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Signals of longitudinal hydrodynamic expansion in ultra-relativistic nucleus-nucleus collisions
Volume 147B, number 6
PHYSICS LETTERS
15 November 1984
SIGNALS OF LONGITUDINAL HYDRODYNAMIC EXPANSION IN ULTRA-RELATIVISTIC NUCLEUS-NUCLEUS COLLISIONS P.V. RUUSKANEN 1
Nuclear Science Division, LBL, University of California, CA, USA and Department of Physics, University of Jyviiskyl~, JyvKskylK,Finland Received 25 May 1984 Revised manuscript received 21 August 1984
The effects of hydrodynamics in the uRra-relativistic collisions of heavy nuclei are studied by using the resuRs of numerical integration of hydrodynamic equations for longitudinal expansion. It is found that hydrodynamics may profoundly change the shape of the rapidity distr~ution of energy, dE/d0. In order to establish the existence of the hydrodynamic stage, both the rapidity distributions of the net baryon number, dnB/d0, and energy, dE/d0, should be studied simultaneously as functions of the size of the colliding nuclei.
1. lntroductiorL Recent studies [1,2] of the energy densities attainable in ultra-relativeistic (Ecm >~ 15 GeV/ nucleon) nucleus-nucleus collisions indicate values, which may be high enough for a phase transition from hadron gas to quark-gluon plasma ,1 to occur. The results o f Busza and Goldhaber [4] on the stopping of the beam nucleon in nucleus-nucleus collisions seem to show that nuclei are less transparent than what was assumed on the basis of earlier data [5]. This could lead to a considerable increase of the estimated energy densities in the central region (CR). If the energy densities are large in the final state one would expect that interactions among the densely produced particles can have a profound effect on the measured distributions which may not reflect the properties of production mechanisms directly [6]. The simplest way to describe such interactions is the use of relativistic hydrodynamics [7]. In this note I show how the longitudinal expansion affects the rapidity distributions dnB[dO and dE/d0, where dnB is the net baryon number and dE the total energy in the rapidity interval dO. This is done by comparing the input and output distributions in the numerical integration of one-dimensional hydrodynamic equations. 1 Permanent address: Department of Physics, University of Jyv/iskyl/i, Seminaarinkatu 15, SF-40100 Jyvgskylg, Finland. ,1 For a general review see ref. [3]. 0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
The numerical integration is carried out as explained in ref. [ 1] but instead of using nucleon-nucleon collision data as input for the source terms, rapidity distributions which are motivated by multichain models [8] have been used. These distributions are shown as smooth curves in figs. 2 and 3. The numerical calculation applies on the collision axis of a head-on (small inpact parameter) collision of two uranium nuclei at Eem = 50 GeV/nucleon.
2. Decoupling. The output of the numerical integration consists of energy and net baryon number densities, e(x, t) and nB(x, t), both defined in the local comoving frame of the hadronic fluid, and the flow velocity o(x, t). The baryon current is ]~=nBu u,
u u=(7,70),
(1)
and the e n e r g y - m o m e n t u m tensor
TUU = _pgUV + (e + p) uUu v ,
(2)
where p is the pressure. The equation of state of ideal relativistic gas, p = el3, is used throughout in the calculations. In order to calculate the rapidity distributions of the emerging particles the decoupling from the hydrodynamic stage, when the densities e(x, t) and nB(x, t) become small enough, must be specified. The following 465
Volume 147B, number 6
PHYSICS LETTERS
simple scenario has been used here. First the hydrodynamic calculation is carried on to arbitrarily low energy and baryon number densities. These results are then used to determine a space-time boundary, t = td(s) and x = Xd(S), on which the decoupling is assumed to take place. The rapidity distributions for the hadronic fluid are then calculated as the baryon number and energy flux through this boundary. This implies the assumption that there is no appreciable change in the flow velocity at the decoupling even though there may be considerable production of entropy [9]. By folding the thermal motion into these fluid quantities the ffmal particle distributions are obtained. If the decoupling boundary is thought of as a onedimensional surface with surface element do u = (dXd, dtd),
(3)
then the baryon number and energy flows through this surface element are given by dn B = da~/~
(4)
15 November 1984 CONTOUR CURVES OF E(y,i) _
,
I
,
3"01
I
.2
,
~ / i
I
J
~;'' i/ '
',///
//,~,.1~
,'
2.o-
~ . / ~ j
~.,,/ , l /
"t
i
O] 0
. 1.0
/ / /~1.~/ /
I -a- ~" I
, i [ t 2.0
/
~
U
"::'.
!
"
','
I
':' 3.0
4.0
Fig. 1. Some of the streamlines (dashed-dotted lines) and curves of constant fluid rapidity Of (dashed lines) are shown together with the contours of energy density. The part of the contour e = 0, 2 where matter is flowing out together with the straight line extending to the end of the source region is used as the de,coupling boundary (heavyline).
and (5)
dE = d o u T uO .
When the equations of the decoupling boundary are expressed in terms of the fluid rapidity Of = { l n { [ 1 +
v(x, t ) ] / [ 1
-
v(x, t ) ] } ,
(6)
eN = mN(1 + 3 T / 2 m N ) n B .
the surface element becomes do u = (dxd/d0t, dtd/dOf) d0f = o u dOf,
(7)
and the rapidity distributions for the hadronic fluid can be written as dnB/d0 f = o u j ~ ,
dE/d0f = ouT u0 .
(8,9)
For numerical calculations the decoupling boundary is taken to be the e(x, t) = eo = 0.2 GeV/fm 3 contour for the outflowing matter. This is shown in fig. 1 together with some of the streamlines. Since the energy density for highest values of rapidity always remains below e0, the boundary must be supplemented there with a line which extends up to the maximum value of the rapidity. A natural choice is the end boundary of the hydrodynamic source region because this allows us to check the conservation of the baryon number and the total energy in the numerical integration. The results are not very sensitive to this choice since both e and n B rapidly decrease outside the e = e0 line in this region.
466
Thermal motion is folded into the fluid motion by assuming that the emerging hadrons are nortinteracting pions and nucleons. The energy density of the nucleons is obtained through the approximate relation (10)
Since the final results are not very sensitive to the details of the thermal distributions, Boltzmann distribution has been used both for the pions and the nucleons. The temperature is fixed to be equal to the pion mass everywhere along the boundary. 3. R e s u l t s and discussion. The final distributions are shown in fig. 2 for the net baryon number and in fig. 3 for the energy. The latter has been divided by cosh 0 for easier presentation. Also the energy associated with the net baryon number has been separated and the remaining part which is presumably dominated by pions is shown by the dashed line. This part contains the baryon-antibaryon pairs, too, even though their ratio to baryons might be less than in the nucleon-nucleon collisions if the hydrodynamic stage exists. All distributions are normalized to one ingoing nucleon-nucleon pair. The smooth curves are the distributions at the production before the hydrodynamic stage.
Volume 147B, number 6
PHYSICS LETTERS
NORMALIZATION IS TO A COLLIDING [ NUCLEON PAIR WHICH MEANS THAT ~ J
0,5
[
7,de ~ .1
U
J
de
J
1
,
2 3 8=RAPIDITY
i
i
4
5
Fig. 2. The smooth curve is the baryon number distribution
at the production (input to the hydrodynamic calculation) and the histogram at the decoupling. The rapidity has been divided into bins of A0 = 0,2 for the numerical calculation at the decoupling because Of is roughly constant at the end part of the boundary (see fig. 1) and thus the derivatives in eqs. (7)-(9) can have a singular behaviour. The normalization is to one ingoing nucleon pair meaning that the integral of the input curve equals one. The results show that the baryon number distribution is essentially the same at the production and the decoupling but a large part of the energy is lost in the expansion from the CR into the FR's. This striking difference comes about because the baryon number can be transferred only through convection but the energy is transferred also as the work which is done by the • solution • [ 10], e cc r -1-Cs2, obpressure. In the scahng tained for the simple equation of state, p = e/3, the effect of the work is the faclor r-Cs2 and should be kept
3-
>
2-
NORMALIZATION IS T0 A C O L LII]ING NUCLEON PAIR W H L C H MEANS THAT
PRODUCTION D{STRtBUTIQN
TO~AL DISTRISU'[ION
NON 8ARYONIC PART
;
"
-
~
l
J
J
i
,
1
2 3 8- RAPIDITY
4
5
Fig. 3. The smooth curve is the energy distribution at the production and the histogram at the decoupling. The normalization is to one ingoing nucleon pair. Thus the integral of
dE]dO equals 50 GeV. The parametrization
in this calculation is such that nucleons loose 2/3 of their energy in the primary collision.
15 November 1984
in mind when estimating the initial energy densities from the measured distributions as pointed out by Gyulassy and Matsui [ 11 ]. The fact that the baryon distribution is slower at the decoupling than at the production is somewhat fortuitous and depends on the details of the treatment of the collision region. In our treatment [ 1 ] the production of final state quanta takes place after a proper time T0 ~ 1 fm/c and they are subsequently thermalized with the previously produced matter. Since the produced quanta can essentially meet only slower matter, they will be slowed down in the thermalization. The amount of slowdown depends on the size of the collision region, but at least within the inside-outside cascade picture the baryon distributions at the production and the decoupling are qualitatively the same. Some more insight can be gained into the properties of the baryon and the energy distributions by noticing that - when expressed in terms of the proper time, r = (t 2 - x2) 1/2 - their expressions, eqs. (8), (9) are proportional to rd, the decoupling time [18]. In the CR the numerical solutions after the formation of matter behave approximately like the scaling solutions [10] ( f o r p = c2e): nB~r -I,
e ~ "-l-c2s.
(11)
As a consequence the net baryon number distribution is essentially independent of the decoup21ing time but the energy distribution decreases as r-Cs. Thus the A dependence of the net baryon amount in CR is a good measure of the A dependence of the production properties, whereas the total energy in CR may be reduced by the hydrodynamic effects and show unexpectedly slow growths with A in light of the energy loss of the nucleons. The hydrodynamic expansion may show up directly as a qualitative change in the A dependence of dE/d0 in the CR [11]. In the collision of light nuclei hydrodynamics is probably insignificant - at least in the energy range considered here - and the A dependence would be that of the production process. With growing A the hydrodynamics becomes increasingly important and slows down the growth of dE/d0 with A. The onset of the hydrodynamics should also be seen in the change of the shape of the energy distribution. For light nuclei we would expect dE/d0 to become steeper in rapidity with growingA. However, when A is large enough this trend may stop and even 467
Volume 147B, number 6
PHYSICS LETTERS
get reversed towards the shape of dE/d0 in the nucleon-nucleon collision. In the above considerations the effects of the viscosity have been neglected. Unfortunately the transport coefficients are not well known [ 13] around the phase transition temperatures which is the relevant region from the point of view of the hydrodynamic expansion. According to a recent estimate [13] the increase due to viscosity of the rapidity density of observed particles might be small. However, these estimates are valid in the high and low temperature limits of the transport coefficients and in the phase transition region the values may differ from these estimates. If the deconfinement phase transition is of the first order it has also another, more direct effect on the expansion because of the reduction of the pressure with respect to the energy density at the transition. This somewhat reduces the energy loss from the CR. The first order phase transition may also lead to instabilities in the expansion but these phenomena would most probably have their own signals [14] through the formation of shock fronts. Another feature which should be included in a realistic decoupling scheme is the flow in transverse direction. Its effect is to shorten the decoupling time and to transfer part of the internal energy into the transverse fluid motion. Thus the onset of the hydrodynamics should be seen as a simultaneous flattening of both the longitudinal and the transverse distribution of the final hadrons. To conclude, the hydrodynamics may have a profound effect on the energy distribution in the central A + A collisions. To disentangle the effects of hydrodynamics from the properties of production of f'mal state matter, the net baryon number distribution and the energy distribution, especially theirA dependence, should be studied simultaneously. Also the dependence of the energy density on the transverse momentum and rapidity should show correlations due to hydrodynamic expansion as A is increased. Many valuable discussions with K. Kajantie, T. Matsui, H. Sumiyoshi and especially with M. Gyulassy
468
15 November 1984
who contributed much to the discussion on theA dependence of the signals are gratefully acknowledged. I would also like to thank the members of Nuclear Science Division of Lawrence Berkeley Laboratory for their warm hospitality. This work was supported by the Academy of Finland and the Director, Office of Energy Research, Division of Nuclear Physics of the Office of High Energy and Nuclear Physics of the U.S. Department of Energy.
References [1] K. Kajantie, R. Raitio and P.V. Ruuskanen, Nucl. Phy~ B222 (1983) 152. [2] H. Sumiyoshi, S. Date, N. Suzuki, O. Miyamura and T. Ochiai, LBL-preprint 16663. [3] M. Jacob and H. Satz, eds., Quark matter formation and heavy ion collisions (World Scientific, Singapore, 1982); Proc. Third Intern. Conf. on Ultra-relativistic nucleusnucleus collisions (Brookhaven National Laboratory, September 1983). [4] W. Busza and A.S. Goldhaber, in: Proc. Third Intern. Conf. on Ultra-relativistic nucleus-nucleus collisions (Brookhaven National Laboratory, September 1983). [5] W. Busza et al., Phys. Rev. Lett. 34 (1976) 836; J.R. Florian et al., Phys. Rev. B13 (1976) 558. [6] A. Capella et aL, contrib. Heavy ion Study (Lawrence Berkeley Laboratory, Berkeley, 1983); H. Sumiyoshi et al., contrib. Heavy ion Study (Lawrence Berkeley Laboratory, Berkeley, 1983). [7] J.D. Bjorken, Phys~ Rev. D27 (1983) 140; IC Kajantie and L. McLerran, Phys. Lett 119B (1982) 203; Nucl. Phys. B214 (1983) 261. [8] A. Capella and A. Krzywicki, Phys. LetL 67B (1977) 84; Phys. Rev. D18 (1978) 3357; K. Kinoshita, A. Minaka and H. Sumiyoshi, Prog. Theor. Phys. 61 (1979) 165; 63 (1980) 1268. [9] G. Baym, Proe. 6th High energy heavy ions Study and 2nd Workshop on anomalies (Lawrence Berkeley Laboratory, Berkeley, June 1983). [10] J.D. Bjorken, Phys. Rev. D27 (1983) 140. [11] M. Gyulassy and T. Matsui, LBL preprint 15947 (1983). [12] F. Cooper, G. Frye and E. Schonberg, Phys~ Rev. Dl12 (1975) 192. [13] A. Hosoya and K. Kajantie, Research Institute for Theoretical Physics (University of Helsinki) reprint HU-TFT83-63. [14] L. Van Hove, Z. Phyg 21 (1983) 93; M. Gyulassy, K. Kajantie, H. Kurki-Suonio and L. McLerran, Nucl. Phys. B 237 (1984) 477.
PHYSICS LETTERS
15 November 1984
SIGNALS OF LONGITUDINAL HYDRODYNAMIC EXPANSION IN ULTRA-RELATIVISTIC NUCLEUS-NUCLEUS COLLISIONS P.V. RUUSKANEN 1
Nuclear Science Division, LBL, University of California, CA, USA and Department of Physics, University of Jyviiskyl~, JyvKskylK,Finland Received 25 May 1984 Revised manuscript received 21 August 1984
The effects of hydrodynamics in the uRra-relativistic collisions of heavy nuclei are studied by using the resuRs of numerical integration of hydrodynamic equations for longitudinal expansion. It is found that hydrodynamics may profoundly change the shape of the rapidity distr~ution of energy, dE/d0. In order to establish the existence of the hydrodynamic stage, both the rapidity distributions of the net baryon number, dnB/d0, and energy, dE/d0, should be studied simultaneously as functions of the size of the colliding nuclei.
1. lntroductiorL Recent studies [1,2] of the energy densities attainable in ultra-relativeistic (Ecm >~ 15 GeV/ nucleon) nucleus-nucleus collisions indicate values, which may be high enough for a phase transition from hadron gas to quark-gluon plasma ,1 to occur. The results o f Busza and Goldhaber [4] on the stopping of the beam nucleon in nucleus-nucleus collisions seem to show that nuclei are less transparent than what was assumed on the basis of earlier data [5]. This could lead to a considerable increase of the estimated energy densities in the central region (CR). If the energy densities are large in the final state one would expect that interactions among the densely produced particles can have a profound effect on the measured distributions which may not reflect the properties of production mechanisms directly [6]. The simplest way to describe such interactions is the use of relativistic hydrodynamics [7]. In this note I show how the longitudinal expansion affects the rapidity distributions dnB[dO and dE/d0, where dnB is the net baryon number and dE the total energy in the rapidity interval dO. This is done by comparing the input and output distributions in the numerical integration of one-dimensional hydrodynamic equations. 1 Permanent address: Department of Physics, University of Jyv/iskyl/i, Seminaarinkatu 15, SF-40100 Jyvgskylg, Finland. ,1 For a general review see ref. [3]. 0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
The numerical integration is carried out as explained in ref. [ 1] but instead of using nucleon-nucleon collision data as input for the source terms, rapidity distributions which are motivated by multichain models [8] have been used. These distributions are shown as smooth curves in figs. 2 and 3. The numerical calculation applies on the collision axis of a head-on (small inpact parameter) collision of two uranium nuclei at Eem = 50 GeV/nucleon.
2. Decoupling. The output of the numerical integration consists of energy and net baryon number densities, e(x, t) and nB(x, t), both defined in the local comoving frame of the hadronic fluid, and the flow velocity o(x, t). The baryon current is ]~=nBu u,
u u=(7,70),
(1)
and the e n e r g y - m o m e n t u m tensor
TUU = _pgUV + (e + p) uUu v ,
(2)
where p is the pressure. The equation of state of ideal relativistic gas, p = el3, is used throughout in the calculations. In order to calculate the rapidity distributions of the emerging particles the decoupling from the hydrodynamic stage, when the densities e(x, t) and nB(x, t) become small enough, must be specified. The following 465
Volume 147B, number 6
PHYSICS LETTERS
simple scenario has been used here. First the hydrodynamic calculation is carried on to arbitrarily low energy and baryon number densities. These results are then used to determine a space-time boundary, t = td(s) and x = Xd(S), on which the decoupling is assumed to take place. The rapidity distributions for the hadronic fluid are then calculated as the baryon number and energy flux through this boundary. This implies the assumption that there is no appreciable change in the flow velocity at the decoupling even though there may be considerable production of entropy [9]. By folding the thermal motion into these fluid quantities the ffmal particle distributions are obtained. If the decoupling boundary is thought of as a onedimensional surface with surface element do u = (dXd, dtd),
(3)
then the baryon number and energy flows through this surface element are given by dn B = da~/~
(4)
15 November 1984 CONTOUR CURVES OF E(y,i) _
,
I
,
3"01
I
.2
,
~ / i
I
J
~;'' i/ '
',///
//,~,.1~
,'
2.o-
~ . / ~ j
~.,,/ , l /
"t
i
O] 0
. 1.0
/ / /~1.~/ /
I -a- ~" I
, i [ t 2.0
/
~
U
"::'.
!
"
','
I
':' 3.0
4.0
Fig. 1. Some of the streamlines (dashed-dotted lines) and curves of constant fluid rapidity Of (dashed lines) are shown together with the contours of energy density. The part of the contour e = 0, 2 where matter is flowing out together with the straight line extending to the end of the source region is used as the de,coupling boundary (heavyline).
and (5)
dE = d o u T uO .
When the equations of the decoupling boundary are expressed in terms of the fluid rapidity Of = { l n { [ 1 +
v(x, t ) ] / [ 1
-
v(x, t ) ] } ,
(6)
eN = mN(1 + 3 T / 2 m N ) n B .
the surface element becomes do u = (dxd/d0t, dtd/dOf) d0f = o u dOf,
(7)
and the rapidity distributions for the hadronic fluid can be written as dnB/d0 f = o u j ~ ,
dE/d0f = ouT u0 .
(8,9)
For numerical calculations the decoupling boundary is taken to be the e(x, t) = eo = 0.2 GeV/fm 3 contour for the outflowing matter. This is shown in fig. 1 together with some of the streamlines. Since the energy density for highest values of rapidity always remains below e0, the boundary must be supplemented there with a line which extends up to the maximum value of the rapidity. A natural choice is the end boundary of the hydrodynamic source region because this allows us to check the conservation of the baryon number and the total energy in the numerical integration. The results are not very sensitive to this choice since both e and n B rapidly decrease outside the e = e0 line in this region.
466
Thermal motion is folded into the fluid motion by assuming that the emerging hadrons are nortinteracting pions and nucleons. The energy density of the nucleons is obtained through the approximate relation (10)
Since the final results are not very sensitive to the details of the thermal distributions, Boltzmann distribution has been used both for the pions and the nucleons. The temperature is fixed to be equal to the pion mass everywhere along the boundary. 3. R e s u l t s and discussion. The final distributions are shown in fig. 2 for the net baryon number and in fig. 3 for the energy. The latter has been divided by cosh 0 for easier presentation. Also the energy associated with the net baryon number has been separated and the remaining part which is presumably dominated by pions is shown by the dashed line. This part contains the baryon-antibaryon pairs, too, even though their ratio to baryons might be less than in the nucleon-nucleon collisions if the hydrodynamic stage exists. All distributions are normalized to one ingoing nucleon-nucleon pair. The smooth curves are the distributions at the production before the hydrodynamic stage.
Volume 147B, number 6
PHYSICS LETTERS
NORMALIZATION IS TO A COLLIDING [ NUCLEON PAIR WHICH MEANS THAT ~ J
0,5
[
7,de ~ .1
U
J
de
J
1
,
2 3 8=RAPIDITY
i
i
4
5
Fig. 2. The smooth curve is the baryon number distribution
at the production (input to the hydrodynamic calculation) and the histogram at the decoupling. The rapidity has been divided into bins of A0 = 0,2 for the numerical calculation at the decoupling because Of is roughly constant at the end part of the boundary (see fig. 1) and thus the derivatives in eqs. (7)-(9) can have a singular behaviour. The normalization is to one ingoing nucleon pair meaning that the integral of the input curve equals one. The results show that the baryon number distribution is essentially the same at the production and the decoupling but a large part of the energy is lost in the expansion from the CR into the FR's. This striking difference comes about because the baryon number can be transferred only through convection but the energy is transferred also as the work which is done by the • solution • [ 10], e cc r -1-Cs2, obpressure. In the scahng tained for the simple equation of state, p = e/3, the effect of the work is the faclor r-Cs2 and should be kept
3-
>
2-
NORMALIZATION IS T0 A C O L LII]ING NUCLEON PAIR W H L C H MEANS THAT
PRODUCTION D{STRtBUTIQN
TO~AL DISTRISU'[ION
NON 8ARYONIC PART
;
"
-
~
l
J
J
i
,
1
2 3 8- RAPIDITY
4
5
Fig. 3. The smooth curve is the energy distribution at the production and the histogram at the decoupling. The normalization is to one ingoing nucleon pair. Thus the integral of
dE]dO equals 50 GeV. The parametrization
in this calculation is such that nucleons loose 2/3 of their energy in the primary collision.
15 November 1984
in mind when estimating the initial energy densities from the measured distributions as pointed out by Gyulassy and Matsui [ 11 ]. The fact that the baryon distribution is slower at the decoupling than at the production is somewhat fortuitous and depends on the details of the treatment of the collision region. In our treatment [ 1 ] the production of final state quanta takes place after a proper time T0 ~ 1 fm/c and they are subsequently thermalized with the previously produced matter. Since the produced quanta can essentially meet only slower matter, they will be slowed down in the thermalization. The amount of slowdown depends on the size of the collision region, but at least within the inside-outside cascade picture the baryon distributions at the production and the decoupling are qualitatively the same. Some more insight can be gained into the properties of the baryon and the energy distributions by noticing that - when expressed in terms of the proper time, r = (t 2 - x2) 1/2 - their expressions, eqs. (8), (9) are proportional to rd, the decoupling time [18]. In the CR the numerical solutions after the formation of matter behave approximately like the scaling solutions [10] ( f o r p = c2e): nB~r -I,
e ~ "-l-c2s.
(11)
As a consequence the net baryon number distribution is essentially independent of the decoup21ing time but the energy distribution decreases as r-Cs. Thus the A dependence of the net baryon amount in CR is a good measure of the A dependence of the production properties, whereas the total energy in CR may be reduced by the hydrodynamic effects and show unexpectedly slow growths with A in light of the energy loss of the nucleons. The hydrodynamic expansion may show up directly as a qualitative change in the A dependence of dE/d0 in the CR [11]. In the collision of light nuclei hydrodynamics is probably insignificant - at least in the energy range considered here - and the A dependence would be that of the production process. With growing A the hydrodynamics becomes increasingly important and slows down the growth of dE/d0 with A. The onset of the hydrodynamics should also be seen in the change of the shape of the energy distribution. For light nuclei we would expect dE/d0 to become steeper in rapidity with growingA. However, when A is large enough this trend may stop and even 467
Volume 147B, number 6
PHYSICS LETTERS
get reversed towards the shape of dE/d0 in the nucleon-nucleon collision. In the above considerations the effects of the viscosity have been neglected. Unfortunately the transport coefficients are not well known [ 13] around the phase transition temperatures which is the relevant region from the point of view of the hydrodynamic expansion. According to a recent estimate [13] the increase due to viscosity of the rapidity density of observed particles might be small. However, these estimates are valid in the high and low temperature limits of the transport coefficients and in the phase transition region the values may differ from these estimates. If the deconfinement phase transition is of the first order it has also another, more direct effect on the expansion because of the reduction of the pressure with respect to the energy density at the transition. This somewhat reduces the energy loss from the CR. The first order phase transition may also lead to instabilities in the expansion but these phenomena would most probably have their own signals [14] through the formation of shock fronts. Another feature which should be included in a realistic decoupling scheme is the flow in transverse direction. Its effect is to shorten the decoupling time and to transfer part of the internal energy into the transverse fluid motion. Thus the onset of the hydrodynamics should be seen as a simultaneous flattening of both the longitudinal and the transverse distribution of the final hadrons. To conclude, the hydrodynamics may have a profound effect on the energy distribution in the central A + A collisions. To disentangle the effects of hydrodynamics from the properties of production of f'mal state matter, the net baryon number distribution and the energy distribution, especially theirA dependence, should be studied simultaneously. Also the dependence of the energy density on the transverse momentum and rapidity should show correlations due to hydrodynamic expansion as A is increased. Many valuable discussions with K. Kajantie, T. Matsui, H. Sumiyoshi and especially with M. Gyulassy
468
15 November 1984
who contributed much to the discussion on theA dependence of the signals are gratefully acknowledged. I would also like to thank the members of Nuclear Science Division of Lawrence Berkeley Laboratory for their warm hospitality. This work was supported by the Academy of Finland and the Director, Office of Energy Research, Division of Nuclear Physics of the Office of High Energy and Nuclear Physics of the U.S. Department of Energy.
References [1] K. Kajantie, R. Raitio and P.V. Ruuskanen, Nucl. Phy~ B222 (1983) 152. [2] H. Sumiyoshi, S. Date, N. Suzuki, O. Miyamura and T. Ochiai, LBL-preprint 16663. [3] M. Jacob and H. Satz, eds., Quark matter formation and heavy ion collisions (World Scientific, Singapore, 1982); Proc. Third Intern. Conf. on Ultra-relativistic nucleusnucleus collisions (Brookhaven National Laboratory, September 1983). [4] W. Busza and A.S. Goldhaber, in: Proc. Third Intern. Conf. on Ultra-relativistic nucleus-nucleus collisions (Brookhaven National Laboratory, September 1983). [5] W. Busza et al., Phys. Rev. Lett. 34 (1976) 836; J.R. Florian et al., Phys. Rev. B13 (1976) 558. [6] A. Capella et aL, contrib. Heavy ion Study (Lawrence Berkeley Laboratory, Berkeley, 1983); H. Sumiyoshi et al., contrib. Heavy ion Study (Lawrence Berkeley Laboratory, Berkeley, 1983). [7] J.D. Bjorken, Phys~ Rev. D27 (1983) 140; IC Kajantie and L. McLerran, Phys. Lett 119B (1982) 203; Nucl. Phys. B214 (1983) 261. [8] A. Capella and A. Krzywicki, Phys. LetL 67B (1977) 84; Phys. Rev. D18 (1978) 3357; K. Kinoshita, A. Minaka and H. Sumiyoshi, Prog. Theor. Phys. 61 (1979) 165; 63 (1980) 1268. [9] G. Baym, Proe. 6th High energy heavy ions Study and 2nd Workshop on anomalies (Lawrence Berkeley Laboratory, Berkeley, June 1983). [10] J.D. Bjorken, Phys. Rev. D27 (1983) 140. [11] M. Gyulassy and T. Matsui, LBL preprint 15947 (1983). [12] F. Cooper, G. Frye and E. Schonberg, Phys~ Rev. Dl12 (1975) 192. [13] A. Hosoya and K. Kajantie, Research Institute for Theoretical Physics (University of Helsinki) reprint HU-TFT83-63. [14] L. Van Hove, Z. Phyg 21 (1983) 93; M. Gyulassy, K. Kajantie, H. Kurki-Suonio and L. McLerran, Nucl. Phys. B 237 (1984) 477.