidity bin −0.65 < y < −0.55 and the next one is for
−0.55 < y < −0.45 scaled by 0.1, etc.
The freeze-out temperature is rather small, Tf < 60
MeV and the expansion velocities in both the longitudi-
nal and transverse direction are quite large. The large
longitudinal expansion velocity(> 0.8c) is needed to fit
the large width of the rapidity distribution. The low
freeze-out temperature together with the large transverse
expansion fits the transverse spectra of pions quite nicely.
If not for the large expansion velocity, one usually gets
much larger freeze-out temperature(Tf > 80 MeV.
Since the freeze-out temperature is small, there are
very few resonances at freeze-out, especially ∆, and thus
the resonance contribution is negligible. This is reason-
able since at this low energy near the pion threshold en-
ergy, production of particles with mass larger than pions
is rare and their contribution to the pion spectrum is
negligible.
The pion transverse momentum spectra looks like that
there are two slopes; one for small momentum region and
another for the higher momentum region near the pion
mass. The smaller slope at lower transverse momentum
is usually attributed to the pions from the resonance de-
cay, especially from the ∆ decay[3, 6]. However, present
calculation shows that this is not the case at low beam
energies.
The shape of the transverse mass spectra of π− and
π+ are different especially in the small mass region. As
the transverse mass mt decreases, the mt spectrum of π
increases sharply while that of π+ saturates showing the
convex shape. This difference is due to the Coulomb in-
teraction of pions leaving the system which has the charge
from the initially bombarding nucleons. The change in
the transverse momentum due to the Coulomb interac-
tion decreases from 25 GeV/c at 2 A·GeV to 15 GeV/c at
8 A·GeV. This behavior can be understood from the in-
crease of the screening effect since the number of charged
pions increase at higher energies. At very high energies
such as RHIC energies, the momentum change from the
Coulomb interaction will be small.
The emerging picture of pion production at low en-
ergy is that the pions are produced through the inter-
mediate ∆ formation and thus they are not in chemical
equilibrium in isospin. They make collisions and ther-
malize to form a fireball which expands and cools until
the freeze-out. Since the fireball has charge which is from
the initially colliding nucleus, the pions leaving the sys-
tem experiences the Coulomb interaction which makes
the difference of the mt spectra of the two oppositely
charged pions.
Acknowledgments
This work is financially supported by Chonnam Na-
tional University and the post-BK21 program. We wish
to acknowledge U.W. Heinz for providing the program
and useful discussions.
[1] A. Wagner et. al., Phys. Lett. B420, 20(1998).
[2] B.J. VerWest and R.A. Arndt, Phys. Rev. C25,
1979(1982)
[3] Phys. Rep. 135, 259(1986).
[4] J.L. Klay, et. al., Phys. Rev. C68, 054905(2003).
[5] B. Pin-zhen and J. Rafelski, nucl-th/0507037.
[6] B.S. Hong, J. Korean Phys. Soc. 46, 1083(2005).
[7] B. Hong et. al., Phys. Lett. B407, 115(1999); ibid., J.
Korean Phys. Soc. 46, 1083(2005).
[8] H. Dobbler, J. Sollfrank and U. Heinz, Phys. Lett.
B457,353(1999) .
[9] Kang S. Lee, U. Heinz, and E. Schnedermann, Z. Phys.
C 48,525 (1990).
[10] E. Schnedermann, J. Sollfrank, and U. Heinz, Phys. Rev.
C 48, 2462 (1993).
[11] H.W. Barz, J.P. Bondorf,J.J. Gaardhoje and H. Heisel-
berg, Phys. Rev. C57, 2536(1998).
[12] M. Gyulassy and S. K. Kaufmann, Nucl. Phys. A362,
503(1981).
[13] A. Ayala and J. Kapusta, Phys. Rev. C56, 407(1997).
[14] F. Cooper and G. Frye, Phys. Rev. D10, 186(1974)
[15] J.D. Bjorken, Phys. Rev. D27, 140(1983)
http://arxiv.org/abs/nucl-th/0507037
