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Coupling of longitudinal and transverse flows in the hydrodynamics of ultrarelativistic nuclear collisions
NUCLEAR PHYSICS A
Nuclear Physics A566 (1994) 473~476~ North-Holland, Amsterdam
Collpling
of longitudinal
ultrarelativist~ic Rajll Venrlgopalan,a “~lrcoretical hlinneapolis,
nuclear Madappa
and transverse
flows in the llydrodynamics
of
collisions* Prakash. b Rlarkku
Physics Institute, University Minnesota 55155, ITSi\
I
of hlinnesota,
“Physics Department. State Tinivcrsity of Nrw York at Stony Stony Hrook, New ‘~‘ork. 11791-3500, USA
Rrook
‘Ph.vsics Department, IJniversity of .J~\~Askyla, PL-35, SF-~10315, .J~v”sl
details
conditions
03759474/94/$07.00
in the cc,r~iIT of mass frame
are as follows:
0 1994 - Elsevier Science B.V. All rights reserved.
R. Venugopalan et al. / Coupling of longitudinal and transverseflows
474c
f,-, is the
Above,
diffuseness tudinal
rapidity
correspond part,ial For
energy
density
parameter.
ye is va.ried
respectively
stopping.
a specific
decoupling
EOS,
strained
such
to the
The
initial
other
temperat
nre
particle
to the
longitudinal
that
“Landall-like”
parameters
initial
between
and
are
multiplicites,
the
2’0, the
0 to -
initial
transverse
1.
longi-
These
limits
These and
in I/,
of full
V, is chosen
longitudinal
energy,
a is a
initial
scenarios
velocity
hypersurface.
2’0 and
maximum
“Bjorken-like”
hydrodynamic
Tdec of the freezePout
temperature
rapidity;
1): ranges
transverse
by the experimental
of charged
corresponding
y is the initial
widt,h
ZL and
parameters the slopes
and
to be zero. the
are conand widths
spectra. S+S (direct
S+S
n-)
5ooI/:/J ---
oLy61.0
-----
1.osys2.0
450
s
-.
2.osyi3.0
2 _^400 ?
350
300
250
Figure
1.
3.
10 20 y (mean
hlean
of direct S+S
00
pions
3.0 4.0 5.0 rapidity bin)
trallsversc
60
momentum
for different
rapidity
Figure
(11~)
collisions.
In general.
hadronic
Iii
longitudinal transverse
Fig.
2, t,lir
(set
are more
Ref.
The
distribution
at 11~-
IJ~
flow is diffrrent
central
pion
sensitilre
[3] f 01 Inore
and transverse
direct,
are shown. in the
spect,ra
EOSs
we show < ~1~> for each yPhin
Evidently.
flows.
The
?T?. the mass
extent
in different,
int,erval
conditions
A significant
~CI’SIIS the mean
cuts
finding
distribufor a bag
of this coupling
than
is shown
1,
pions.
interva.ls. different
rapidity
, 2 < y < 3, exhihib
a clear
shoulder
of t,he particle.
are
in Fig.
rapic1it.y of the bin for direct
rapidity
integratrcl
they
is the coupling
over
pt clistril)ntions
rapidit,y
to the initial
details).
It. is well known
[5] that
intervals in the pt
such a structure
spectrum
Tdec = int.0
momentum rapidity
results
to the different, where
with different
EOS.
Qualitative
between
2. Transverse
tions
I)ins in
“two
1111nqml“
structure
finally
relaxes
MeV,
a gaussian
the
Tdec =
evolves R’IeV.
R. Venugopalan et al. I Coupling of longitudinal and transverse
At early
times,
increa.sing
most
time,
cent,ral
rapidities
steady
state
of the energy-moment,um
tra.nsverse to conserve
when
Figure with
3.
1.0
energy
flow is in the longitudinal significant,
thereby
and transverse
directions
2.0
3.0 y (rapidity)
result,s
5.0
40
of NA35
S+S
dat,a
Figure
for a bag
EOS
with
with calculat.ed
4. C!omparison
with E; =
from
resonance
from
c
and kaons:
decays;
da&cd the
ramifications
We illustra.te
the esperimrntal
flows
conditions 200
if we choose
2.27
GeV/fm3,
yr. =
resonance K-
and
1.3,
decays; kaons;
~1, =
Dot-dash dashed
solid
1.X
fm
curve:
K-
curve:
d-
the
net
curve:
negatives.
Experimental
frame.
= 250 MeV;
rect
net negatives.
Furthermore,
B”.’
data
To = 182.2 RfcV,
7r-
c.m.
a
for a hag EOS
direct
solid curve:
of NA3.5 S+S
results
curve:
For To -
reaches
6.0
curve:
initial
toward
are comparable.
and Tdec = 140 hleV.
transverse
With
particles
At lower Tdec, the system
and momentum.
B’/’ = 250 MeV (T,=lSl.SS hIcV): To = 200 hleV, E% = 3.36 GeV/fm3, ye = 0.0, ZL = 1.12 fm and Tdec = l-40 XleV. Dot-dash
4.
direction.
shifting
s (200 GGV/lI)
Comparison
calculated
hecomes
flows in longitudinal 5 +
“0.0
flow
475c
flows
for
and
large
initial
at
we find
ZL x
rlAJ/rly spectra
the slopes
ramifications
collisions
(!y~=O.o),
Me\’ the
S+S
1 fm are
of 11~spectra temperatures,
200
of the coupling GeV/nucleon.
a. significant 2 -
dependence
3 times
t,oo narrow are larger ‘To E ‘SO
the
Lorentz
in comparison than
between
longitudinal
If we assume
and
“Landau-like”
of &N/&J on ZL, and contracted with
the experimental
the
radius data.
results.
To.
in the
(Fig.
3).
However,
R. Venugopalan et al. 1 Coupling of longitudinal and transverse flows
4162
3
I
01
&
5.
Nh:jr,
s+s data for 03
1
01 0.0
00
’
__
dir.(n-+K-)+res.Z
------
dir.(n-+K-)
10 PT
L ?/ < 2.0
Figure compared wit.11 calculatetl results for a hag EOS. Initial conditions as ill Fig. 4.
’
1
05
I
’ 15
1
:\ 20
(cev/c)
Figure 6. NA35 S+S data for 2.0 5 y 6 3.0 compared with calculated rrsult,s for a hag EOS. Initial conditions as in Fig. 4.
intervals OS < ~1< 2 and 2 < I/ < 3, SIIOWI~ in Fig. 5 and Fig. 6, respectively. In the latter case, there is a clear csccss at 10~ pt. It is nnlikely that bett,er agrcrmcnt for :! < y < 3 can he ohtainctl 1,~ a fiiic tuning of the paramctcrs. For finite bar?-on chemical pot,entials. the contributions of baryonic resonance tlecays to tile spectrum of chargvtl negatives ma)- lx significa,nt [6]. t,hereby explaining the 1~ excess. This possibility is u~ltler investigation in 011r model. IIo~ever, microscopic considerations sl~o\v that, bar!.ons arr unlikely to have t IIP same flow as pions [7]. Therefore, issues related to the rquililxation and tliffereirt ial decoupling of thr reaction process merit further study.
of tile various
species
at the dilut,e st,agcs
5. Acknowledgements WC thank Svigfrirtl \\;enig of the X:13.5 collaboration NA:J5 data 011 S+S collisions.
for supplying
IIS
\vith the CERN
REFERENCES 1. :‘. 3. 4.
5. fi. T.
Venngopalan ant1 iU. Prakasll, n’ucl. Phy.5. A546 (1992) 715. Zalcsal;. .I. C’o~p. f’hy.q. 31 (1979) 335: Al. Iiat,aja, Z. Phys. C38 (1958) 419. Venugopalan, PII. D. thesis, SUNY at Stony Brook, Augnst 1992. Vemlgopalan, hl. Prakash. M. Kataja and V. Ruuskanen, in Ploccrdiny.s of VZZth Il’itltcr TI~or4d~op orr Muclcnr~ D~/nnrnic.s. editors; W. Bauer and J. Kapusta., Worltl Scientific, Singapor~~ I991 I’. 3. Sicmcns and J. 0. Rasnlusscn. Phyi;. Hev. Mt. 42 (1979) SSO. J. Bols, I.-. Ornil< ant1 R. Wirier. 1’hu.s. Rev. C46 (1992) 2037; E. Schnrtlermann, .J. Sollfrauk and I’. IIc-inz. Rrgensh~lrg preprint TPR-93.lfj. Sladappa Prakash. nIall,ju Praliasll, R. Vrnugopalan and G. Welke, Phyq. Rca. Lrtt. 70 (1999) 12%: P/IV.?. Kfp. 227 (1993) 3’1. R. S. R. R.
Nuclear Physics A566 (1994) 473~476~ North-Holland, Amsterdam
Collpling
of longitudinal
ultrarelativist~ic Rajll Venrlgopalan,a “~lrcoretical hlinneapolis,
nuclear Madappa
and transverse
flows in the llydrodynamics
of
collisions* Prakash. b Rlarkku
Physics Institute, University Minnesota 55155, ITSi\
I
of hlinnesota,
“Physics Department. State Tinivcrsity of Nrw York at Stony Stony Hrook, New ‘~‘ork. 11791-3500, USA
Rrook
‘Ph.vsics Department, IJniversity of .J~\~Askyla, PL-35, SF-~10315, .J~v”sl
details
conditions
03759474/94/$07.00
in the cc,r~iIT of mass frame
are as follows:
0 1994 - Elsevier Science B.V. All rights reserved.
R. Venugopalan et al. / Coupling of longitudinal and transverseflows
474c
f,-, is the
Above,
diffuseness tudinal
rapidity
correspond part,ial For
energy
density
parameter.
ye is va.ried
respectively
stopping.
a specific
decoupling
EOS,
strained
such
to the
The
initial
other
temperat
nre
particle
to the
longitudinal
that
“Landall-like”
parameters
initial
between
and
are
multiplicites,
the
2’0, the
0 to -
initial
transverse
1.
longi-
These
limits
These and
in I/,
of full
V, is chosen
longitudinal
energy,
a is a
initial
scenarios
velocity
hypersurface.
2’0 and
maximum
“Bjorken-like”
hydrodynamic
Tdec of the freezePout
temperature
rapidity;
1): ranges
transverse
by the experimental
of charged
corresponding
y is the initial
widt,h
ZL and
parameters the slopes
and
to be zero. the
are conand widths
spectra. S+S (direct
S+S
n-)
5ooI/:/J ---
oLy61.0
-----
1.osys2.0
450
s
-.
2.osyi3.0
2 _^400 ?
350
300
250
Figure
1.
3.
10 20 y (mean
hlean
of direct S+S
00
pions
3.0 4.0 5.0 rapidity bin)
trallsversc
60
momentum
for different
rapidity
Figure
(11~)
collisions.
In general.
hadronic
Iii
longitudinal transverse
Fig.
2, t,lir
(set
are more
Ref.
The
distribution
at 11~-
IJ~
flow is diffrrent
central
pion
sensitilre
[3] f 01 Inore
and transverse
direct,
are shown. in the
spect,ra
EOSs
we show < ~1~> for each yPhin
Evidently.
flows.
The
?T?. the mass
extent
in different,
int,erval
conditions
A significant
~CI’SIIS the mean
cuts
finding
distribufor a bag
of this coupling
than
is shown
1,
pions.
interva.ls. different
rapidity
, 2 < y < 3, exhihib
a clear
shoulder
of t,he particle.
are
in Fig.
rapic1it.y of the bin for direct
rapidity
integratrcl
they
is the coupling
over
pt clistril)ntions
rapidit,y
to the initial
details).
It. is well known
[5] that
intervals in the pt
such a structure
spectrum
Tdec = int.0
momentum rapidity
results
to the different, where
with different
EOS.
Qualitative
between
2. Transverse
tions
I)ins in
“two
1111nqml“
structure
finally
relaxes
MeV,
a gaussian
the
Tdec =
evolves R’IeV.
R. Venugopalan et al. I Coupling of longitudinal and transverse
At early
times,
increa.sing
most
time,
cent,ral
rapidities
steady
state
of the energy-moment,um
tra.nsverse to conserve
when
Figure with
3.
1.0
energy
flow is in the longitudinal significant,
thereby
and transverse
directions
2.0
3.0 y (rapidity)
result,s
5.0
40
of NA35
S+S
dat,a
Figure
for a bag
EOS
with
with calculat.ed
4. C!omparison
with E; =
from
resonance
from
c
and kaons:
decays;
da&cd the
ramifications
We illustra.te
the esperimrntal
flows
conditions 200
if we choose
2.27
GeV/fm3,
yr. =
resonance K-
and
1.3,
decays; kaons;
~1, =
Dot-dash dashed
solid
1.X
fm
curve:
K-
curve:
d-
the
net
curve:
negatives.
Experimental
frame.
= 250 MeV;
rect
net negatives.
Furthermore,
B”.’
data
To = 182.2 RfcV,
7r-
c.m.
a
for a hag EOS
direct
solid curve:
of NA3.5 S+S
results
curve:
For To -
reaches
6.0
curve:
initial
toward
are comparable.
and Tdec = 140 hleV.
transverse
With
particles
At lower Tdec, the system
and momentum.
B’/’ = 250 MeV (T,=lSl.SS hIcV): To = 200 hleV, E% = 3.36 GeV/fm3, ye = 0.0, ZL = 1.12 fm and Tdec = l-40 XleV. Dot-dash
4.
direction.
shifting
s (200 GGV/lI)
Comparison
calculated
hecomes
flows in longitudinal 5 +
“0.0
flow
475c
flows
for
and
large
initial
at
we find
ZL x
rlAJ/rly spectra
the slopes
ramifications
collisions
(!y~=O.o),
Me\’ the
S+S
1 fm are
of 11~spectra temperatures,
200
of the coupling GeV/nucleon.
a. significant 2 -
dependence
3 times
t,oo narrow are larger ‘To E ‘SO
the
Lorentz
in comparison than
between
longitudinal
If we assume
and
“Landau-like”
of &N/&J on ZL, and contracted with
the experimental
the
radius data.
results.
To.
in the
(Fig.
3).
However,
R. Venugopalan et al. 1 Coupling of longitudinal and transverse flows
4162
3
I
01
&
5.
Nh:jr,
s+s data for 03
1
01 0.0
00
’
__
dir.(n-+K-)+res.Z
------
dir.(n-+K-)
10 PT
L ?/ < 2.0
Figure compared wit.11 calculatetl results for a hag EOS. Initial conditions as ill Fig. 4.
’
1
05
I
’ 15
1
:\ 20
(cev/c)
Figure 6. NA35 S+S data for 2.0 5 y 6 3.0 compared with calculated rrsult,s for a hag EOS. Initial conditions as in Fig. 4.
intervals OS < ~1< 2 and 2 < I/ < 3, SIIOWI~ in Fig. 5 and Fig. 6, respectively. In the latter case, there is a clear csccss at 10~ pt. It is nnlikely that bett,er agrcrmcnt for :! < y < 3 can he ohtainctl 1,~ a fiiic tuning of the paramctcrs. For finite bar?-on chemical pot,entials. the contributions of baryonic resonance tlecays to tile spectrum of chargvtl negatives ma)- lx significa,nt [6]. t,hereby explaining the 1~ excess. This possibility is u~ltler investigation in 011r model. IIo~ever, microscopic considerations sl~o\v that, bar!.ons arr unlikely to have t IIP same flow as pions [7]. Therefore, issues related to the rquililxation and tliffereirt ial decoupling of thr reaction process merit further study.
of tile various
species
at the dilut,e st,agcs
5. Acknowledgements WC thank Svigfrirtl \\;enig of the X:13.5 collaboration NA:J5 data 011 S+S collisions.
for supplying
IIS
\vith the CERN
REFERENCES 1. :‘. 3. 4.
5. fi. T.
Venngopalan ant1 iU. Prakasll, n’ucl. Phy.5. A546 (1992) 715. Zalcsal;. .I. C’o~p. f’hy.q. 31 (1979) 335: Al. Iiat,aja, Z. Phys. C38 (1958) 419. Venugopalan, PII. D. thesis, SUNY at Stony Brook, Augnst 1992. Vemlgopalan, hl. Prakash. M. Kataja and V. Ruuskanen, in Ploccrdiny.s of VZZth Il’itltcr TI~or4d~op orr Muclcnr~ D~/nnrnic.s. editors; W. Bauer and J. Kapusta., Worltl Scientific, Singapor~~ I991 I’. 3. Sicmcns and J. 0. Rasnlusscn. Phyi;. Hev. Mt. 42 (1979) SSO. J. Bols, I.-. Ornil< ant1 R. Wirier. 1’hu.s. Rev. C46 (1992) 2037; E. Schnrtlermann, .J. Sollfrauk and I’. IIc-inz. Rrgensh~lrg preprint TPR-93.lfj. Sladappa Prakash. nIall,ju Praliasll, R. Vrnugopalan and G. Welke, Phyq. Rca. Lrtt. 70 (1999) 12%: P/IV.?. Kfp. 227 (1993) 3’1. R. S. R. R.