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Intermittent rapidity distributions in nucleus-nucleus collisions and scaling indices
Volume 247, number 4
PHYSICS LETTERS B
20 September 1990
Intermittent rapidity distributions in nucleus-nucleus collisions and scaling indices Mikulfig Bla2ek
L vman Laboratoo' of Physics, ttarvard Untversity. ('amhridge. MA 02138, US.t Received 27 May 1990
Scaling indices characterizing the intermittent rapidity distributions of ~O+ t2C and ( ~60, 32S)+Ag/Br nuclear collisions are analyzed within the frame of the two-scale Cantor set. Several qualitative as well as quantitative conclusions are formulated and it is called here for experimental results corresponding also to non-integer (positive as well as negative) values of the order parameter.
1. Recently published data on rapidity distributions observed in high energy collisions indicate with sufficient reliability intermittent behavior. It concerns, for instance, the e l e c t r o n - p o s i t r o n annihilations at the average CM energy ( x / ~ ) = 35 GeV [ 1 ], m u o n - p r o t o n collisions at x / s = 4 - 2 0 GeV [2], the 250 G e V / c It + and K + interactions with protons [ 3 ], and also the nucleus-nucleus interactions observed in the cosmic rays as well as at the accelerator energies [4,5] (a well-fitted survey can be found in refs. [2,6] ). The presence of the intermittency in a rapidity interval AY=y,,ax -Ymi, ( d i v i d e d into M bins each o f length By, 53:= A Y / M ) is deduced there from the fact that the conveniently normalized factorial m o m e n t s o f the order q, Fq,
Fq=norm.( n!/ ( n - q ) ! )
(1)
are related to 8y in the following way: In/'~
, aq-bqlnSy.
(2)
~v , 0
The slopes bq allow the introduction of the generalized (Hausdorff, fractal) dimensions Do:
Du= l - b u / ( q - 1 ) ,
(3)
which have been considered by several authors (more details e.g. in ref. [ 7 ] ). On leave of absence from the Institute of Physics, Slovak Academy of Sciences, CS-84228 Bratislava, Czechoslovakia. 576
However, the dimensions ( 3 ) themselves do not represent a complete characterization o f the intermittency (different models can lead to the same generalized dimensions [8] ). In the present contribution we show that also for the intermittent rapidity distributions, the basic properties of the scaling indices represent an important source of additional information. 2. The regular (continuous and sufficiently s m o o t h ) parts of the probability distributions (involved in the averaging procedures ) entering the factorial m o m e n t s ( 1 ) cannot be responsible for the appearance o f the intermittency. Their singular parts which are i m p o r t a n t for us just now, can be characterized by two indices, namely o~, which determines the strength of their singularities, and f which describes how densely they are distributed. The spectrum o f singularities is described by giving the possible range o f o~ values and the function.f( o~) [ 8,9 ]. The experimentally d e t e r m i n e d slopes bu (by means o f ( 2 ) ) allow the deduction o f the generalized dimensions (3) which in turn lead to the following relations [9]: d c~-aq= ~ [(q-l)Dql uq
(4)
and
0370-2693/90/$ 03.50 cc) 1990 - Elsevier Science Publishers B.V. ( Norlh-Holland )
Volume 247, number 4
PHYSICS LETTERS B
dDq
f ( a ) =D,~+q(q- 1 ) dq
(5)
20 September 1990
Dq= (In R q ) l [ ( I - q )
In 2] ,
(9)
where On the other hand, as far as there is known a suitable theoretical model allowing the determination of the generalized dimension D,
D = D ( q ) =Dq,
Rq=p~ + p~ + ... + p~ . In this case,
(6)
relation (3) provides a theoretical value of the slopes
(10)
With respect to the relation
bq= ( l - q ) ( D q -
1) ,
(7)
and relations (4) and ( 5 ) give the strength a and the d i s t r i b u t i o n f ( a ) of the corresponding singularities, respectively. At this place we introduce a new quantity, G, /)_~
G=
lnp, + ... +p~.lnpa)/(Rqlnit).
=-(pq
bq:
J .f(aq) daq,
(8)
O+~
f(au) = q a q - ( q - 1)Dq,
(11)
which follows from the results ofref. [9], we obtain either
J-- f ( a ) .
(12)
as far as in ( 11 ) the order q is expressed by means of (10) in the form q = q ( a ) and then Dq. relation (9) in the form D = D ( q ( a ) ) - - / ) ( a ) , or
which characterizes the global intensity of those singularities. Let us note that the definition of the factorial moments under consideration contains the order q only in the factorial entering the denominator of relation ( 1 ). Expressing this factorial in terms of the gamma function, ( n - q ) ! = l - ' ( n - q + 1 ), relation ( 1 ) allows the consideration of the corresponding factorial moments for arbitrary (real) orders.
f = f ( q ) =fq,
3. To obtain a theoretical expression for Dq, let us start by a regular exact division of the rapidity interval AY (cf. refs. [ 10,11 ] ) where we assume a uniform rapidity density (and let it be equal to unity). In the first step, AY is divided into it paras of equal length, each part being characterized by the rapidity density pj ( j = 1, 2 ..... it), with ~pj= 1. In the next step, each part is again divided into it parts of equal length, each part being characterized by the rapidity density pjp~ ( k = 1, 2 ..... it). Let us continue this Cantor construction n times, where n is defined in such a way that the last, nth, division part length is small compared with the detector resolution. Now, the derivation of D~ follows essentially the lines of ref. [12] with the exception that in our case the Bernoulli multinomial distribution is involved (instead of the binomial one of ref. [ 12 ] ). Our result can be presented in the form
=-(Pl
(13)
if relations (10) and (9) are applied directly in ( 11 ). Let us recall the symmetry property of the function (13)
,7(q)=f(-q) .
(14)
Moreover, in the limit q ~ 1, we obtain
Dq = oQ =fq lnpl + ... + p a l n p a ) / l n 2 ,
(15)
and in the limit I q l - - ' ~ (with p~ >pz...>pa) lira Dq= lim a q = - ( l n p ~ ) / l n 2 , q - . ~z,
q
lim q
*
--
:Jc,
(16)
,oc
Dq=
lim a , ~ = - ( l n p x ) / l n 2 ,
(17)
q~-oo
and lim f ( a q ) = I ql ~ ' ~ o
lim f ( q ) = 0 . Iql
(18)
*o¢
4. With respect to the results presented in refs. [ I 5 ], the most straightforward application of our approach requires the data from refs. [4,5]. Especially we consider the collisions of 200 GeV/nucleon ~60 beams with the ~2C nuclei (where the energy of the projectile fragments and produced particles as mea577
Volume 247, number 4
PHYSICS LETTERS B
20 September 1990
Table 1 Experimentally observed and theoretically calculated slopes evidencing the appearance of the intermitteney, relation (2) for three different nucleus-nucleus collisions. Parameters p. ( -~p ) and P2 ( -= 1- p ) give the probabilities which describe the rapidity flow during the collisions. The numbers in parentheses give the uncertainty in the last digits of the associated values. Collision
b2
b3
h4
bs
b6
Reference
'60+ '-'C
experiment theory, (p=0.572)
0.029(1) 0.029
0.087(3) 0.087
0.171(5) 0.170
0.278(12) 0.274
0.399(25) 0.398
[4] relation (7)
~60+Ag/Br
experiment theory, (p=0.543)
0.010(1) 0.010
0.034(3) 0.032
0.062(5) 0.062
0.106(15) 0.103
0.163(26) 0.153
[51 relation (7)
32S+Ag/Br
cxperiment theory (p=0.524)
0.004( 1) 0.003
0.014( 1) 0.010
0.023(3) 0.020
0.032(5) 0.032
0.040(9) 0.049
[5] relation (7)
sured by the zero-degree calorimeter is in the range 2.2-2.4 TeV and the corresponding p s e u d o r a p i d i t y window is 2 . 4 - 4 . 0 ) [4] and the 200 G e V / n u c l e o n ' 6 0 and n S nuclei with ")TAg/snBr nuclei in nuclear emulsion (with thc pseudorapidity window 0.5-5.5 ) [5]. In this case, a sufficiently accurate description o f the slopes bq ( q = 2 , 3, 4, 5, 6) is obtained by means of the two-scale Cantor set, as is seen in table 1 (2 = 2 and we denote p~ = p , p2 = 1 -p, i.e., p is the only free p a r a m e t e r there). For other types o f collisions, the higher values o f 2, even with some vanishing probabilities p,, cannot be excluded. We note that when solving the problems o f the turbulent energy dissipation, the generalized dimensions Do~ and D _ ~ represent the cxperimcntally accessible quantities and then relations (16) and (17) allow the determination o f the probabilities Pt and P2 = 1 - p ~ [ 12 ]. With the values o f the p a r a m e t e r p given in table 1, thc dependcncc o f the gcneralized dimensions Dq on the order q, relation (9), is seen in fig. 1, and that of the slopes bq o n q, relation (7), in figs. 2 and 3. Except the points where q = 2 , 3, 4, 5, 6, these figures present our predictions involving even non-integer (positive as well as negative) values o f the o r d e r parameter q. The gencralized dimensions are positive and, as is d c m o n s t r a t c d by fig. 1, they decrease with increasing order q; for positive values of q, they increase with the increasing complexity of the colliding objects. The slopes bq might be negative for 0 < q < 1 (fig. 3) and outside o f this region they decrease with the complexity o f the colliding objects. We call the attention o f experimental groups to verify those conclusions, as well as in other cases. 578
1.2
-1551 111
Dq LO
0.8
-6()
-410
-2'0
b q
210
4()
60
Fig. 1. The dependence of the generalized dimensions Dqon the order parameter q, relation (9), with A=2. The corresponding value of the parameter p is given in table 1. The dotted curve corresponds to the '60 + ]~C, the dashed one to the '60 + Ag/Br. and the full one to the ~"S+Ag/Br collisions. Thc dependence o f the strength o f singularities aq on the order p a r a m e t e r q is seen in fig. 4. Like Dq, aq also decreases with increasing q and for positive values o f that order p a r a m e t e r it increases with the complexity of the colliding objects, i.e., collisions of more complex nuclei give rise to bigger discontinuities. Also the density of discontinuities increases with the complexity o f the colliding objects (fig. 5 ). The spectrum o f singularities is presented in the form o f t h e f - a dependence in fig. 6. As is seen therc, for instance, the strength o f singularities is b o u n d e d from below as well as from abovc and the global intensity, relation (8), decrcases with the increasing complexity o f the colliding objects. The density distribution f ( a ) gets its maximal value (equal to unity) at a=Ot~xtr=[ln(plp2)-tn]/ln2 which decreases to unity with increasing complexity o f the colliding objects. 5. In summary, we have presented here some fun-
Volume 247, number 4
PHYSICS LETTERS B
20 September 1990
i I01
/ •.. ".
%%~. 8i
bq
\~\
//
".'.
\
6
/
//
/
.. %•~
//
/ ¢
':;.
/
~
4
2
%.. %-..
../ .~
0 I
- 8 '0
-100
- 6'0
- 4' 0
' -20
b
' 20
4' 0
;0
' 80
' 100
Fig. 2. The dependence of the slope bq on the order q, relation ( 7 ) (the notation is the same as in fig. 1 ).
0.004
1.2 I
\
0003
\ \
0.002
I s
/ /
',
aq
; /
1.0
08 T
0.001
,
bq
-4'0
,
-1o
a
20
, ,o
T
q
j
Fig. 4. T h e dependence o f the strength o f the singularities otq on
o
the order q, relation ( I 0 )
(the notation is the same as in fig. l ).
/ :
-0.001
1.0
I: :
-0.007
li
0.8 -0.003
..
." "....."
i -I
J 0
q
7(q) 0.6 i I
0.4
///IIll."I:I2~/~i
Fig. 3. A detail of the dependence b,~versus q, seen in fig. 2. 0.2
d a m e n t a l properties o f the scaling indices with respect to the c o m p l e x i t y o f c o l l i d i n g nuclei. T h e experimental input for our c o n c l u s i o n s is rcpresented by the values o f the slope parameter bq. A n y extension o f the presently available range ( w h i c h consists o f five isolated points q = 2 , 3, 4, 5, 6) can specify
-60
-40
-20
0 q
20
40
i
60
Fig. 5. The dependence of the density of singulariticsf(q) on the order parameter q, relation (13) (the notation is the same as in fig. 1).
579
Volume 247, number 4
PHYSICS LETTERS B
1.0
0.8
\
"..
"
"
::
\
•:
f(aq)
t h e i r s u p p o r t and fruitful discussions. S t i m u l a t i n g c o n v e r s a t i o n s with Professor W. Kittel are also highly a c k n o w l e d g e d . T h i s research was s u p p o r t e d by the U n i t e d States D e p a r t m e n t o f Energy, u n d e r G r a n t D E A C 0 2 7 6 ER 03064.
,~.. ,.'1
0.6
•-
' \ i 0.8
, |.0
References
".
~
f
I ,,t, 0.9
:-
I
~
°2
-
I~ i
04
';. I
-
, I ]. ~ ,,k - 1,1 -
i : ,12-
aq
Fig. 6. The dependence of the density of singularities J(c~,a) on their strength, relation ( 12 ). The a,~ value of the points denoted by circles (squarcs) is given by D~ (D_y), relations (16) and ( 17 ) (the notation is the samc as in fig. 1).
m o r e a d e q u a t e l y the m e c h a n i s m o f the rapidity flow d u r i n g the d e v e l o p m e n t o f the collision process. A s i m i l a r analysis o f the scaling indices d e p e n d i n g on the energy o f the colliding objects will bring a n o t h e r piece o f i m p o r t a n t i n f o r m a t i o n . T h e a u t h o r expresses his gratitude to Profcssor Roy J. G l a u b e r , L y m a n Laboratory, H a r v a r d U n i v e r s i t y , and Professor P. Carruthers, D e p a r t m e n t o f Physics, U n i v e r s i t y o f A r i z o n a ( w h e r e this w o r k s t a r t e d ) for
580
20 September 1990
[ 1 ] TASSO Collab., W. Braunschweig ctal., Phys. Lett. 13 231 (1989)548. [ 2 ] I. Derado. G. Jancso. N. Schmitz and P. Stora. Investigation of intermittency in muon-proton scattering at 280 GeV/c, preprint MPI-PAE/Exp. El. 221 (February 1990). [3] EHS/NA 22 Collab., I.V. Ajinenko et al., Phys. Let(. B 235 (1990) 373. [ 4 ] WA 80 Collab., R. Albrecht et al., Phys. Lett. B 221 (1989) 427. [ 5 ] KLM Collab., R. Holynski et al., One- and two-dimensional analysis of the factorial moments in 200 GeV/nucleon p. ~60 and 3'S interactions with Ag/Br nuclei, KrakowLouisiana-Minnesota preprint. [6] P. Carruthers, H.C. Eggcrs, Q. Gao and 1. Sarcevic. Correlations and intermittency in high energy multihadron distributions, Arizona University preprint AZPH-TH/909 (March 1990). [ 7 ] P. Lipa and B. Buschbcck, Phys. Left. B 223 ( 1989 ) 465. [ 8 ] T.C. Halsey and M.H. Jensen. Physica D 23 ( 1986 ) I 12. [9] T.C. Halsey, M.H. Jcnsen, L.P. Kadanoff, I. Procaccia and B.I. Shraiman. Phys. Rev. A 33 (1986) 1141. [ 10 ] A. Bialas and R. Peschanski. Nucl. Phys. B 273 ( 1986 ) 703. [ 11 ] A. Bialas and R. Peschanski, Nucl. Phys. B 308 ( 1988 ) 857: Phys. Lett. B 207 ( 1988 ) 59. [ 12] C. Meneveau and K.R. Sreenivasan. Phys. Rcv. Len. 59 (1987) 1424.
PHYSICS LETTERS B
20 September 1990
Intermittent rapidity distributions in nucleus-nucleus collisions and scaling indices Mikulfig Bla2ek
L vman Laboratoo' of Physics, ttarvard Untversity. ('amhridge. MA 02138, US.t Received 27 May 1990
Scaling indices characterizing the intermittent rapidity distributions of ~O+ t2C and ( ~60, 32S)+Ag/Br nuclear collisions are analyzed within the frame of the two-scale Cantor set. Several qualitative as well as quantitative conclusions are formulated and it is called here for experimental results corresponding also to non-integer (positive as well as negative) values of the order parameter.
1. Recently published data on rapidity distributions observed in high energy collisions indicate with sufficient reliability intermittent behavior. It concerns, for instance, the e l e c t r o n - p o s i t r o n annihilations at the average CM energy ( x / ~ ) = 35 GeV [ 1 ], m u o n - p r o t o n collisions at x / s = 4 - 2 0 GeV [2], the 250 G e V / c It + and K + interactions with protons [ 3 ], and also the nucleus-nucleus interactions observed in the cosmic rays as well as at the accelerator energies [4,5] (a well-fitted survey can be found in refs. [2,6] ). The presence of the intermittency in a rapidity interval AY=y,,ax -Ymi, ( d i v i d e d into M bins each o f length By, 53:= A Y / M ) is deduced there from the fact that the conveniently normalized factorial m o m e n t s o f the order q, Fq,
Fq=norm.( n!/ ( n - q ) ! )
(1)
are related to 8y in the following way: In/'~
, aq-bqlnSy.
(2)
~v , 0
The slopes bq allow the introduction of the generalized (Hausdorff, fractal) dimensions Do:
Du= l - b u / ( q - 1 ) ,
(3)
which have been considered by several authors (more details e.g. in ref. [ 7 ] ). On leave of absence from the Institute of Physics, Slovak Academy of Sciences, CS-84228 Bratislava, Czechoslovakia. 576
However, the dimensions ( 3 ) themselves do not represent a complete characterization o f the intermittency (different models can lead to the same generalized dimensions [8] ). In the present contribution we show that also for the intermittent rapidity distributions, the basic properties of the scaling indices represent an important source of additional information. 2. The regular (continuous and sufficiently s m o o t h ) parts of the probability distributions (involved in the averaging procedures ) entering the factorial m o m e n t s ( 1 ) cannot be responsible for the appearance o f the intermittency. Their singular parts which are i m p o r t a n t for us just now, can be characterized by two indices, namely o~, which determines the strength of their singularities, and f which describes how densely they are distributed. The spectrum o f singularities is described by giving the possible range o f o~ values and the function.f( o~) [ 8,9 ]. The experimentally d e t e r m i n e d slopes bu (by means o f ( 2 ) ) allow the deduction o f the generalized dimensions (3) which in turn lead to the following relations [9]: d c~-aq= ~ [(q-l)Dql uq
(4)
and
0370-2693/90/$ 03.50 cc) 1990 - Elsevier Science Publishers B.V. ( Norlh-Holland )
Volume 247, number 4
PHYSICS LETTERS B
dDq
f ( a ) =D,~+q(q- 1 ) dq
(5)
20 September 1990
Dq= (In R q ) l [ ( I - q )
In 2] ,
(9)
where On the other hand, as far as there is known a suitable theoretical model allowing the determination of the generalized dimension D,
D = D ( q ) =Dq,
Rq=p~ + p~ + ... + p~ . In this case,
(6)
relation (3) provides a theoretical value of the slopes
(10)
With respect to the relation
bq= ( l - q ) ( D q -
1) ,
(7)
and relations (4) and ( 5 ) give the strength a and the d i s t r i b u t i o n f ( a ) of the corresponding singularities, respectively. At this place we introduce a new quantity, G, /)_~
G=
lnp, + ... +p~.lnpa)/(Rqlnit).
=-(pq
bq:
J .f(aq) daq,
(8)
O+~
f(au) = q a q - ( q - 1)Dq,
(11)
which follows from the results ofref. [9], we obtain either
J-- f ( a ) .
(12)
as far as in ( 11 ) the order q is expressed by means of (10) in the form q = q ( a ) and then Dq. relation (9) in the form D = D ( q ( a ) ) - - / ) ( a ) , or
which characterizes the global intensity of those singularities. Let us note that the definition of the factorial moments under consideration contains the order q only in the factorial entering the denominator of relation ( 1 ). Expressing this factorial in terms of the gamma function, ( n - q ) ! = l - ' ( n - q + 1 ), relation ( 1 ) allows the consideration of the corresponding factorial moments for arbitrary (real) orders.
f = f ( q ) =fq,
3. To obtain a theoretical expression for Dq, let us start by a regular exact division of the rapidity interval AY (cf. refs. [ 10,11 ] ) where we assume a uniform rapidity density (and let it be equal to unity). In the first step, AY is divided into it paras of equal length, each part being characterized by the rapidity density pj ( j = 1, 2 ..... it), with ~pj= 1. In the next step, each part is again divided into it parts of equal length, each part being characterized by the rapidity density pjp~ ( k = 1, 2 ..... it). Let us continue this Cantor construction n times, where n is defined in such a way that the last, nth, division part length is small compared with the detector resolution. Now, the derivation of D~ follows essentially the lines of ref. [12] with the exception that in our case the Bernoulli multinomial distribution is involved (instead of the binomial one of ref. [ 12 ] ). Our result can be presented in the form
=-(Pl
(13)
if relations (10) and (9) are applied directly in ( 11 ). Let us recall the symmetry property of the function (13)
,7(q)=f(-q) .
(14)
Moreover, in the limit q ~ 1, we obtain
Dq = oQ =fq lnpl + ... + p a l n p a ) / l n 2 ,
(15)
and in the limit I q l - - ' ~ (with p~ >pz...>pa) lira Dq= lim a q = - ( l n p ~ ) / l n 2 , q - . ~z,
q
lim q
*
--
:Jc,
(16)
,oc
Dq=
lim a , ~ = - ( l n p x ) / l n 2 ,
(17)
q~-oo
and lim f ( a q ) = I ql ~ ' ~ o
lim f ( q ) = 0 . Iql
(18)
*o¢
4. With respect to the results presented in refs. [ I 5 ], the most straightforward application of our approach requires the data from refs. [4,5]. Especially we consider the collisions of 200 GeV/nucleon ~60 beams with the ~2C nuclei (where the energy of the projectile fragments and produced particles as mea577
Volume 247, number 4
PHYSICS LETTERS B
20 September 1990
Table 1 Experimentally observed and theoretically calculated slopes evidencing the appearance of the intermitteney, relation (2) for three different nucleus-nucleus collisions. Parameters p. ( -~p ) and P2 ( -= 1- p ) give the probabilities which describe the rapidity flow during the collisions. The numbers in parentheses give the uncertainty in the last digits of the associated values. Collision
b2
b3
h4
bs
b6
Reference
'60+ '-'C
experiment theory, (p=0.572)
0.029(1) 0.029
0.087(3) 0.087
0.171(5) 0.170
0.278(12) 0.274
0.399(25) 0.398
[4] relation (7)
~60+Ag/Br
experiment theory, (p=0.543)
0.010(1) 0.010
0.034(3) 0.032
0.062(5) 0.062
0.106(15) 0.103
0.163(26) 0.153
[51 relation (7)
32S+Ag/Br
cxperiment theory (p=0.524)
0.004( 1) 0.003
0.014( 1) 0.010
0.023(3) 0.020
0.032(5) 0.032
0.040(9) 0.049
[5] relation (7)
sured by the zero-degree calorimeter is in the range 2.2-2.4 TeV and the corresponding p s e u d o r a p i d i t y window is 2 . 4 - 4 . 0 ) [4] and the 200 G e V / n u c l e o n ' 6 0 and n S nuclei with ")TAg/snBr nuclei in nuclear emulsion (with thc pseudorapidity window 0.5-5.5 ) [5]. In this case, a sufficiently accurate description o f the slopes bq ( q = 2 , 3, 4, 5, 6) is obtained by means of the two-scale Cantor set, as is seen in table 1 (2 = 2 and we denote p~ = p , p2 = 1 -p, i.e., p is the only free p a r a m e t e r there). For other types o f collisions, the higher values o f 2, even with some vanishing probabilities p,, cannot be excluded. We note that when solving the problems o f the turbulent energy dissipation, the generalized dimensions Do~ and D _ ~ represent the cxperimcntally accessible quantities and then relations (16) and (17) allow the determination o f the probabilities Pt and P2 = 1 - p ~ [ 12 ]. With the values o f the p a r a m e t e r p given in table 1, thc dependcncc o f the gcneralized dimensions Dq on the order q, relation (9), is seen in fig. 1, and that of the slopes bq o n q, relation (7), in figs. 2 and 3. Except the points where q = 2 , 3, 4, 5, 6, these figures present our predictions involving even non-integer (positive as well as negative) values o f the o r d e r parameter q. The gencralized dimensions are positive and, as is d c m o n s t r a t c d by fig. 1, they decrease with increasing order q; for positive values of q, they increase with the increasing complexity of the colliding objects. The slopes bq might be negative for 0 < q < 1 (fig. 3) and outside o f this region they decrease with the complexity o f the colliding objects. We call the attention o f experimental groups to verify those conclusions, as well as in other cases. 578
1.2
-1551 111
Dq LO
0.8
-6()
-410
-2'0
b q
210
4()
60
Fig. 1. The dependence of the generalized dimensions Dqon the order parameter q, relation (9), with A=2. The corresponding value of the parameter p is given in table 1. The dotted curve corresponds to the '60 + ]~C, the dashed one to the '60 + Ag/Br. and the full one to the ~"S+Ag/Br collisions. Thc dependence o f the strength o f singularities aq on the order p a r a m e t e r q is seen in fig. 4. Like Dq, aq also decreases with increasing q and for positive values o f that order p a r a m e t e r it increases with the complexity of the colliding objects, i.e., collisions of more complex nuclei give rise to bigger discontinuities. Also the density of discontinuities increases with the complexity o f the colliding objects (fig. 5 ). The spectrum o f singularities is presented in the form o f t h e f - a dependence in fig. 6. As is seen therc, for instance, the strength o f singularities is b o u n d e d from below as well as from abovc and the global intensity, relation (8), decrcases with the increasing complexity o f the colliding objects. The density distribution f ( a ) gets its maximal value (equal to unity) at a=Ot~xtr=[ln(plp2)-tn]/ln2 which decreases to unity with increasing complexity o f the colliding objects. 5. In summary, we have presented here some fun-
Volume 247, number 4
PHYSICS LETTERS B
20 September 1990
i I01
/ •.. ".
%%~. 8i
bq
\~\
//
".'.
\
6
/
//
/
.. %•~
//
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Fig. 2. The dependence of the slope bq on the order q, relation ( 7 ) (the notation is the same as in fig. 1 ).
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Fig. 4. T h e dependence o f the strength o f the singularities otq on
o
the order q, relation ( I 0 )
(the notation is the same as in fig. l ).
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1.0
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Fig. 3. A detail of the dependence b,~versus q, seen in fig. 2. 0.2
d a m e n t a l properties o f the scaling indices with respect to the c o m p l e x i t y o f c o l l i d i n g nuclei. T h e experimental input for our c o n c l u s i o n s is rcpresented by the values o f the slope parameter bq. A n y extension o f the presently available range ( w h i c h consists o f five isolated points q = 2 , 3, 4, 5, 6) can specify
-60
-40
-20
0 q
20
40
i
60
Fig. 5. The dependence of the density of singulariticsf(q) on the order parameter q, relation (13) (the notation is the same as in fig. 1).
579
Volume 247, number 4
PHYSICS LETTERS B
1.0
0.8
\
"..
"
"
::
\
•:
f(aq)
t h e i r s u p p o r t and fruitful discussions. S t i m u l a t i n g c o n v e r s a t i o n s with Professor W. Kittel are also highly a c k n o w l e d g e d . T h i s research was s u p p o r t e d by the U n i t e d States D e p a r t m e n t o f Energy, u n d e r G r a n t D E A C 0 2 7 6 ER 03064.
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References
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~
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:-
I
~
°2
-
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04
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-
, I ]. ~ ,,k - 1,1 -
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Fig. 6. The dependence of the density of singularities J(c~,a) on their strength, relation ( 12 ). The a,~ value of the points denoted by circles (squarcs) is given by D~ (D_y), relations (16) and ( 17 ) (the notation is the samc as in fig. 1).
m o r e a d e q u a t e l y the m e c h a n i s m o f the rapidity flow d u r i n g the d e v e l o p m e n t o f the collision process. A s i m i l a r analysis o f the scaling indices d e p e n d i n g on the energy o f the colliding objects will bring a n o t h e r piece o f i m p o r t a n t i n f o r m a t i o n . T h e a u t h o r expresses his gratitude to Profcssor Roy J. G l a u b e r , L y m a n Laboratory, H a r v a r d U n i v e r s i t y , and Professor P. Carruthers, D e p a r t m e n t o f Physics, U n i v e r s i t y o f A r i z o n a ( w h e r e this w o r k s t a r t e d ) for
580
20 September 1990
[ 1 ] TASSO Collab., W. Braunschweig ctal., Phys. Lett. 13 231 (1989)548. [ 2 ] I. Derado. G. Jancso. N. Schmitz and P. Stora. Investigation of intermittency in muon-proton scattering at 280 GeV/c, preprint MPI-PAE/Exp. El. 221 (February 1990). [3] EHS/NA 22 Collab., I.V. Ajinenko et al., Phys. Let(. B 235 (1990) 373. [ 4 ] WA 80 Collab., R. Albrecht et al., Phys. Lett. B 221 (1989) 427. [ 5 ] KLM Collab., R. Holynski et al., One- and two-dimensional analysis of the factorial moments in 200 GeV/nucleon p. ~60 and 3'S interactions with Ag/Br nuclei, KrakowLouisiana-Minnesota preprint. [6] P. Carruthers, H.C. Eggcrs, Q. Gao and 1. Sarcevic. Correlations and intermittency in high energy multihadron distributions, Arizona University preprint AZPH-TH/909 (March 1990). [ 7 ] P. Lipa and B. Buschbcck, Phys. Left. B 223 ( 1989 ) 465. [ 8 ] T.C. Halsey and M.H. Jensen. Physica D 23 ( 1986 ) I 12. [9] T.C. Halsey, M.H. Jcnsen, L.P. Kadanoff, I. Procaccia and B.I. Shraiman. Phys. Rev. A 33 (1986) 1141. [ 10 ] A. Bialas and R. Peschanski. Nucl. Phys. B 273 ( 1986 ) 703. [ 11 ] A. Bialas and R. Peschanski, Nucl. Phys. B 308 ( 1988 ) 857: Phys. Lett. B 207 ( 1988 ) 59. [ 12] C. Meneveau and K.R. Sreenivasan. Phys. Rcv. Len. 59 (1987) 1424.