- Email: [email protected]
Dynamics of parton cascades in highly relativistic nuclear collisions
Nuclear Physics A544 (1992) 467c-470c North-Holland, Amsterdam
Y amics of Part Klaus Geiger
d
sca es i
and Berndt Müller
YS elativistic Nude
b
a
School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455 b Physics Department, Duke University, Durham, NC 27706 We present a QCD based relativistic kinetic model for high energy nuclear collisions, which is inspired by the parton picture of hadronic interactions. The nuclear dynamics is traced back to the microscopic level of quark and gluon interactions in the framework of perturbative QCD . The time evolution of the nuclear system is described in real time by a relativistic transport equation for the parton distributions that is solved by Monte Carlo methods. UCTI®N The properties of a quark-gluon plasma, expected to be achieved in ultrarelativistic nuclear collisions (Een1 > 50 GeV/A), and various possible signals for its detection as well as its final dissolution into individual hadrons have been extensively studied [1] . On the other hand, the processes leading up to the formation of a thermally equilibrated QCD plasma have received only rather scant attcat :on. It is for this reason that we have begun to develop a comprehensive description for the dissipative processes occurring in the pre-equilibrium phase of a nuclear collision at very high energy [2] . We emphasize that our approach is not designed to yield a complete description of the nuclear reaction up to the prediction of details of hadronic. spectra, at least in its present state of development . Rather we wish to answer the question whether, and how, a local thermodyaamïc equilibrium will be established in the central rapidity region . Since there is little experimental information to guide the theoretical analysis, our strategy has been to use as few modelling assumptions as possible, and to base our analysis almost exclusively on the firmly established framework of perturbative QCD . Wherever we reach the limits of its validity, we pa.rametrize our lack of better knowledge by cut-off parameters, rather than trying to model nonperturbative QCD . Although models similar in spirit to ours have been studied earlier [3], we feel that this is the first attempt to apply- the full knowledge of perturbative QCD to the description of the early stage of relativistic nuclear reactions . 2. T
QCI) TRANSPORT EQUATIONS
We start from a relativistic transport equation for the phase space density of partons which we can solve by means of Monte Carlo techniques, i.e. by simulating many collision events and taking their average . The basic idea is to embody the perturbative parton cascade picture into a relativistic version of the semiclassical kinetic theory. We represent the partons as classical point particles of various types as specified by their internal degrees of freedom . The state of a parton is characterized by its flavor a = of , qf, g (quark, antiquark, or gluon), its momentum P and position r Its energy 0375-9474/92/$05 .00 © 1992 - Elsevier Science Publishers E.V. A.11 rights reserved.
468C
K. Geiger, B.
filler /
ynainics of parion cascades
is determined by Em = p2 + 172 a + 11,12 where m,,, is the rest mass and M a possible space- or time-like virtual mass. The spin and color degrees of freedom are taken .-, amd in the composition of irao account only implicitly in the interaction amplitude partons as hadrons. 'kVe define a Lorentz invariant single particle distribution function (j)', F, t) dpd3 r. for the bine-dependent phase space density of partons of species a. The transport equation for this single particle distribution function has the manifestly covariant Orm. P po'
=
processes
AC« ( Pl - it) = .
processes
with Lorentz invariant collision integrals 4, = E,,C. . 1"he left-hand side of this Boltzmann equation describes the free propagation of partons, here generally taken to be on mass shell in order to avoid violations of gauge invariance, whereas the collision term on the right-hand side represents a suni over all possible interaction processes involving partons of type a . In order to solvn act. 0 ) Or the time evolution of parton distributions in 6-dimensional phase-space, we need to specify, a) the initial distribution of partons ;n the incoming nuclei, and, b) the form of the collision term . 3.
INITIAL PAN DISTRIBUTIONS
The initial distribution functions F,,(g. represented in the factorized form NAB w,o~a wa
'NI
W
r, t o ) of the partons in the incoming nuclei are -
P, P) - R.N ° (T, -
-
,1Z)
t=to
Here P~V'i and R-'V i give the initial momentum and spatial distributions, respectively, for partons of type a in each individual nucleon Ni, and the sum runs over all NA -1-NB nucleons in nuclei A and B. The vectors #, P, J5 denote the momenta of partons, nucleons and nuclei, respectively, and similarly do the positions i:, .fi and fz. Our ansatz for the initial momentum distribution Pai of parton flavors a in each nucleon Ni is where r = QP is the faction of the nucleons longitudinal momentum carried by 2 + M.2)/p2]1/2 the parton (P = Mill l ,j and r R = [ 3.2 + (PI denotes the energy a f a Vi ( X. Q2) measured fraction . The o are the nucleon structure functions and the distribution g(g_L ) with width ho ~ 0 .4 GeV1c takes into account the transverse momentuin 9-L of the initial partons which results from their confinement within the radius of a nucleon (of the order 0.8 fm). The scale Q20 ,. at which the structure functions .f Ni ( .r1 Q2) MVISt be evaluated, is computed event by event on a statistical basis by the a 0 Q2 -evolution of the structure functions by including (space-like) radiation processes for each such primary parton that encounters a scattering .
K. Geiger, B. Müller l Dynamics ofparson cascades
469e
The spatial distribution RQi of the partons depends on the initial coordinates 1Z of the two incoming nuclei A and B, and the positions of the nucleons R inside the volume of A, respectively B. Experimental information about its specific form is rather incomplete, so we have employed the following phenomenological ansatz: the individual nucleons are assigned positions around the centers of the colliding nuclei A and B according to a Fermi distribution in the rest frame of each nucleus. Next the partons coordinates are distributed in this reference frame around the nucleons centers with an exponential distribution corresponding to the measured nucleons electromagnetic form factor. Finally, the positions of nucleons and their valence quarks are boosted into the c.m. frame, while the sea quarks and gluons are smeared out longitudinally by an amount (Oz) ti h/pll < 2Rnuc around the valence quarks . This procedure is called distributed Lorentz contraction and yields a `fuzzy' pancake shape of the incoming nuclei [4]. 4. T E COLLISION TEEM The collision tern in eq. (1), Fa Ca , in general, describes all possible m -+ n interaction processes where at least one parton of type a is involved. All the information about these processes is contained in invariant matrix elements of the type M,,,-,, = (1',2.. . . . . n 111111, 2, . . . m), which give the amplitudes for processes with m particles in the initial state and n particles in the final state. Since it is impractical to include ail possible (in --> n) processes in the collision term of the transport equation, we restrict ourselves to a subset of processes that give the major contributions in perturbative QCD. In accordance with the intuitive picture of the evolution of a nuclear collision, we consider the important class of asymptotically (in energy) dominant (2 --+ n) processes that can be obtained from elementary (2 --~ 2) scatterings including associated space-like and time-like branchings [5] . For this set of (2 --+ n) processes, we formally write the invariant amplitudes squared as ~2 = 1(caia2 . . .bib2 . . .CIC2 . . .dld2 . . .cdIMIab)12
IM, 2-n
[sa (Ta, Q2 ,Q0) Sb ÉXb,Q2+QÛ)l 2 IMab-cd(S,t,u~ 4G 2 )I ~T~(Q2~p0)'Td(Q2~p0)],
where a,-, bt are the partons radiated from a, b ~)efore the elementary scattering a + b ) c+d, and ci, di are the partons radiates' from the final state partons c, d. Eq. (4) expresses that the total squared amplitude can be factorized into three components, ) namely: the Sudakov form factors S,, ;, determining the pro abilities Pb â = 1 - Sab that the scattering partons a, b have emitted a number of partons before the scattering, a -+ a + c.1 + . . . , b -~ b + b i . . . (space-like branchings), the squared matrix element IMab-cd1 2 for the elementary (2 --+ 2) subprocess a+b - 3 c+d at as scale of interaction Q2 , and the Sudaks;?) form factors T~.d giving the probabilities P~ d' = 1 - T~,d that the scatf?red partons c, d have initiated a sequence oY branchings after the scattering, c -; c + eI + . . . , d --> d + a1 + . . . (time-like branchings) . With the initial parton distribution and the collision term specified, the time-evolution of the distribution functions F(t,f,il according to the transport equation (1), is
IC. Geiger,
70r
,
ller / I~yna~nics of parton cascades
computed numerically by means of Monte Carlo methods. For a detailed presentation of the topics touched here tive refer to our extended manuscript [2]. S
s outlined above, we have simulated and analyzed the evolution of energetic collisions at ttivo energies ~/A = 100 ( 200 ) C~eV/A, respectively, for a large variety of symmetric collision systems from A = 1, . . . , 238 [2]. Fig. 1 shows, as an example of our results, the transverse energy l produced in nuclear collisions, which is a direct measure of how much of the beam energy has been redirected in interactions and characterizes the violence of the collision. It may also serve as an indicator of the energy density achieved during the collision . .El is found to increase approximately with A3 , i.e. it roughly scales with the nuclear radius, which is close to what is observed in nuclear collisions at lower energies. A certain saturation is seen to occur around A = 100. ~~Tost, interesting for the question in how to achieve maximum v1o1e11Ce 111 a c®111S1o11 nlavbe that an increase of the beam energy for a given system is obviously nluCh nlOre e ectlve than an increase in the mass of the beam particles at filed energy . Similarly the thrust T = ~_ ~ pll= I / ~= e â~z I of the partons (or the inelasticity 1 - ?') 1- e ects the degree of nuclear stopping and transverse expansion. tOD c
o _._ . ..
c
9D
ë c
so c
(200 +
=
0.7
so =
z
T®Fa!P.!/ :R
O.B
20D) GeV/A
7D
z
5D : r H c5
o.s
ao ` ' c~.
iD00 + 100) Gev/A
30 -
ä
Zo =
=
o.s
_-'
OS
-'_
04
~' ~ ~- a ~
i
c.7
C' 20
. ... .. . .. . ...
(20D + 200) GeV/A
D2
70 ~
0
.. . .
_0 .3 ~
aC
6C
BO
120 74D mass aumber A 700
16p
180
20C
220
240
~0
~
~
20
40
v 60
80
i
v
100 72C 740 76D mass number A
iYQ. b n
v
780 2DD
22C
240
Fig. 1 : Production of a) transverse energy E1 and b) thrust T as filnction of mass ntllnber .4 at Ec,rt = 100 ( ?00 ) GeV/.~ per collision event for central A ~- A collisions . .F
I~tC
S
1 See e.g . uar~-Ghaon Plasma, edited 1~y R. C. Hwa (World Scientific, Singapore 1990) 2 h . Geiger a.nd H. l~~IÜller, Parton Cascades in Relativistic Nuclear Collisions, preprint I)IJhE-TH-91-15, to appear in Nucl . Phys. )~. 3 oal, Phys. Rev. C33, 2?06 (1966); S. M1~®wczynski and J. Rafelski, Phys. Ret~. C , 107ô (1989) . 4 R . C. wa. and h. hajant,ie, Phys. Rev. Lett. ,SF, 696 (1986) . 5 A . assetto, ~~I . Ciafalcnii, and C~. l~~Iarchesini, Phys. Rep. 1®®, 203 (1983).
Y amics of Part Klaus Geiger
d
sca es i
and Berndt Müller
YS elativistic Nude
b
a
School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455 b Physics Department, Duke University, Durham, NC 27706 We present a QCD based relativistic kinetic model for high energy nuclear collisions, which is inspired by the parton picture of hadronic interactions. The nuclear dynamics is traced back to the microscopic level of quark and gluon interactions in the framework of perturbative QCD . The time evolution of the nuclear system is described in real time by a relativistic transport equation for the parton distributions that is solved by Monte Carlo methods. UCTI®N The properties of a quark-gluon plasma, expected to be achieved in ultrarelativistic nuclear collisions (Een1 > 50 GeV/A), and various possible signals for its detection as well as its final dissolution into individual hadrons have been extensively studied [1] . On the other hand, the processes leading up to the formation of a thermally equilibrated QCD plasma have received only rather scant attcat :on. It is for this reason that we have begun to develop a comprehensive description for the dissipative processes occurring in the pre-equilibrium phase of a nuclear collision at very high energy [2] . We emphasize that our approach is not designed to yield a complete description of the nuclear reaction up to the prediction of details of hadronic. spectra, at least in its present state of development . Rather we wish to answer the question whether, and how, a local thermodyaamïc equilibrium will be established in the central rapidity region . Since there is little experimental information to guide the theoretical analysis, our strategy has been to use as few modelling assumptions as possible, and to base our analysis almost exclusively on the firmly established framework of perturbative QCD . Wherever we reach the limits of its validity, we pa.rametrize our lack of better knowledge by cut-off parameters, rather than trying to model nonperturbative QCD . Although models similar in spirit to ours have been studied earlier [3], we feel that this is the first attempt to apply- the full knowledge of perturbative QCD to the description of the early stage of relativistic nuclear reactions . 2. T
QCI) TRANSPORT EQUATIONS
We start from a relativistic transport equation for the phase space density of partons which we can solve by means of Monte Carlo techniques, i.e. by simulating many collision events and taking their average . The basic idea is to embody the perturbative parton cascade picture into a relativistic version of the semiclassical kinetic theory. We represent the partons as classical point particles of various types as specified by their internal degrees of freedom . The state of a parton is characterized by its flavor a = of , qf, g (quark, antiquark, or gluon), its momentum P and position r Its energy 0375-9474/92/$05 .00 © 1992 - Elsevier Science Publishers E.V. A.11 rights reserved.
468C
K. Geiger, B.
filler /
ynainics of parion cascades
is determined by Em = p2 + 172 a + 11,12 where m,,, is the rest mass and M a possible space- or time-like virtual mass. The spin and color degrees of freedom are taken .-, amd in the composition of irao account only implicitly in the interaction amplitude partons as hadrons. 'kVe define a Lorentz invariant single particle distribution function (j)', F, t) dpd3 r. for the bine-dependent phase space density of partons of species a. The transport equation for this single particle distribution function has the manifestly covariant Orm. P po'
=
processes
AC« ( Pl - it) = .
processes
with Lorentz invariant collision integrals 4, = E,,C. . 1"he left-hand side of this Boltzmann equation describes the free propagation of partons, here generally taken to be on mass shell in order to avoid violations of gauge invariance, whereas the collision term on the right-hand side represents a suni over all possible interaction processes involving partons of type a . In order to solvn act. 0 ) Or the time evolution of parton distributions in 6-dimensional phase-space, we need to specify, a) the initial distribution of partons ;n the incoming nuclei, and, b) the form of the collision term . 3.
INITIAL PAN DISTRIBUTIONS
The initial distribution functions F,,(g. represented in the factorized form NAB w,o~a wa
'NI
W
r, t o ) of the partons in the incoming nuclei are -
P, P) - R.N ° (T, -
-
,1Z)
t=to
Here P~V'i and R-'V i give the initial momentum and spatial distributions, respectively, for partons of type a in each individual nucleon Ni, and the sum runs over all NA -1-NB nucleons in nuclei A and B. The vectors #, P, J5 denote the momenta of partons, nucleons and nuclei, respectively, and similarly do the positions i:, .fi and fz. Our ansatz for the initial momentum distribution Pai of parton flavors a in each nucleon Ni is where r = QP is the faction of the nucleons longitudinal momentum carried by 2 + M.2)/p2]1/2 the parton (P = Mill l ,j and r R = [ 3.2 + (PI denotes the energy a f a Vi ( X. Q2) measured fraction . The o are the nucleon structure functions and the distribution g(g_L ) with width ho ~ 0 .4 GeV1c takes into account the transverse momentuin 9-L of the initial partons which results from their confinement within the radius of a nucleon (of the order 0.8 fm). The scale Q20 ,. at which the structure functions .f Ni ( .r1 Q2) MVISt be evaluated, is computed event by event on a statistical basis by the a 0 Q2 -evolution of the structure functions by including (space-like) radiation processes for each such primary parton that encounters a scattering .
K. Geiger, B. Müller l Dynamics ofparson cascades
469e
The spatial distribution RQi of the partons depends on the initial coordinates 1Z of the two incoming nuclei A and B, and the positions of the nucleons R inside the volume of A, respectively B. Experimental information about its specific form is rather incomplete, so we have employed the following phenomenological ansatz: the individual nucleons are assigned positions around the centers of the colliding nuclei A and B according to a Fermi distribution in the rest frame of each nucleus. Next the partons coordinates are distributed in this reference frame around the nucleons centers with an exponential distribution corresponding to the measured nucleons electromagnetic form factor. Finally, the positions of nucleons and their valence quarks are boosted into the c.m. frame, while the sea quarks and gluons are smeared out longitudinally by an amount (Oz) ti h/pll < 2Rnuc around the valence quarks . This procedure is called distributed Lorentz contraction and yields a `fuzzy' pancake shape of the incoming nuclei [4]. 4. T E COLLISION TEEM The collision tern in eq. (1), Fa Ca , in general, describes all possible m -+ n interaction processes where at least one parton of type a is involved. All the information about these processes is contained in invariant matrix elements of the type M,,,-,, = (1',2.. . . . . n 111111, 2, . . . m), which give the amplitudes for processes with m particles in the initial state and n particles in the final state. Since it is impractical to include ail possible (in --> n) processes in the collision term of the transport equation, we restrict ourselves to a subset of processes that give the major contributions in perturbative QCD. In accordance with the intuitive picture of the evolution of a nuclear collision, we consider the important class of asymptotically (in energy) dominant (2 --+ n) processes that can be obtained from elementary (2 --~ 2) scatterings including associated space-like and time-like branchings [5] . For this set of (2 --+ n) processes, we formally write the invariant amplitudes squared as ~2 = 1(caia2 . . .bib2 . . .CIC2 . . .dld2 . . .cdIMIab)12
IM, 2-n
[sa (Ta, Q2 ,Q0) Sb ÉXb,Q2+QÛ)l 2 IMab-cd(S,t,u~ 4G 2 )I ~T~(Q2~p0)'Td(Q2~p0)],
where a,-, bt are the partons radiated from a, b ~)efore the elementary scattering a + b ) c+d, and ci, di are the partons radiates' from the final state partons c, d. Eq. (4) expresses that the total squared amplitude can be factorized into three components, ) namely: the Sudakov form factors S,, ;, determining the pro abilities Pb â = 1 - Sab that the scattering partons a, b have emitted a number of partons before the scattering, a -+ a + c.1 + . . . , b -~ b + b i . . . (space-like branchings), the squared matrix element IMab-cd1 2 for the elementary (2 --+ 2) subprocess a+b - 3 c+d at as scale of interaction Q2 , and the Sudaks;?) form factors T~.d giving the probabilities P~ d' = 1 - T~,d that the scatf?red partons c, d have initiated a sequence oY branchings after the scattering, c -; c + eI + . . . , d --> d + a1 + . . . (time-like branchings) . With the initial parton distribution and the collision term specified, the time-evolution of the distribution functions F(t,f,il according to the transport equation (1), is
IC. Geiger,
70r
,
ller / I~yna~nics of parton cascades
computed numerically by means of Monte Carlo methods. For a detailed presentation of the topics touched here tive refer to our extended manuscript [2]. S
s outlined above, we have simulated and analyzed the evolution of energetic collisions at ttivo energies ~/A = 100 ( 200 ) C~eV/A, respectively, for a large variety of symmetric collision systems from A = 1, . . . , 238 [2]. Fig. 1 shows, as an example of our results, the transverse energy l produced in nuclear collisions, which is a direct measure of how much of the beam energy has been redirected in interactions and characterizes the violence of the collision. It may also serve as an indicator of the energy density achieved during the collision . .El is found to increase approximately with A3 , i.e. it roughly scales with the nuclear radius, which is close to what is observed in nuclear collisions at lower energies. A certain saturation is seen to occur around A = 100. ~~Tost, interesting for the question in how to achieve maximum v1o1e11Ce 111 a c®111S1o11 nlavbe that an increase of the beam energy for a given system is obviously nluCh nlOre e ectlve than an increase in the mass of the beam particles at filed energy . Similarly the thrust T = ~_ ~ pll= I / ~= e â~z I of the partons (or the inelasticity 1 - ?') 1- e ects the degree of nuclear stopping and transverse expansion. tOD c
o _._ . ..
c
9D
ë c
so c
(200 +
=
0.7
so =
z
T®Fa!P.!/ :R
O.B
20D) GeV/A
7D
z
5D : r H c5
o.s
ao ` ' c~.
iD00 + 100) Gev/A
30 -
ä
Zo =
=
o.s
_-'
OS
-'_
04
~' ~ ~- a ~
i
c.7
C' 20
. ... .. . .. . ...
(20D + 200) GeV/A
D2
70 ~
0
.. . .
_0 .3 ~
aC
6C
BO
120 74D mass aumber A 700
16p
180
20C
220
240
~0
~
~
20
40
v 60
80
i
v
100 72C 740 76D mass number A
iYQ. b n
v
780 2DD
22C
240
Fig. 1 : Production of a) transverse energy E1 and b) thrust T as filnction of mass ntllnber .4 at Ec,rt = 100 ( ?00 ) GeV/.~ per collision event for central A ~- A collisions . .F
I~tC
S
1 See e.g . uar~-Ghaon Plasma, edited 1~y R. C. Hwa (World Scientific, Singapore 1990) 2 h . Geiger a.nd H. l~~IÜller, Parton Cascades in Relativistic Nuclear Collisions, preprint I)IJhE-TH-91-15, to appear in Nucl . Phys. )~. 3 oal, Phys. Rev. C33, 2?06 (1966); S. M1~®wczynski and J. Rafelski, Phys. Ret~. C , 107ô (1989) . 4 R . C. wa. and h. hajant,ie, Phys. Rev. Lett. ,SF, 696 (1986) . 5 A . assetto, ~~I . Ciafalcnii, and C~. l~~Iarchesini, Phys. Rep. 1®®, 203 (1983).