Extended Longitudinal Scaling: Direct evidence of
saturation
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Citation
Busza, Wit. “Extended Longitudinal Scaling: Direct Evidence of
Saturation.” Nuclear Physics A 854, no. 1 (March 2011): 57–63.
As Published
http://dx.doi.org/10.1016/j.nuclphysa.2010.12.015
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Elsevier
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Author's final manuscript
Accessed
Thu May 26 15:12:28 EDT 2016
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http://hdl.handle.net/1721.1/99186
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Extended Longitudinal Scaling: direct evidence of
saturation*
arXiv:1102.3921v1 [nucl-ex] 18 Feb 2011
Wit Busza
Department of Physics and Laboratory for Nuclear Science, Massachusetts Institute of
Technology, 77 Massachusetts Avenue, Cambridge, MA USA
Abstract
Multiparticle production of charged particles at high energies exhibit the
phenomenon of Limiting Fragmentation. Furthermore, the region in rapidity
over which the production of particles appears to be independent of energy,
increases with energy. It is argued that this phenomenon, known as Extended
Longitudinal Scaling, is a direct manifestation of some kind of saturation,
akin to that in the Color Glass Condensate picture of particle production.
PACS: 25.75.-q, 25.75.Dw, 13.85.-t
*Invited paper presented at the Glasma Workshop, BNL, May 2010
Keywords: Saturation, Limiting Fragmentation, Extended Longitudinal
Scaling, Color Glass Condensate
In 1969 Benecke, Chou, Yang and Yen [1] proposed the Hypothesis of
Limiting Fragmentation. It was based on the “two-fireball model” used to
explain cosmic ray data and the “intuitive picture of a high-energy collision process as two extended objects going through each other, breaking into
fragments in the process...”. In essence, the Hypothesis of Limiting Fragmentation, which is a consequence of Feynman Scaling, states that if in the
collision of two particles one views the collision in the rest frame of one of the
incident particles, the production process of the soft particles is independent
of the energy or rapidity of the other particle. Manifestations of Limiting
Fragmentation are apparent for example in figures 1-3, where pseudorapidity
distributions of charged particles produced in p+p, p+A and A+A collisions
at various energies are shown in the rest frame of one of the incident particles.
It should be noted that throughout this paper no distinction is made
p
between the pseudorapidity η = −ln tan 2θ = sinh−1 ( p⊥k ) and the rapidity
Preprint submitted to Nuclear Physics A
February 22, 2011
dNch
dη
7
pp CMS NSD 7000 GeV
pp CMS NSD 2360 GeV
pp ALICE NSD 2360 GeV
6
pp CMS NSD 900 GeV
pp ALICE NSD 900 GeV
pp CDF NSD 1800 GeV
pp UA5 NSD 900 GeV
5
pp CDF NSD 630 GeV
pp P238 NSD 630 GeV
pp UA5 NSD 546 GeV
4
pp UA1 NSD 540 GeV
pp UA5 NSD 200 GeV
pp UA5 NSD 53 GeV
3
2
1
0
0
1
2
3
4
5
6
7
8
9
η+y
10
Beam
Figure 1: (color on line) Example of Limiting Fragmentation and of Extended Longitudinal
ch
Scaling as seen in non-single diffractive (NSD) p+p and p+ p̄ collisions. dN
dη is the number
of produced charged particles per unit of pseudorapidity η plotted as a function of η in
the rest frame of the incident proton. In boosting the data from the center of mass to the
proton rest frame the difference in pseudorapidity and rapidity y is ignored. The figure is
courtesy of Yen-Jie Lee [2].
2
7
200 GeV (1/1.6) d+Au (50-70%) : PHOBOS
dNch/d!
200 GeV (1/1.6) d+Au (50-70%) :PHOBOS
pEmulsion
sNN
38.7 GeV
23.7 GeV
19.4 GeV
11.2 GeV
6.7 GeV
6
5
4
pEmulsion
sNN
38.7 GeV
23.7 GeV
19.4 GeV
11.2 GeV
6.7 GeV
3
2
1
7
200 GeV (1/1.83) d+Au (40-50%) : PHOBOS
dNch/d!
200 GeV (1/1.83) d+Au (40-50%) : PHOBOS
pPb: E178
6
pPb: E178
sNN
5
4
sNN
19.4 GeV
19.4 GeV
13.8 GeV
13.8 GeV
9.8 GeV
9.8 GeV
3
2
1
0
-2
0
2
4
6
8
!+ytarget
10
12 -10
-8
-6
-4
-2
!-ybeam
0
2
Figure 2: Example of Limiting Fragmentation and of Extended Longitudinal Scaling seen
ch
in p+A and d+A collisions. dN
dη is the total number of produced charged particles per unit
of pseudorapidity η plotted as a function of η in the rest frame of one or the other incident
particle. In boosting the data from one frame to another the difference in pseudorapidity η
and rapidity y is ignored. The d+Au data are appropriately normalized to be comparable
with the corresponding p+A data. The figure is from [3].
3
dNch/d!
800
a) 0-6%
b) 0-6%
c) 0-6%
Au+Au
200 GeV
130 "
19.6 "
600
400
200
dNch/d!
0
200
d) 35-40%
e) 35-40%
f) 35-40% PHOBOS
150
100
50
0
-2
0
2
4
6
!+ybeam
8 10 -10 -8 -6 -4 -2
!-ybeam
0
2
-4
-2
0
2
|!|-ybeam
Figure 3: Example of Limiting Fragmentation and of Extended Longitudinal Scaling seen
in the most central (0-6%) and more peripheral (35-40% centrality) Au+Au collisions.
dNch
dη is the total number of produced charged particles per unit of pseudorapidity η plotted
as a function of η in the rest frame of one or other Au nucleus. In boosting data from the
center of mass frame to the Au rest frame, the difference in pseudorapidity η and rapidity
y is ignored. The figure is from [3].
p
y = sinh−1 ( mk⊥ ), where θ is the polar angle of the produced particle, p⊥
and pk are respectively the perpendicular and parallel components of its
momentum and m2⊥ = m2 + p2⊥ where m is the mass of the produced particle.
At high energies, to a precision relevant for the discussion here, the difference between η and y is not material. Examples of rapidity rather than
pseudorapidity distributions which exhibit Limiting Fragmentation can be
found in [4, 5].
Figures 4-7 illustrate the universal nature of Limiting Fragmentation. It
is reflected not only in the energy independence of the rapidity distributions
of particle densities but also of particle ratios, of centrality or A-dependence
of particle production in A+A collisions, and even of v1 and v2 , measures of
the azimuthal unisotropy of particles produced in such collisions.
It should be pointed out that in fig 5 the independence of the A-dependence
of the data on energy follows from Limiting Fragmentation. However the values of the exponent α in the Aα parametrization of the particle density and
the apparent independence of α on the nature of the produced particle is a
new feature that, to the best of my knowledge is still not well understood.
Interestingly, it has characteristics which are strikingly similar to those of jet
4
Npart
RPC (!’)
200 GeV
PHOBOS
130 GeV
10
62.4 GeV
19.6 GeV
1
-2
-1
0
1
2
!’
Figure 4: Illustration of the degree to which Limiting Fragmentation seems to be exact
in some processes. In the figure the ratio of the number of charged particles produced
in peripheral (35-40% centrality) to central (0-6% centrality) Au+Au collisions at various
energies, normalized to the same number of participants Npart , is plotted as a function of
the pseudorapidity η ‘ of the produced particle, in the rest frame of one of the Au nuclei.
In boosting the data from the center of mass frame to the Au rest frame the difference in
pseudorapidity and rapidity is ignored. The figure is from [6].
!INEL
24 GeV
0.7
300 GeV
0.8
24 GeV
"
p$ #
p$ %+ 100 GeV
p$ p
p$ n 400 GeV
p$ &
p$ &
p$ K0s
p$ '0 400 GeV
p$ %+p$ %
+
p$ Kp$ K
p$ p
p$ p
0.6
0.5
0.4
0
0.2
0.4
xF
0.6
0.8
1
Figure 5: Compilation of the A-dependence in the forward fragmentation region of the
production of various particles in p+A collisions at a variety of energies. α is the exponent
in the parametrization of p + A → h + anything data in the form σA = σo Aα . It is plotted
as a function of xF , the ratio of the momentum of the hadron h to the momentum of the
incident proton. All the data are for low pt (≤ 300 MceV ). The figure is from [7].
5
0.05
0.04
v2
0.03
0.02
0.01
0
-10
-8
-6
-4
-2
! - ybeam
2 -2
0
2
4
6
! + ybeam
8
10
Au+Au
0.05
v2
0
0.04
0.03
200 GeV
0.02
130 GeV
62.4 GeV
0.01
19.6 GeV PHOBOS
0
-5
-4
-3
-2
-1
|!| - ybeam
0
1
2
Figure 6: Example of Limiting Fragmentation and Extended Longitudinal Scaling exhibited in the azimuthal distribution (in the so-called elliptic flow) of charged particles
produced in Au+Au collisions. v2 is the second coefficient in the Fourier expansion of
the azimuthal distribution of the charged particles. In the top two panels v2 is plotted
as a function of the pseudorapidity in the rest frames of the two incident Au nuclei. In
boosting the data from the center of mass frame to the Au rest frame the difference in
pseudorapidity and rapidity is ignored. In the bottom panel the positive and negative
rapidity data in the top panels are averaged. The figure is from [3].
0.15
v1
Au+Au 200 GeV
Au+Au 130 GeV
Au+Au 62.4 GeV
0.1
Au+Au 19.6 GeV
0.05
0
PHOBOS
-4
-2
!’
!’
0
2
Figure 7: Same as caption to figure 6 except that, in the data above, v1 is the first
Fourier coefficient. The data show how directed flow exhibit Limiting Fragmentation and
Extended Longitudinal Scaling. The figure is from [5].
6
quenching.
Figures 1-7 show in the data another prominent feature related to Limiting Fragmentation. It is that the region in rapidity, over which the production
of particles appears to be independent of energy, increases with energy. This
“increase in the region of Limiting Fragmentation”, first seen in p+A collisions in the 1970s [8, 9, 10], was rechristen a few years ago by Mark Baker
of the PHOBOS collaboration as “Extended Longitudinal Scaling”.
The main goal of this brief paper is to point out that Extended Longitudinal Scaling is direct evidence that some kind of saturation takes place in
high energy collisions.
The easiest way for me to explain how Extended Longitudinal Scaling
implies saturation is through a discussion of a “gedanken experiment”, where
the results of measurements are known from existing experimental data.
Suppose we construct a Au+Au collider where the energies of the two
beams in the laboratory frame of reference can be independently adjusted.
We call one of the two beams the “target” and the other the “beam”. The
corresponding rapidity of the two beams in our laboratory frame we call
y“target” and y“beam” . With this collider we proceed to study the dependence
of particle production at y = 0 (in our rest frame) as we change y“target” and
y“beam” .
|
the density of
What will be the outcome of our measurements of dN
dη y=0
charged particles produced at y = 0, and of v2 |y=0 , the measure of elliplic flow
at y = 0? Examples of results that will be obtained in such measurements
are shown in figures 8 and 9.
Figure 8 shows, for the most central (0-6% centrality) Au+Au collisions,
the y“beam” dependence of
dN
|
dη y=0
Npart
( 2 )
for various values of y“target00 . The data
points shown are taken directly from figure 10, that gives the results of PHOBOS studies of Extended Longitudinal Scaling at RHIC.
As an illustration, in our gedanken experiment results in fig 8, the value
of
dN
|
dη y=0
N
( part
)
2
for y“target” = 2.0 and y“beam” = 7.06 is obtained from the PHOBOS
√
measurement in figure 10 at [|η| − ybeam ] = −2.0 for s = 130GeV (yAu =
±4.93).
Figure 9 shows, for the 0-40% Au+Au collisions the y“beam” dependence
of v2 for various values of y“target” . In a similar manner in which the results
shown in figure 8 are obtained from data in figure 10, the results in figure 9
are obtained from data in figure 6.
7
dN/dη / 〈N
Part
/2〉
4
3
ytarget=3.0
ytarget=2.0
y
=1.0
target
y
=0.5
target
2
1
0
4
6
-y
8
10
"beam"
Figure 8: (color on line) The
dN
dη |y=0
N
( part
)
2
that would be measured in the “gedanken experiment”
(see text) if at y = 0 in the laboratory frame of reference, a Au beam with a rapidity y“beam”
collided with a Au beam with rapidity y“target” . The data show that, for a given value
of y“target” the produced particle density reaches a saturated value as y“beam” increases.
Furthermore, that this saturated value is higher the larger is the the value of y“target” .
The figure is from [11].
y
=3.0
target
y
=2.0
target
y
=1.0
0.05
target
v2
0.04
0.03
0.02
0.01
0
4
6
-y
8
10
"beam"
Figure 9: (color on line)The elliptic flow parameter v2 that would be measured in the
“gedanken experiment” (see text) if at y = 0, in the laboratory frame of references, a Au
beam with a rapidity y“beam” collided with a Au beam with rapidity y“target” . The data
show that, for a given value as y“target” , v2 reaches a saturated value as y“beam00 increases.
Furthermore that this saturated value is higher the larger is the value of y“target” . The
figure is for [11].
8
PHOBOS
Figure 10: (color on line) Compilation of PHOBOS data on the charged particle density
dNch
dη per participant pair produced in A+A collisions at various energies. The data are
plotted in the rest frame of one of the nuclei. In boosting the data from the A+A center
of mass system to he rest frame of one of the nuclei, the difference in pseudorapidity η
and rapidity y is ignored. The figure is from [11].
The results of the “gedanken experiment” shown in figures 8 and 9 are the
essence of this paper. In words, they state that if, for a fixed value of y“target” ,
we increase y“beam” , from zero to very high values, at first the rapidity density
of produced particles and the elliptic flow v2 , at y=0 , increase from zero until
they reach a maximum value. Further increases of y“beam” , no matter how
large, do not produce further increases of dN
|
or of v2 |y=0 . In short,
dη y=0
the “potential” of the target particle to produce more particles is saturated.
|
and v2 |y=0 is to raise the value of
The only way to increase further dN
dη y=0
y“target” .
Viewed in this light, we see that Extended Longitudinal Scaling is a direct
manifestation of some kind of saturation taking place in high energy collisions. It should be noted that the existence of a saturated “potential” of the
slower (“target”) particle to produce particles which can only be increased by
increasing its rapidity, is reminiscent of the Color Glass Condensate picture
of high energy collisions [12]. As data at higher and higher energies from the
LHC becomes available, it will be interesting to see if saturation continues
to persist up to the highest energies.
9
References
[1] J. Benecke et al., Phys. Rev. 188, 2159 (1969)
[2] Yen-Jie Lee, MIT Ph.D. Thesis
[3] B.B. Back et al. (PHOBOS), Nucl. Phys. A757, 28 (2005)
[4] J. Whitmore, Physics Reports 10c (1974)
[5] G. Veres, Nucl. Phys. A774, 287 (2006)
[6] B.B. Back et al. (PHOBOS), Phys. Rev. C74, 021901(R) (2006)
[7] W. Busza, Nucl. Phys. A544, 49c (1992)
[8] C. Halliwell et al., (E178), Phys. Rev. Lett. 39, 1499 (1977)
[9] I. Otterlund et al., Nucl. Phys. B142, 445 (1978)
[10] J. Elias et al., (E178), Phys. Rev. D22, 13 (1980)
[11] W. Busza et al. (PHOBOS), Nucl. Phys. A30, 35c (2009)
[12] L. McLerran, Nucl. Phys. A787, 1c (2007)
10