PHASE-SPACE CONSIDERATIONS IN THE TRANSVERSE MOMENTUM ANALYSIS

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PHASE-SPACE CONSIDERATIONS IN THE TRANSVERSE MOMENTUM ANALYSIS

P. Danielewicz

To cite this version:

P. Danielewicz. PHASE-SPACE CONSIDERATIONS IN THE TRANSVERSE MOMEN- TUM ANALYSIS. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-233-C2-239.

�10.1051/jphyscol:1987233�. �jpa-00226500�

PHASE-SPACE CONSIDERATIONS IN THE TRANSVERSE MOMENTUM ANALYSIS

P. DANIELEWICZ

Institute of Theoretical Physics, Warsaw University, ul. Hoza 69, PL-00-681 Warsaw, Poland

Abstract:

Transverse momentum method of analysing collective sideward motion in heavy-ion collisions at intermediate energies is reviewed. The emphasis is put on the phase space aspects. The method is tested in the Cugnon cascade model.

1. INTRODUCTION

Much of experimental effort in the intermediate energy collisions of heavy nuclei has been devoted to the detection and quantification of the nuclear collective flow. The flow has been first observed in the Nb+Nb reaction at 400 MeV/nucl [I 1 and in the Ar+Pb reaction at 800 MeU/nucl C23. The sphericity matrix method [3,41 utilized then in data analysisgave no conclusive results in other reactions C1,5,61. Later the data have been analysed using the transverse momentum method C7-101. The method attempting to establish the reaction plane in the reactions, has proven to be more sensitive than the sphericity matrix method and permitted a better quantification of the flow effects. The talk discusses the transverse momentum analysis with emphasis put on the theoretic aspects. The method is tested in the cascade model.

2. REACTION PLANE AND COLLECTIVE MOTION

The state of nuclei approaching each other to undergo a collision involves a specific symmetry. This is a reflection symmetry with respect to thereaction plane defined, for finite impact parameters, by the direction of the beam axis and a line joining the centers of nuclei. Though the structure of nuclei gets destroyed at intermediate bombarding energies, the symmetry present in the initial state should be preserved in the course of a collision. Due to the finite number of products, still, the symmetry may not be evident in every single final state measurement. (We may remark here that presumably for large deformations of initial nuclei, and sufficiently low impact parameters, a symmetry with respect to the reaction plane cannot be claimed.)

The momentum distribution of nucleons in the nuclei is isotropic in transverse directions. With the spatial distribution of matter not having that property (for finite impact parameters), the momentum distribution should in general evolve in the collision process from isotropy, complying with the overall reflection symmetry with respect to the reaction plane. A consequence of that symmetry for the momenta is the fact that any vector constructed from transverse particle momenta, in a manner that does not a priori distinguish any particular transverse direction, must on the average point in the direction of the reaction plane.

The generation of transverse anisotropies in collisions of heavy nuclei at intermediate energies, that may be directly viewed in model calculations [Ill, is intimately related with the question of creation and properties of highly excited

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987233

C2-234 JOURNAL DE PHYSIQUE

dense nuclear matter. During the decompression matter should be repelled from an excited region, evidenced, in association with the reaction plane, in an opposite sideward deflection of the forward and backward going fragments. The preference for sideward emission has been first seen in the data of Refs. C11 and C21.

The experimental results have not been reproduced by the calculations in the cascade model [12,13]. Theoretically it is known that the stiffness of the equation of

state enhances a collective motion [14,151. At the other hand, an enhancement of the transport coefficients C16,171 or, equivalently, a reduction of the particle cross sections C 181 reduces the flow effects.

The transverse momentum method of data analysis exploits and quantifies the transverse anisotropies associated with the reaction plane, generated in the collision process.

3. THE TRANSVERSE MOMENTUM METHOD

As described in Ref. [71, the method consisted in the construction of a transverse reference vector from momenta pt of particles from a reaction,

Q = z wvp+, (1

with the weights wv chosen to minimize the fluctuations of the vector around the direction of the reaction plane in collision events. The weights wv = 0 were chosen for pions, while for the baryons wV = 1 for yv > yc + ^{6 , wv = -1} for yV < yc - 6 ,

and g = 0 otherwise. For the symmetric collision C71 it was natural to take the rapidity yc equal to the value for the overall c.m. system, yc = yB/2. The quantity

6 was inserted to remove particles having negligeable correlation with the reaction plane.

The direction of the vector Q has been used C7-101 to estimate the direction of the reaction plane in the collision events. On this direction the particle

transverse momenta were projected to yield the mean components of transverse momenta per nucleon in the estimated reaction plane, as a function of the rapidity. The magnitude of the vector Q has been used [7,81 to evaluate mean components of momenta

in the true reaction plane, at rapidities ly - ^{ycJ > 6 ,} according to the formula

with x denoting the in-plane pL component, and A a total mass at ly - ycl > ^6, A = ZVav. The mean <wpX/a> has served to normalize the components in the estimated reaction plane reduced from components in the true reaction plane by

a factor <cos rp>. The angle rp is here the azimuthal deviation of the vector Q from the true reaction plane. The magnitude of the mean in-plane momenta in the

data C7-101 appeared to be systematically larger than in the cascade model for the same reactions, but still low as compared with average r.m.s. transverse momenta in the reactions.

Equation (2) is based on a presumption that the dominant correlation between

$article transverse momenta is that associated with the reaction planed and due to the anisotropies of single particle distributions. I.e., if we knew the direction of the reaction plane, the particle emission might be Considered uncorrelated.

On writing an invariant 2-particle distribution function as a product of the single-particle distributions associated with a reaction plane,

we find for the average scalar product of transverse momenta at fixed rapidities

<pL(yl)*pL(y2)> = .fd2p:d2pi P;*P~ P ~ ~ ) ( P ~ . Y ~ . P ~ ~ Y ~ ) / s ~ ~ P ~ ~ P ~ P(z)

= [.fd2pi :P P ( 4 , ~ ~ 1 /.fd2~: P] [$d2p: :P p(F$,yy) /.fd2p$ P)

some proper weighting, and summation may carried over pairs of fragments in the collision events, as is actually done in (2).

In recent investigations [I91 of the Streamer Chamber data C2,201, Eq. (4) is extensively used to exhibit the sideward motion, rather than the previous procedure.

In that way no reference is made to the distribution of the estimated directions of the reaction plane with respect to its true direction. Further, an effort has been made to isolate an additional source of correlation between transverse momenta from the transverse momentum conservation.

4. EFFECT OF ENERGY-MOMENTUM CONSERVATION

We consider here a general uncorrelated emission subject to an overall energy-momentum conservation constraint assuming a fixed direction of the reaction plane. We study the effect of the constraint on the value of a scalar product of transverse momenta.

The exclusive distribution of particles for the discussed emission is of the form

N

P(~)(P;,Y,, . . - . P ~ . Y ~ ) = P(P;.Y~)]~(~)( z P, - P)

v= 1 (5)

with the 4-momentum P = (0,E) in the system c.m. In (5) it may be assumed that

2 2

where <pt? are the single-particle averages <p% = Id pLdy papl$d pLdy p, and a = 0,1,2,3 designates components of a 4-vector. This is because otherwise the single-particle distributions may be renormalized for (6) with a factor ~X~(-B,\~;) at the r.h.s. of (5), and no alteration to the 1.h.s. of (5).

With a central-limit theorem one gets from (5) for the two-particle distribution

P(~)(P+,Y~ ,P+,y2) = P(P+,Y~ )P(P+,Y~)

x exp[-I Z U T A (p 1+P - < P ~ > - < P ~ > ) ~ ( P ~ + P ~ - < P ~ > - < P ~ > ) ~ ] . (7) 2 where AaT satisfies

AaT xy+v,p<( P ~ - < P ~ > I ~ ( P ~ - < P ~ > ) ~ > = 6; 9 (8) and the sum runs over particles in the final state. The energy-momentum

conservation constraint introduces an anticorrelation between the particle 4-momenta. This is of the order of 1/N. Of our concern is the anticorrelation induced on the transverse particle momenta. On dropping other than the leading term from conservation affecting the finally deduced mean values of in-plane momenta, Eq. (7) can be rewritten into

~(,)(P+,Y, ,4,y2) P(P+,Y~) P(P;,Y~) exp(-2aP+*p;) 9 (9) with a-' = Z <p12> = ^x <pl'>. On multiplying both sides of (9) with a scalar

r#v,ll y Y' 7

product of transverse momenta, integrating over the momenta and expanding the exponential, one then obtains

<P'(Y, ).P'(Y,)> = <pX(yl )><pX (~2)) - ^a <P'~(Y, )><P~~(Y~)>, ⁽¹⁰⁾ that replaces Eq. (4).

An origin of the recoil correction term at the r.h.s. of (10) can be easily understood. As the transverse momentum pi of particle 1 fluctuates around the average <pt.> = (<pl>,O), the rest of the system absorbs the recoil. Per any given particle the share of the recoil is c (q - <*)IN, a-d the contribution from the

C2-236 JOURNAL DE PHYSIQUE

recoil to the average scalar product of momenta of two particles is then

% <pL*(pL 1 1 - 1 <pl>)>/~ = <p:2>/~. More properly, however, the result should be symmetric in the two particles, and this is accounted for in Eq. (10).

Although the recoil correction term in (10) is of the order of 1/N, it may be comparable with the dynamic term in (10) for low values of mean in-plane momenta, like in the cascade model, and lighter systems. When the weighting in rapidity is used, such as in the vector 4, the role of the recoil can be much reduced (if not nullified). This is because the weighting causes a constructive addition of the dynamic contributions from the forward and backward rapidity regions and

a destructive addition of the recoil contributions. This can be explicitly studied in such equations as (2).

Equation (10) obtained from a consideration of the uncorrelated emission in a coordinate system associated with the reaction plane, has been found to adequately describe the Streamer Chamber Group data from the reactions at 0.8 GeV/nucl [191.

The coefficient in (10) has been treated as a parameter determined from the data, and sensible results were obtained for that coefficient. Here we shall present an outcome of the analysis, with Eqs. (4) and (lo), of the collision events generated in the cascade model 1121 in which the reaction plane is under control.

5. ANALYSIS OF CASCADE EVENTS

Figure 1 displays a contour plot of proton <pL(y, )*pL(y2)> for the La+La reaction at 0.8 GeV/nucl simulated in the Cugnon cascade model [I21 for impact parameters b < 6 fm. The averages have been evaluated for 7 rapidity intervals of

width Ay = yYg/6, to yield 28 independant values for the plot. The lack of symmetry 1

0.8--

m 0.6- x

%

in the plot with respect to a diagonal running from the upper left corner to the lower right corner, reflects a finite event statistics. A fit to the values of the average product using Eq. (4) yields the values of <pX(y)> shown in Fig. 2, and an acceptable x2 = 29. A subsequent use of Eq. (10) in a fit gives then virtually

- 1 2

unchanged values of <pX(y)>, ^a = 42 + ^{1 1} (~ev/c)', and lowers x down to 15.9.

Inclusion of the recoil correction is much more crucial in the second studied reaction of lighter nuclei .Ar+KCl at 0.8 GeVInucl. An analogous plot of

<~'(~~)'p'(y~)> as for the La+La reaction, evaluated for the Ar+KCl events at b < 3.3 fm, is shown in Fig. 3. Negative values of the average product dominate.

- - ^{4 / I 1}

I t

, - , 1 I I

, ^{- - ,} - 2 , I

0 Fig. 1. Contour plot of the

average scalar product of

- 2 - - -

/

/ ^C -

\ /

\ \ ? ,'

I I

/ ' I

, 1 * ,

proton transverse momenta as a function of two rapidities, for the La+La reaction at 0.8 GeV/nucl simulated in the cascade model. Contour lines are labelled with values of the product in units of I O ~ * M ~ V ~ / C ~ , and those

0 012 O.i ^0j6 0 1 ~ ¹ corresponding to negative

values are drawn as dashed.

Y ~ I Y B

0 , +i Fig. 2. Mean transverse

T momenta in the reaction plane

as a function of the rapidity -25 -

t

from the transverse momentum analysis ( r ) , and from the

-50 - - known direction of the

- 5 - ++tli reaction plane (x), for the

- La+La reaction at

-100 - 0.8 GeV/nucl studied in the

I I I I I 1 cascade model.

-0.2 0 0.2 0.4 0.6 0.8 1.0 1.2

Fig. 3. Contour plot of the average scalar product of proton transverse momenta as a function of two rapidities, for the Ar+KCI reaction at 0.8 GeV/nucl simulated in the cascade model. Contour lines are labelled with values of the product in units of 1 03*~ev2/c2, and those corresponding to negative values are drawn as dashed.

A fit using Eq. (4) brings a failure with x2 = 180. _A fit using Eq. (10) gives however x2 = 21. The values of <pX(y)> obtained from the fit are shown in Fig. 4

(incidentally the fit with Eq. (2) yields essentially identical values), and d 1 =

11.5 * ^{1.0 (GeVIc)} 2 .

For comparison the results for ipX(y)> obtained using the known direction of the reaction plane in the model are indicated in Figs. 2 and 4 with the crosses. It can be seen that the transverse momentum method reproduces fairly well these values.

As to the magnitude of a-I, it has to be mentioned that the total initial transverse momentum in the Cugnon cascade model [12], is not constrained to zero, but varies with the initial Fermi momenta of individual nucleons. Further, the nucleons that have not undergone a collision do not participate in the momentum exchange. Under these conditions, the inverse of should correspond to average sum of final

C2-238 JOURNAL DE P H Y S I Q U E

Y/Y,

initial transverse momenta squared, excluding nucleons that have not undergone -

50

- 25-

- X ^{0 -}

X

-25

- 5

-75 -100.

a collision. This sum is, respectively, equal to 44.7 + 0.9 ( G ~ V I C ) ~ and 12.00 + 0.09 (Ge~lc)~, in the studied La+La and Ar+KCI events.

6. EVALUATION OF THE SPHERICITY MATRIX PARAMETERS

The transverse momentum method can be used to evaluate the parameters of the i j per-particle sphericity matrix associated with the reaction plane <siJ> = <wp p >.

Here w is a weight usually taken as 1/2m or llp, and the momenta are in the -0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 cascade model.

-

+ $*$ ^{- -}

*+ *+ -

ti*'*+$ ^-

- -

1 I I I I I

system c.m. The problem with the matrix boils down to evaluating an element <sXZ>

Fig. 4. Mean transverse momenta in the reaction plane as a function of the rapidity from

analysis the transverse (r), and from momentum the known direction of the reaction plane (x), for the Ar+KCl reaction at

0.8 GeV/nucl studied in the

and resolving the ^I in1 and 'out of plane1 transverse elements <sXX> and <sYY>, as otherwise <sZZ> = <wpZ2>, and <sXX> + <sYy> = <wf12>. (The elements <sXY> and <sYZ>

vanish by the reflection symmetry.)

The element <sXZ> can be obtained from a formula that follows from (10) on carrying a weighted averaging over rapidity:

2 2

< ~ ~ p t ~ ; . W ~ ~ + > = <sxz><wpx> -

.

<w 1 2 3 w u {p12pl.pl 1 2 3 - 2(p+-p$)(p+*pf)b = {<sYY> - <sx~)<~px>2. (12) The recoil corrections cancel out in (12) to the leading order.

When the sphericity matrix elements are known the diagonalization for the flow parameters C31 may proceed in the usual manner. Table 1 compares the results for the flow parameters in the cascade model obtained using the known direction of the reaction plane and within the transverse method discussed.

7. CONCLUDING REMARKS

We have reviewed the features of the transverse momentum method of the analysis of sideward collective motion in collisions of heavy nuclei. The method has been tested using events generated in the Cugnon cascade model C121, and has been found to reproduce well the results from the known direction of the reaction plane.

In general the method relies on a presumption that the strongest correlation between

momentum analysis (B). A weight llp is used, and the eigenvalue ratios are rij = fi/fj, r = 2f3/(fl + ^f2). The flow angle ef is the polar angle of the longest axis in the reaction plane, corresponding to the eigenvalue f3, and f2 = <sYY>.

particles in transverse directions is that associated with the reaction plane.

As Fig. 3 demonstrates, effects from momentum conservation in the Ar+KCl reaction are stronger in the cascade model than the dynamic effects. The performance of the method for that reaction may be considered surprising. Quite generally the method cannot be considered valid to any arbitrary degree of accuracy. However, the integrity of the results within a given accuracy may be tested using methods of statistical analysis, such as applied to Eqs. (4) and (10). Looking forward, one may hope that to the extent that such equations as (3) and (9) are valid, one can unfold more features of single particle distributions associated with the reaction plane. Incidentally it is plausible that, for high event statistics, the in-plane triple differential cross sections in the collisions of very heavy nuclei [I] can be re1 iably established even now, without further developments in the methods of analysis. This can utilize the smallness of the fluctuations of the estimated reaction plane with respect to true plane, and the fact that by reflection symmetry the planes of constant differential cross section must be perpendicular to the reaction plane.

This work was partially supported by the Program of Basic Research CPBP 01.09.

ef Cdegl r3 1 r32 ' 2 1

r

lef erences

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La+La 0.8 GeVInucl b < 6.0 fm

A 9.0 + ^0.5

1.88 + ^0.02

1.88 + ^0.03

1-00 + ^0.02

1.88 + ^0.02

A 14.4 + 0.8 2.07 + 0.05

1.98 + ^0.04

1.05 + ^0.03

2.02 + ^0.03

B 7.9 + ^1.9

2.05 + ^0.17

1.73 + ^0.12

1.19 + ^0.18

1.88 + ^0.03

B 13.2 + ^1.7

2.17 + ^0.08

1.87 + ^0.07

1.16 + ^0.07

2.01 + ^0.05