Scaling pT distributions for p and p̄ produced in Au+Au collisions at RHIC
W.C. Zhang, Y. Zeng, W.X. Nie, L.L. Zhu and C.B. Yang
Institute of Particle Physics, Hua-Zhong Normal University, Wuhan 430079, P.R. China
With the experimental data from STAR and PHENIX on the centrality dependence of the pT
spectra of protons and anti-protons produced at mid-rapidity in Au+Au collisions at 200 GeV,
we show that for protons and anti-protons there exists a scaling distribution independent of the
colliding centrality. The scaling functions can also describe data from BRAHMS for both proton
and anti-proton spectra at y = 2.2 and 3.2. The scaling behaviors are shown to be incompatible
with the usual string fragmentation scenario for particle production.
PACS numbers: 25.75.Dw,13.85.Ni
I. INTRODUCTION
One of the most important quantities in investigating
properties of the medium produced in high energy col-
lisions is the particle distribution for different species of
final state particles. RHIC experiments have found a lot
of novel phenomena from the particle spectra, such as the
unexpectedly large p/π ratio at pT ∼ 3 GeV/c [1], the
constituent quark number scaling of the elliptic flows [2],
and strong nuclear suppression of the pion spectrum in
central Au+Au collisions [3], etc. From the spectrum one
can learn a lot on the dynamics for particle production.
In many studies, searching for a scaling behavior of
some quantities vs suitable variables is useful for unveil-
ing potential universal dynamics. A typical example is
the proposal of the parton model from the x-scaling of
the structure functions in deep-inelastic scatterings [4].
Quite recently, a scaling behavior [5] of the pion spectrum
at mid-rapidity in Au+Au collisions at RHIC was found,
which related spectra with different collision centralities.
In [6] the scaling behavior was extended to non-central
region, up to η = 3.2 for both Au+Au and d+Au colli-
sions. The same scaling function can be used to describe
pion spectra for pT up to a few GeV/c from different col-
liding systems at different rapidities and centralities. The
shape of pion spectrum in those collisions is determined
by only one parameter 〈pT 〉, the mean transverse momen-
tum of the particle. It is very interesting to ask whether
similar scaling behaviors can be found for spectra of other
particles produced in Au+Au collisions at RHIC. In this
paper, the scaling property of the spectra for protons and
anti-protons is investigated and compared with that for
pions.
The organization of this paper is as follows. In Sec. II
we will address the procedures for searching the scaling
behaviors. Then in Sec. III the scaling properties of the
spectra for protons and anti-protons produced in Au+Au
collisions at RHIC at
sNN = 200 GeV will be studied.
We discuss mainly the centrality scaling of the spectra at
mid-rapidity and extend the discussion to very forward
region with rapidity y = 2.2 and 3.2 briefly. Sec. IV is for
discussions on the relation between the scaling behaviors
and the string fragmentation scenario.
II. METHOD FOR SEARCHING THE SCALING
BEHAVIOR OF THE SPECTRUM
As done in [5, 6], the scaling behavior of a set of spectra
at different centralities can be searched in a few steps.
First, we define a scaled variable
z = pT /K , (1)
and the scaled spectrum
Φ(z) = A
2πpTdpTdy
pT=Kz
, (2)
with K and A free parameters. As a convention, we
choose K = A = 1 for the most central collisions. With
this choice Φ(z) is nothing but the pT distribution for
the most central collisions. For the spectra with other
centralities, we try to coalesce all data points to one curve
by choosing proper parameters A and K. If this can
be achieved, a scaling behavior is found. The detailed
expression of the scaling function depends, of course, on
the choice of A and K for the most central collisions.
This arbitrary can be overcome by introducing another
scaling variable
u = z/〈z〉 = pT /〈pT 〉 , (3)
and the normalized scaling function
Ψ(u) = 〈z〉2Φ(〈z〉u)/
Φ(z)zdz . (4)
Here 〈z〉 is defined as
〈z〉 ≡
zΦ(z)zdz/
Φ(z)zdz . (5)
By definition,
Ψ(u)udu =
uΨ(u)udu = 1. This
scaled transverse momentum distribution is in essence
similar to the KNO-scaling [7] on multiplicity distribu-
tion.
III. SCALING BEHAVIORS OF PROTON AND
ANTI-PROTON DISTRIBUTIONS
Now we focus on the spectra of protons and anti-
protons produced at mid-rapidity in Au+Au collisions at
http://arxiv.org/abs/0704.1062v2
sNN = 200 GeV. STAR and PHENIX Collaborations
at RHIC published spectra for protons and anti-protons
at mid-rapidity for a set of colliding centralities [8, 9].
STAR data have a pT coverage larger than PHENIX ones.
As shown in Fig. 1, all data points for proton spectra at
different centralities can be put to the same curve with
suitably chosen A and K, by the procedure explained
in last section. The parameters are shown in Table I.
Except a few points for very peripheral collisions (cen-
tralities 60-92% for PHENIX data and 60-80% for STAR
data), all points agree well with the curve in about six
orders of magnitude. The larger deviation of data at
centralities 60-92% for PHENIX and 60-80% for STAR
from the scaling curve may be due to the larger central-
ity coverage, because the size of colliding system changes
dramatically in those centrality bins. For simplicity we
define v = ln(1+ z), and the curve can be parameterized
Φp(z) = 0.052 exp(14.9v − 16.2v2 + 3.3v3) . (6)
2 4 6 8 10 12
PHENIX
0−10%
20−30%
40−50%
60−92%
0−12%
10−20%
20−40%
40−60%
60−80%
p in Au+Au
FIG. 1: Scaling behavior of the spectrum for protons pro-
duced at mid-rapidity in Au+Au collisions at RHIC. The data
are taken from [8, 9]. Feed-down corrections are considered
in the data. The solid curve is from Eq. (6).
Similarly, one can put all data points for anti-proton
spectra at different centralities to a curve with other sets
of parameters A and K which are given also in TABLE I.
The agreement is good, as can be seen from Fig. 2, with
only a few points in small pT region for peripheral colli-
sions departing a little from the curve. For anti-proton
the scaling function is
Φp̄(z) = 0.16 exp(13v − 14.9v2 + 2.9v3) , (7)
with v defined above.
To see how good is the agreement between the fitted
curves in Figs. 1 and 2 and the experimental data, one
can calculate a ratio
B = experimental data/fitted results ,
2 4 6 8 10 12
STARPHENIX
0−12%0−10%
10−20%20−30%
20−40%40−50%
40−60%60−92%
60−80%
40−80%
 pbar in Au+Au
FIG. 2: Scaling behavior of the spectrum for anti-protons
produced at mid-rapidity in Au+Au collisions at RHIC. The
data are taken from [8, 9]. Feed-down effects are not corrected
in the STAR data for p̄. The solid curve is from Eq. (7).
STAR p p̄
centrality K A K A
0-12% 1 1 1 1
10-20% 0.997 1.203 1.005 1.417
20-40% 0.986 2.009 0.991 2.305
40-60% 0.973 4.432 0.993 5.414
60-80% 0.941 13.591 0.959 16.686
40-80% 0.986 8.126
PHENIX p p̄
centrality K A K A
0-10% 1.042 1.226 1.068 2.404
20-30% 1.026 2.532 1.045 4.901
40-50% 1.031 6.253 1.013 11.754
60-92% 0.934 39.056 0.935 69.31
BRAHMS p p̄
centrality K A K A
y = 2.2 0.930 0.921
y = 3.2 1.079 0.754 1.153 6.985
TABLE I: Parameters for coalescing all data points to the
same curves in Figs. 1 and 2.
and show B as a function of pT in linear scale for all
the data sets, as shown in Fig. 3 for the case of proton.
From the figure one can see that almost all the points
have values of B within 0.7 to 1.3, which means that the
scaling is true within an accuracy of 30%. This is quite
a good fit, considering the fact that the data cover about
6 orders of magnitude. For anti-protons, the agreement
is better than for protons.
Now one can see that the transverse momentum dis-
tributions for protons and anti-protons satisfy a scaling
law. For large pT (thus large z) the scaling functions in
Eqs. (6) and (7) behave as powers of pT , though the
2 4 6 8 10 12
FIG. 3: Ratio between experimental data and the fitted re-
sults shown in Fig. 1. STAR and PHENIX data are taken
from [8, 9]. Symbols are the same as in Fig. 1.
2 4 6 8 10 12
PHENIX
0−10%
20−30%
40−50%
60−92%
BRAHMS
y=3.2
p in Au+Au
0−12%
10−20%
20−40%
40−60%
60−80%
FIG. 4: Normalized scaling distribution for protons produced
at mid-rapidity and very forward direction in Au+Au col-
lisions at RHIC with the scaling variable u. STAR and
PHENIX data are taken from [8, 9] and BRAHMS data from
[10].
expressions are not in powers of z or pT . The scaling
functions in Eqs. (6) and (7) depend on the choices of A
and K for the case with centrality 0-12% for STAR data.
With the variable u defined in Eq. (3) this dependence
can be circumvented. 〈z〉’s for protons and anti-protons
are 1.14 and 1.08, respectively, with integration over z in
the range from 0 to 12, roughly corresponding to the pT
range measured by STAR. The normalized scaling func-
tions Ψ(u) for protons and anti-protons can be obtained
easily from Eqs. (6) and (7) and are shown in Figs. 4
and 5, respectively together with scaled data points as
in Figs. 1 and 2. A simple parameterization for the two
normalized scaling functions in Figs. 4 and 5 can be given
as follows
Ψp(u) = 0.064 exp(13.6v − 16.67v2 + 3.6v3) ,
Ψp̄(u) = 0.086 exp(12.41v − 15.31v2 + 3.16v3) ,
with v = ln(1 + u).
2 4 6 8 10 12
PHENIX
0−10%
20−30%
40−50%
60−92%
BRAHMS
y=3.2
y=2.2
pbar in Au+Au
0−12%
10−20%
20−40%
40−60%
60−80%
40−80%
FIG. 5: Normalized scaling distribution for anti-protons pro-
duced at mid-rapidity and very forward direction in Au+Au
collisions at RHIC with the scaling variable u. STAR and
PHENIX data are taken from [8, 9] and BRAHMS data from
[10].
As in the case for pion distributions, one can also in-
vestigate the pT distributions of protons and anti-protons
in non-central rapidity regions in Au+Au collisions. The
only data set we can find is from BRAHMS [10] at rapid-
ity y = 2.2 and 3.2 with centrality 0-10%. It is found that
the BRAHMS data can also be put to the same scaling
curves, as shown in Figs. 4 and 5. The values of corre-
sponding parameters A and K are also given in TABLE
I. Thus the scaling distributions found in this paper may
be valid in both central and very forward regions for pro-
tons and anti-protons produced in Au+Au collisions at
RHIC at
sNN=200 GeV.
Now one can ask for the difference between the scaling
functions for protons and anti-protons. After normaliza-
tion to 1 the difference between the scaling distributions
Ψ(u) for protons and anti-protons is shown in Fig. 6. In
log scale the difference between the two scaling functions
is invisible at low u. To show the difference clearly a ra-
tio r = Ψp(u)/Ψp̄(u) is plotted in the inset of Fig. 6 as
a function of u. The increase of r with u is in agreement
qualitatively with data shown in [9] where it is shown that
p̄/p decreases with pT monotonically. The difference in
the two scaling functions can be understood physically.
In Au+Au collisions there are much more quarks u, d
than ū and d̄ in the initial state. In the central region in
the state just before hadronization, more u and d quarks
can be found because of the nuclear stopping effect in
the interactions. As a consequence, more protons can
be formed from the almost thermalized quark medium
than anti-protons in the small pT regime. Experimen-
tal data show that in low pT region the yield of anti-
proton is about 80% that of protons in central Au+Au
collisions at RHIC. This difference contributes to the net
baryon density in the central region in Au+Au collisions
at RHIC. On the other hand, in the large pT region, pro-
tons and anti-protons are formed mainly from fragmen-
tation of hard partons produced in the QCD interactions
with large momentum transfer. As shown in [11], the
gluon yield from hard processes is about five times that
of u and d quarks. The fragmentation from a gluon to
p and p̄ is the same. The amount of u, d quarks from
hard processes is about 10 times that of ū, d̄ when the
hard parton’s transverse momentum is high enough. It
is well-known that the fragmentation function for a gluon
to p or p̄ is much smaller than that for a u or d (ū or d̄)
to p (p̄) because of the dominant valence quark contribu-
tion to the latter process. As a result, the ratio of yields
of proton over anti-proton at large pT is even more than
that at small pT . After normalizing the distributions to
the scaling functions the yield ratio of proton over anti-
proton increases approximately linearly with u when u is
large. It should be mentioned that no such difference for
π+, π− and π0, because they all are composed of a quark
and an antiquark.
2 4 6 8 10 12
2 4 6 8 10 12
FIG. 6: Comparison between the scaling functions for pro-
tons and anti-protons produced at mid-rapidity in Au+Au
collisions at RHIC with the scaling variable u. The inset if
for the ratio Ψp(u)/Ψp̄(u).
The scaling behaviors of the pT distribution functions
for protons and anti-protons can be tested experimen-
tally from studying the ratio of moments of the mo-
mentum distribution, 〈pnT 〉/〈pT 〉n =
unΨ(u)udu for
n = 2, 3, 4, · · ·. From the determined normalized distri-
butions, the ratio can be calculated by integrating over
u in the range from 0 to 12, as mentioned above, and
the results are tabulated in TABLE II. The values of the
ratio are independent of the parameters A and K in the
fitting process but only on the functional form of the
scaling distributions. If the scaling behaviors of particle
distributions are true, such ratios should be constants in-
n p p̄ π
2 1.194 1.215 1.65
3 1.717 1.775 4.08
4 2.978 3.064 14.4
5 6.415 6.417 64.73
6 19.045 17.253 373.82
TABLE II: Ratio of moments 〈pnT 〉/〈pT 〉
n for protons, anti-
protons and pions produced in Au+Au collisions at RHIC.
dependent of the colliding centralities and rapidities. For
comparison, the corresponding values of the ratio for pi-
ons produced in the same interactions, calculated in [6],
are also given in TABLE II. Because of very small dif-
ference in the scaling distributions for protons and anti-
protons at small u, the ratio for protons increases with
n at about the same rate as for anti-proton for small n.
For large n, the ratio for p becomes larger than that for p̄
because of the big difference in the scaling functions for p
and p̄ at large u. Because of the very strong suppression
of high transverse momentum proton production relative
to that of pions, the ratio for pions increases with n much
more rapidly than for p and p̄.
Another important question is about the difference be-
tween the scaling functions for protons in this paper and
for pions in [5, 6]. Experiments at RHIC have shown that
the ratio of proton yield over that of pion increases with
pT up to 1 in the region pT ≤ 3 GeV/c and saturates
in large pT region. This behavior should be seen from
the scaling functions for these two species of particles.
For the purpose of comparing the scaling distributions
we define a ratio
R = Ψp(u)/Ψπ(u) , (8)
and plot the ratio R as a function of u in Fig. 7. The ratio
increases with u, when u is small, reaches a maximum at
u about 1 and then decreases. Finally it decreases slowly
to about 0.1 for very large u. The highest value of R
is about 1.6, while the experimentally observed p over π
ratio is about 1 at pT ∼ 3 GeV/c. The reason for this
difference is two-fold. One is the normalization difference
in defining R and the experimental ratio. Another lies in
the different mean transverse momenta 〈pT 〉’s for pions
and protons with which the scaling variable u is defined
and used in getting the ratio R.
The existence of difference in the scaling distributions
for different species of particles produced in high energy
collisions is not surprising, because the distributions re-
flect the particle production dynamics which may be dif-
ferent for different particles. In the quark recombina-
tion models [12, 13, 14] pions are formed by combining a
quark and an anti-quark while protons by three quarks.
Because different numbers of (anti)quarks participate in
forming the particles, their scaling distributions must be
different. In this sense, our investigation results urge
more studies on particle production mechanisms.
0 1 2 3 4 5
FIG. 7: Ratio Ψp(u)/Ψπ(u) between the scaling functions for
protons and pions produced in Au+Au collisions at RHIC as
a function of the scaling variable u. The pion scaling distri-
bution is from [5, 6].
IV. DISCUSSIONS
From above investigation we have found scaling distri-
butions for protons and anti-protons produced in Au+Au
collisions at RHIC in both mid-rapidity and forward re-
gion. The difference between those two scaling distri-
butions is quite small, but they differ a lot from that for
pions and the ratio Ψp/Ψπ exhibits a nontrivial behavior.
Investigations in [5, 6] and in this paper have shown
that particle distributions can be put to the same curve
by linear transformation on pT . Though we have not
yet a uniform picture for the particle productions in high
energy nuclear collisions, the scaling behaviors can, in
some sense, be compared to that from the string frag-
mentation picture [15]. In that picture if there are n
strings, they may overlap in an area of Sn and the
average area for a string is then Sn/n. It is shown
that the momentum distributions can be related to the
case in pp collisions also by a linear variable change
pT → pT ((Sn/n)AuAu/(Sn/n)pp)1/4. Viewed from that
picture, our fitted K gives the degree of string overlap.
The average area for a string in most central Au+Au
collisions is about 70 percent of that in peripheral ones
from the values of K obtained from fitting the spectra of
proton. If string fragmentation is really the production
mechanism for all species of particles in the collisions, one
can expect that the overlap degree obtained is the same
from the changes of spectrum of any particle. In the lan-
guage in this work, values of K are expected the same
for pions, protons and other particles in the string frag-
mentation picture for particle production. Our results
show the opposite. Comparing the values of K from [5]
and this work, one can see that for pion spectrum K is
larger for more peripheral collisions but smaller for pro-
ton and anti-proton spectra. Our results indicate that
other particle production mechanisms may also provide
ways to the scaling distributions. Obviously more de-
tailed studies, both theoretically and experimentally, are
needed.
Acknowledgments
This work was supported in part by the National
Natural Science Foundation of China under Grant Nos.
10635020 and 10475032, by the Ministry of Education of
China under Grant No. 306022 and project IRT0624.
[1] S.S. Adler et al., PHENIX Collaboration, Phys. Rev. C
69, 034909 (2004).
[2] D. Molnár and S.A. Voloshin, Phys. Rev. Lett. 91,
092301 (2003); P.Sorensen,STAR Collaboration, J.Phys.
G30, S217 (2004).
[3] See, for example, S.S.Adler,PHENIX Collaboration,
Phys. Rev. Lett. 91, 072301 (2003).
[4] J.D. Bjorken and E.A. Paschos, Phys. Rev. 185, 1975
(1969).
[5] R.C. Hwa and C.B. Yang, Phys. Rev. Lett. 90, 212301
(2003).
[6] L.L. Zhu and C.B. Yang, Phys. ReV. C 75, 044904
(2007).
[7] Z. Koba, H.B. Nielsen and P. Olesen, Nucl. Phys. B 40,
317 (1972).
[8] S.S. Adler et al., PHENIX Collaboration, Phys. Rev.
Lett. 91, 172301 (2003).
[9] B.I. Abelev et al., STAR Collaboration, Phys. Rev. Lett.
97, 152301 (2006).
[10] R. Karabowicz for the BRAHMS Collaboration, talk at
Quark Matter 2005, Budapest, Hungary, Nucl. Phys. A
774, 447 (2006); I. Arsene et al., BRAHMS collaboration,
nucl-ex/0610021.
[11] D.K. Srivastava, C. Gale and R.J. Fries, Phys. Rev. C
67, 034903 (2003).
[12] R.C. Hwa, and C.B. Yang, Phys. Rev. C 67, 034902
(2003).
[13] V. Greco, C.M. Ko, and P. Lévai, Phys. Rev. Lett. 90,
202302 (2003); Phys. Rev. C 68, 034904 (2003).
[14] R.J. Fries, B. Müller, C. Nonaka and S.A. Bass, Phys.
Rev. Lett. 90, 202303 (2003); Phys. Rev. C 68, 044902
(2003).
[15] M.A. Braun, F. del Moral, and C. Pajares, Nucl. Phys.
A 715, 791 (2003).
http://arxiv.org/abs/nucl-ex/0610021

  With the experimental data from STAR and PHENIX on the centrality dependence
of the $p_T$ spectra of protons and anti-protons produced at mid-rapidity in
Au+Au collisions at 200 GeV, we show that for protons and anti-protons there
exists a scaling distribution independent of the colliding centrality. The
scaling functions can also describe data from BRAHMS for both proton and
anti-proton spectra at $y=2.2$ and 3.2. The scaling behaviors are shown to be
incompatible with the usual string fragmentation scenario for particle
production.

Scaling pT distributions for p and p̄ produced in Au+Au collisions at RHIC
W.C. Zhang, Y. Zeng, W.X. Nie, L.L. Zhu and C.B. Yang
Institute of Particle Physics, Hua-Zhong Normal University, Wuhan 430079, P.R. China
With the experimental data from STAR and PHENIX on the centrality dependence of the pT
spectra of protons and anti-protons produced at mid-rapidity in Au+Au collisions at 200 GeV,
we show that for protons and anti-protons there exists a scaling distribution independent of the
colliding centrality. The scaling functions can also describe data from BRAHMS for both proton
and anti-proton spectra at y = 2.2 and 3.2. The scaling behaviors are shown to be incompatible
with the usual string fragmentation scenario for particle production.
PACS numbers: 25.75.Dw,13.85.Ni
I. INTRODUCTION
One of the most important quantities in investigating
properties of the medium produced in high energy col-
lisions is the particle distribution for different species of
final state particles. RHIC experiments have found a lot
of novel phenomena from the particle spectra, such as the
unexpectedly large p/π ratio at pT ∼ 3 GeV/c [1], the
constituent quark number scaling of the elliptic flows [2],
and strong nuclear suppression of the pion spectrum in
central Au+Au collisions [3], etc. From the spectrum one
can learn a lot on the dynamics for particle production.
In many studies, searching for a scaling behavior of
some quantities vs suitable variables is useful for unveil-
ing potential universal dynamics. A typical example is
the proposal of the parton model from the x-scaling of
the structure functions in deep-inelastic scatterings [4].
Quite recently, a scaling behavior [5] of the pion spectrum
at mid-rapidity in Au+Au collisions at RHIC was found,
which related spectra with different collision centralities.
In [6] the scaling behavior was extended to non-central
region, up to η = 3.2 for both Au+Au and d+Au colli-
sions. The same scaling function can be used to describe
pion spectra for pT up to a few GeV/c from different col-
liding systems at different rapidities and centralities. The
shape of pion spectrum in those collisions is determined
by only one parameter 〈pT 〉, the mean transverse momen-
tum of the particle. It is very interesting to ask whether
similar scaling behaviors can be found for spectra of other
particles produced in Au+Au collisions at RHIC. In this
paper, the scaling property of the spectra for protons and
anti-protons is investigated and compared with that for
pions.
The organization of this paper is as follows. In Sec. II
we will address the procedures for searching the scaling
behaviors. Then in Sec. III the scaling properties of the
spectra for protons and anti-protons produced in Au+Au
collisions at RHIC at
sNN = 200 GeV will be studied.
We discuss mainly the centrality scaling of the spectra at
mid-rapidity and extend the discussion to very forward
region with rapidity y = 2.2 and 3.2 briefly. Sec. IV is for
discussions on the relation between the scaling behaviors
and the string fragmentation scenario.
II. METHOD FOR SEARCHING THE SCALING
BEHAVIOR OF THE SPECTRUM
As done in [5, 6], the scaling behavior of a set of spectra
at different centralities can be searched in a few steps.
First, we define a scaled variable
z = pT /K , (1)
and the scaled spectrum
Φ(z) = A
2πpTdpTdy
pT=Kz
, (2)
with K and A free parameters. As a convention, we
choose K = A = 1 for the most central collisions. With
this choice Φ(z) is nothing but the pT distribution for
the most central collisions. For the spectra with other
centralities, we try to coalesce all data points to one curve
by choosing proper parameters A and K. If this can
be achieved, a scaling behavior is found. The detailed
expression of the scaling function depends, of course, on
the choice of A and K for the most central collisions.
This arbitrary can be overcome by introducing another
scaling variable
u = z/〈z〉 = pT /〈pT 〉 , (3)
and the normalized scaling function
Ψ(u) = 〈z〉2Φ(〈z〉u)/
Φ(z)zdz . (4)
Here 〈z〉 is defined as
〈z〉 ≡
zΦ(z)zdz/
Φ(z)zdz . (5)
By definition,
Ψ(u)udu =
uΨ(u)udu = 1. This
scaled transverse momentum distribution is in essence
similar to the KNO-scaling [7] on multiplicity distribu-
tion.
III. SCALING BEHAVIORS OF PROTON AND
ANTI-PROTON DISTRIBUTIONS
Now we focus on the spectra of protons and anti-
protons produced at mid-rapidity in Au+Au collisions at
http://arxiv.org/abs/0704.1062v2
sNN = 200 GeV. STAR and PHENIX Collaborations
at RHIC published spectra for protons and anti-protons
at mid-rapidity for a set of colliding centralities [8, 9].
STAR data have a pT coverage larger than PHENIX ones.
As shown in Fig. 1, all data points for proton spectra at
different centralities can be put to the same curve with
suitably chosen A and K, by the procedure explained
in last section. The parameters are shown in Table I.
Except a few points for very peripheral collisions (cen-
tralities 60-92% for PHENIX data and 60-80% for STAR
data), all points agree well with the curve in about six
orders of magnitude. The larger deviation of data at
centralities 60-92% for PHENIX and 60-80% for STAR
from the scaling curve may be due to the larger central-
ity coverage, because the size of colliding system changes
dramatically in those centrality bins. For simplicity we
define v = ln(1+ z), and the curve can be parameterized
Φp(z) = 0.052 exp(14.9v − 16.2v2 + 3.3v3) . (6)
2 4 6 8 10 12
PHENIX
0−10%
20−30%
40−50%
60−92%
0−12%
10−20%
20−40%
40−60%
60−80%
p in Au+Au
FIG. 1: Scaling behavior of the spectrum for protons pro-
duced at mid-rapidity in Au+Au collisions at RHIC. The data
are taken from [8, 9]. Feed-down corrections are considered
in the data. The solid curve is from Eq. (6).
Similarly, one can put all data points for anti-proton
spectra at different centralities to a curve with other sets
of parameters A and K which are given also in TABLE I.
The agreement is good, as can be seen from Fig. 2, with
only a few points in small pT region for peripheral colli-
sions departing a little from the curve. For anti-proton
the scaling function is
Φp̄(z) = 0.16 exp(13v − 14.9v2 + 2.9v3) , (7)
with v defined above.
To see how good is the agreement between the fitted
curves in Figs. 1 and 2 and the experimental data, one
can calculate a ratio
B = experimental data/fitted results ,
2 4 6 8 10 12
STARPHENIX
0−12%0−10%
10−20%20−30%
20−40%40−50%
40−60%60−92%
60−80%
40−80%
 pbar in Au+Au
FIG. 2: Scaling behavior of the spectrum for anti-protons
produced at mid-rapidity in Au+Au collisions at RHIC. The
data are taken from [8, 9]. Feed-down effects are not corrected
in the STAR data for p̄. The solid curve is from Eq. (7).
STAR p p̄
centrality K A K A
0-12% 1 1 1 1
10-20% 0.997 1.203 1.005 1.417
20-40% 0.986 2.009 0.991 2.305
40-60% 0.973 4.432 0.993 5.414
60-80% 0.941 13.591 0.959 16.686
40-80% 0.986 8.126
PHENIX p p̄
centrality K A K A
0-10% 1.042 1.226 1.068 2.404
20-30% 1.026 2.532 1.045 4.901
40-50% 1.031 6.253 1.013 11.754
60-92% 0.934 39.056 0.935 69.31
BRAHMS p p̄
centrality K A K A
y = 2.2 0.930 0.921
y = 3.2 1.079 0.754 1.153 6.985
TABLE I: Parameters for coalescing all data points to the
same curves in Figs. 1 and 2.
and show B as a function of pT in linear scale for all
the data sets, as shown in Fig. 3 for the case of proton.
From the figure one can see that almost all the points
have values of B within 0.7 to 1.3, which means that the
scaling is true within an accuracy of 30%. This is quite
a good fit, considering the fact that the data cover about
6 orders of magnitude. For anti-protons, the agreement
is better than for protons.
Now one can see that the transverse momentum dis-
tributions for protons and anti-protons satisfy a scaling
law. For large pT (thus large z) the scaling functions in
Eqs. (6) and (7) behave as powers of pT , though the
2 4 6 8 10 12
FIG. 3: Ratio between experimental data and the fitted re-
sults shown in Fig. 1. STAR and PHENIX data are taken
from [8, 9]. Symbols are the same as in Fig. 1.
2 4 6 8 10 12
PHENIX
0−10%
20−30%
40−50%
60−92%
BRAHMS
y=3.2
p in Au+Au
0−12%
10−20%
20−40%
40−60%
60−80%
FIG. 4: Normalized scaling distribution for protons produced
at mid-rapidity and very forward direction in Au+Au col-
lisions at RHIC with the scaling variable u. STAR and
PHENIX data are taken from [8, 9] and BRAHMS data from
[10].
expressions are not in powers of z or pT . The scaling
functions in Eqs. (6) and (7) depend on the choices of A
and K for the case with centrality 0-12% for STAR data.
With the variable u defined in Eq. (3) this dependence
can be circumvented. 〈z〉’s for protons and anti-protons
are 1.14 and 1.08, respectively, with integration over z in
the range from 0 to 12, roughly corresponding to the pT
range measured by STAR. The normalized scaling func-
tions Ψ(u) for protons and anti-protons can be obtained
easily from Eqs. (6) and (7) and are shown in Figs. 4
and 5, respectively together with scaled data points as
in Figs. 1 and 2. A simple parameterization for the two
normalized scaling functions in Figs. 4 and 5 can be given
as follows
Ψp(u) = 0.064 exp(13.6v − 16.67v2 + 3.6v3) ,
Ψp̄(u) = 0.086 exp(12.41v − 15.31v2 + 3.16v3) ,
with v = ln(1 + u).
2 4 6 8 10 12
PHENIX
0−10%
20−30%
40−50%
60−92%
BRAHMS
y=3.2
y=2.2
pbar in Au+Au
0−12%
10−20%
20−40%
40−60%
60−80%
40−80%
FIG. 5: Normalized scaling distribution for anti-protons pro-
duced at mid-rapidity and very forward direction in Au+Au
collisions at RHIC with the scaling variable u. STAR and
PHENIX data are taken from [8, 9] and BRAHMS data from
[10].
As in the case for pion distributions, one can also in-
vestigate the pT distributions of protons and anti-protons
in non-central rapidity regions in Au+Au collisions. The
only data set we can find is from BRAHMS [10] at rapid-
ity y = 2.2 and 3.2 with centrality 0-10%. It is found that
the BRAHMS data can also be put to the same scaling
curves, as shown in Figs. 4 and 5. The values of corre-
sponding parameters A and K are also given in TABLE
I. Thus the scaling distributions found in this paper may
be valid in both central and very forward regions for pro-
tons and anti-protons produced in Au+Au collisions at
RHIC at
sNN=200 GeV.
Now one can ask for the difference between the scaling
functions for protons and anti-protons. After normaliza-
tion to 1 the difference between the scaling distributions
Ψ(u) for protons and anti-protons is shown in Fig. 6. In
log scale the difference between the two scaling functions
is invisible at low u. To show the difference clearly a ra-
tio r = Ψp(u)/Ψp̄(u) is plotted in the inset of Fig. 6 as
a function of u. The increase of r with u is in agreement
qualitatively with data shown in [9] where it is shown that
p̄/p decreases with pT monotonically. The difference in
the two scaling functions can be understood physically.
In Au+Au collisions there are much more quarks u, d
than ū and d̄ in the initial state. In the central region in
the state just before hadronization, more u and d quarks
can be found because of the nuclear stopping effect in
the interactions. As a consequence, more protons can
be formed from the almost thermalized quark medium
than anti-protons in the small pT regime. Experimen-
tal data show that in low pT region the yield of anti-
proton is about 80% that of protons in central Au+Au
collisions at RHIC. This difference contributes to the net
baryon density in the central region in Au+Au collisions
at RHIC. On the other hand, in the large pT region, pro-
tons and anti-protons are formed mainly from fragmen-
tation of hard partons produced in the QCD interactions
with large momentum transfer. As shown in [11], the
gluon yield from hard processes is about five times that
of u and d quarks. The fragmentation from a gluon to
p and p̄ is the same. The amount of u, d quarks from
hard processes is about 10 times that of ū, d̄ when the
hard parton’s transverse momentum is high enough. It
is well-known that the fragmentation function for a gluon
to p or p̄ is much smaller than that for a u or d (ū or d̄)
to p (p̄) because of the dominant valence quark contribu-
tion to the latter process. As a result, the ratio of yields
of proton over anti-proton at large pT is even more than
that at small pT . After normalizing the distributions to
the scaling functions the yield ratio of proton over anti-
proton increases approximately linearly with u when u is
large. It should be mentioned that no such difference for
π+, π− and π0, because they all are composed of a quark
and an antiquark.
2 4 6 8 10 12
2 4 6 8 10 12
FIG. 6: Comparison between the scaling functions for pro-
tons and anti-protons produced at mid-rapidity in Au+Au
collisions at RHIC with the scaling variable u. The inset if
for the ratio Ψp(u)/Ψp̄(u).
The scaling behaviors of the pT distribution functions
for protons and anti-protons can be tested experimen-
tally from studying the ratio of moments of the mo-
mentum distribution, 〈pnT 〉/〈pT 〉n =
unΨ(u)udu for
n = 2, 3, 4, · · ·. From the determined normalized distri-
butions, the ratio can be calculated by integrating over
u in the range from 0 to 12, as mentioned above, and
the results are tabulated in TABLE II. The values of the
ratio are independent of the parameters A and K in the
fitting process but only on the functional form of the
scaling distributions. If the scaling behaviors of particle
distributions are true, such ratios should be constants in-
n p p̄ π
2 1.194 1.215 1.65
3 1.717 1.775 4.08
4 2.978 3.064 14.4
5 6.415 6.417 64.73
6 19.045 17.253 373.82
TABLE II: Ratio of moments 〈pnT 〉/〈pT 〉
n for protons, anti-
protons and pions produced in Au+Au collisions at RHIC.
dependent of the colliding centralities and rapidities. For
comparison, the corresponding values of the ratio for pi-
ons produced in the same interactions, calculated in [6],
are also given in TABLE II. Because of very small dif-
ference in the scaling distributions for protons and anti-
protons at small u, the ratio for protons increases with
n at about the same rate as for anti-proton for small n.
For large n, the ratio for p becomes larger than that for p̄
because of the big difference in the scaling functions for p
and p̄ at large u. Because of the very strong suppression
of high transverse momentum proton production relative
to that of pions, the ratio for pions increases with n much
more rapidly than for p and p̄.
Another important question is about the difference be-
tween the scaling functions for protons in this paper and
for pions in [5, 6]. Experiments at RHIC have shown that
the ratio of proton yield over that of pion increases with
pT up to 1 in the region pT ≤ 3 GeV/c and saturates
in large pT region. This behavior should be seen from
the scaling functions for these two species of particles.
For the purpose of comparing the scaling distributions
we define a ratio
R = Ψp(u)/Ψπ(u) , (8)
and plot the ratio R as a function of u in Fig. 7. The ratio
increases with u, when u is small, reaches a maximum at
u about 1 and then decreases. Finally it decreases slowly
to about 0.1 for very large u. The highest value of R
is about 1.6, while the experimentally observed p over π
ratio is about 1 at pT ∼ 3 GeV/c. The reason for this
difference is two-fold. One is the normalization difference
in defining R and the experimental ratio. Another lies in
the different mean transverse momenta 〈pT 〉’s for pions
and protons with which the scaling variable u is defined
and used in getting the ratio R.
The existence of difference in the scaling distributions
for different species of particles produced in high energy
collisions is not surprising, because the distributions re-
flect the particle production dynamics which may be dif-
ferent for different particles. In the quark recombina-
tion models [12, 13, 14] pions are formed by combining a
quark and an anti-quark while protons by three quarks.
Because different numbers of (anti)quarks participate in
forming the particles, their scaling distributions must be
different. In this sense, our investigation results urge
more studies on particle production mechanisms.
0 1 2 3 4 5
FIG. 7: Ratio Ψp(u)/Ψπ(u) between the scaling functions for
protons and pions produced in Au+Au collisions at RHIC as
a function of the scaling variable u. The pion scaling distri-
bution is from [5, 6].
IV. DISCUSSIONS
From above investigation we have found scaling distri-
butions for protons and anti-protons produced in Au+Au
collisions at RHIC in both mid-rapidity and forward re-
gion. The difference between those two scaling distri-
butions is quite small, but they differ a lot from that for
pions and the ratio Ψp/Ψπ exhibits a nontrivial behavior.
Investigations in [5, 6] and in this paper have shown
that particle distributions can be put to the same curve
by linear transformation on pT . Though we have not
yet a uniform picture for the particle productions in high
energy nuclear collisions, the scaling behaviors can, in
some sense, be compared to that from the string frag-
mentation picture [15]. In that picture if there are n
strings, they may overlap in an area of Sn and the
average area for a string is then Sn/n. It is shown
that the momentum distributions can be related to the
case in pp collisions also by a linear variable change
pT → pT ((Sn/n)AuAu/(Sn/n)pp)1/4. Viewed from that
picture, our fitted K gives the degree of string overlap.
The average area for a string in most central Au+Au
collisions is about 70 percent of that in peripheral ones
from the values of K obtained from fitting the spectra of
proton. If string fragmentation is really the production
mechanism for all species of particles in the collisions, one
can expect that the overlap degree obtained is the same
from the changes of spectrum of any particle. In the lan-
guage in this work, values of K are expected the same
for pions, protons and other particles in the string frag-
mentation picture for particle production. Our results
show the opposite. Comparing the values of K from [5]
and this work, one can see that for pion spectrum K is
larger for more peripheral collisions but smaller for pro-
ton and anti-proton spectra. Our results indicate that
other particle production mechanisms may also provide
ways to the scaling distributions. Obviously more de-
tailed studies, both theoretically and experimentally, are
needed.
Acknowledgments
This work was supported in part by the National
Natural Science Foundation of China under Grant Nos.
10635020 and 10475032, by the Ministry of Education of
China under Grant No. 306022 and project IRT0624.
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http://arxiv.org/abs/nucl-ex/0610021
