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Intermittency in branching models
Volume 236, number 4
PHYSICS LETTERS B
1 March 1990
I N T E R M I T T E N C Y IN B R A N C H I N G M O D E L S Charles B. C H I U Centerfor Particle Theory and Department of Physics, Universityof Texas, Austin, TX 78712, USA
and Rudolph C. HWA Institute of Theoretical Science and Department of Physics, Universityof Oregon. Eugene, OR 97403-5203, USA
Received 7 November 1989
The intermittency properties of three branching models have been investigated. The factorial moments show power-lawbehavior as function of small rapidity width. The slopes and energy dependences reveal different characteristics of the models. The gluon model has the weakest intermittency.
Recent experiments on multiparticle production in various collision processes [ 1,2 ] have stirred up considerable interest in the possibility that the underlying dynamics gives rise to the property of self-similarity under the scale transformation of measurement resolution [ 3,4]. It were Bialas and Peschanski [ 5 ] who first suggested the study of factorial moments of multiplicity distributions in decreasing widths of rapidity windows; they referred to a power-law behavior as intermittency, in analogy to similar behavior in turbulent flow and other nonlinear dynamics. There is currently some controversy on whether the observed power-law behavior is indeed the manifestation of a new dynamical mechanism beyond the conventional physics of 1ow-pT particle production. It has recently been shown [6,7] that the present data on factorial moments for hadronic collisions can be fitted by calculations using only the usual short-range rapidity correlations. However, if the power-law behavior is to persist at even narrower width, 5y, of the rapidity window, then the conventional approach would fail and a more specific dynamical mechanism must be invoked. To explore such a mechanism for intermittency in this paper we investigate various branching models for particle production and show the relationship between the nature of branching and the intermittency indices. 466
A cascade model of massive clusters decaying into two smaller clusters at each branching has been shown to exhibit power-law behavior [8 ]. At the partonic level a cascade model [9] based on the dipole approximation of multigluon emission shows hints of some sort of self-similarity, but the power-law behavior does not refer to the rapidities of the final-state hadrons. By means of a multifractal analysis it is shown that the model possesses an unusual scaling behavior, which is a property of the Q C D cascade [10]. In a loose usage of the term intermittency as described in ref. [ 5 ], these models can be said to exhibit intermittent behavior. However, it is not transparent what aspects of the models are responsible for the various properties of intermittency, such as the degree of the power behavior, the range of~y in which there is linearity in the log-log plot, the dependence on energy, etc. In order to understand the origin of those properties, we have made a systematic study of branching models with a range of characteristics of the splitting function P ( z ) , where z is the momentum fraction of a daughter. An analytical description o f branching in the ~3 theory in six dimensions was given earlier [ 11 ]; however, since the coupled integral equations in the formulation were difficult to solve, the existence of intermittency was not proven rigorously, although it was shown to be a possible self-
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
Volume 236, number 4
PHYSICS LETTERSB
consistent property. Here we take the Monte Carlo approach to the problem and generalize P(z) to other possibilities in exploring their consequences on the nature of intermittency. Our purpose in this paper is not to confront data but to examine theoretical issues as a prelude to studying the fractal properties of multiparticle production. For hadronic collisions there is the complication associated with impact-parameter fluctuation, which is outside our scope here. For e+e - annihilation, present data in the 30 GeV range can justify neither the leading log approximation nor our focus on the tree diagrams only. Nevertheless, we believe that it is important to investigate branching models because they serve as a key link between the mathematical problem of fractal structures of geometrical objects and the physical problem of particle production in high-energy collisions. For convenience, we refer to the lines in a tree diagram as partons. We consider only trilinear coupling, so a (mother) parton branches into only two (daughter) partons, all partons being of one species. We work in the infinite-momentum frame with longitudinal momentum of the initial parton being 1, and its daughters having momentum fractions z and 1 - z. The probability for a daughter to have z is specified by the splitting function P(z), which characterizes the branching model. We shall consider three types of P(z): (I) Twin model:
P(z) = ~ ( z - ½),
(1)
(II) 03 model:
P(z)=6z(l-z),
(2)
(III) Gluon model:
P(z)=c
(zl____z+ ~ z
+z(1-z)
).
(3)
In these models the daughter partons at a vertex become increasingly dissimilar in their partition of the mother's momentum, starting from the twin model in which both daughters have identical z, ending with the gluon model in which there is divergence at z = 0 and 1. Eq. (2) is the splitting function for the ~3 theory in six dimensions. We shall use q2 as a generic notation for the vir-
1 March 1990
tuality (four-momentum squared) of an arbitrary parton in the tree, and use q Z and q~ to denote the virtualities of its daughters. The evolution parameter of the branching process is
{ln(q2/AZ)~ t=A In \ln(q2/A2) j ,
(4)
where A is a constant, depending on the model. The branching process continues until the virtuality of a parton reaches q2< q2, at which point the parton is identified as a final-state particle. Given the q2 of the mother parton, the maximum virtualities of the daughters are qZm==Zq2 and q2m.x = ( 1 --z)q 2. Subject to these bounds, kinematically q2 and q2 can have any values, although strictly they are not totally independent of z [ 12 ]. The kinematical boundary near z = 0 and 1 is troublesome to handle, especially in the case of the gluon model, for which ( 3 ) diverges there. Since we want to treat the three models on equal footing using the same algorithm for branching, and since we do not intend to treat QCD branching seriously, we choose to modify (3) slightly by imposing an infrared cut-off, thereby availing ourselves to a very simple procedure for implementing the branching that is common for all three models. For O<~z<~zo and 1-Zo~
(5)
We have now completed the definition of our three 467
Volume 236, number 4
PHYSICS LETTERS B
branching models, except for the specification of A, A and qo in (4). Since our interest is in comparing the three models in their implications on intermittency, we shall adjust those parameters so that all three models have (essentially) the same average multiplicity at the end of each branching process. The values of A, A and qo have no intrinsic significance in our model analyses here; similarly, the value of the initial maximum virtuality, Q, should not be identified seriously with a corresponding quantity in a physical process, such as the CM energy in e+e - annihilation. After all, there is no physical model with twin daughters at each vertex. However, it will be instructive to compare the consequences of these models, if we demand that in all models the extents of branching are the same, at least at one value of Q, leading to the same ( n ) . The values of Q are therefore only convenient labels to denote the extents of branching. We shall also consider the Q dependence of these models, to which we shall loosely refer as the "energy" dependence. Following ref. [ 5], we calculate the factorial moments of multiplicity distributions in narrow rapidity windows, whose width 8y shall be abbreviated by 6. Define Ft(6) =
(k(k-l)...(k-l+l)) (k)t
,
(6)
where k is the number of particles in 6. In this theoretical calculation we can afford the luxury of not making horizontal averages (i.e. averaging over many bins with the same 6), since we are not limited by experimental statistics. Thus we focus our attention at one bin of width 6 situated at y = 0 , and average over all events. This is the (vertical) averaging process indicated in (6), which differs from those considered in ref. [5] and in subsequent experimental analyses [1,2]. Since the three models give widely different y distributions, horizontal averaging combined with vertical averaging will confuse the issues. The subject of horizontal averaging at various scales (coarse and fine) is interesting in its own right, and will be discussed at length elsewhere. In our calculation we work with the m o m e n t u m fraction variables until the branching process is complete. Then for each particle with m o m e n t u m fraction x, the corresponding rapidity is determined by y= sinh-l(xQ/mT). We take the transverse mass to 468
1 March 1990
be rnT=qo/2. Thus for small 6 at y = 0 , we have 6= (2Q/qo)Xo, where Xo is an incremental x interval at x = 0 . A power-law behavior in Xo is therefore also a power-law behavior in 6. We have chosen qo = 2 GeV and adjusted A so that at Q = 5 0 GeV the calculated ( n ) is about 15. For the models I, II, and III, we use A = 0.5, 0.5, and 0.15, and A = 5, 5, and 11.94, respectively, the last number being determined from the usual Q C D formula for four flavors. We have done the calculations for Q = 0 . 0 5 , 0.5, 2, 8, and 20 TeV ~. In each case a power-law behavior
Ft( Q )ac6 -a'~Q)
(7)
has been found. Specifically, for model III a wide range oflinearity exists in In Fl versus In 6 - ~for each of the moments l = 2, ..., 6. In order to establish some relationship among the various results for different Q values, we have made a series of shifts of the horizontal axis and found that all the moments overlap (except the lowest Q case), exhibiting energy independence in the slopes aj. Furthermore, the amount of shift is found to be proportional to the excess of the total rapidity Y over some threshold value lit, where Y = c o s h -~ Q/mT. That is, when l n F t is plotted against In [ ( Y - Yt)/6], as shown in fig. 1, the results for all Q >t 0.5 TeV fall on a universal straight line for each l. The universality is achieved for Yt=6.7. Deviation from the c o m m o n asymptote is expected when 6 gets to be too small, since for every fixed Q the multiplicity k in a bin of width 6 must decrease as 6 is decreased; when k gets below the value of l, Ft must vanish according to (6), resulting in a precipitous drop away from the solid straight line in fig. 1, when 6 is small enough. Nevertheless, before such an effect sets in, the scaling property of the intermittent behavior is striking. In models I and II, the linear portions in the loglog plots are not as extensive as in fig. 1. However, by plotting In Ft versus In [ ( Y - Yt)/6 ] again, but with Yt = 5.4 and 4.5 for models I and II, respectively, the overlap can be achieved in such a manner that a common straight asymptote can still be identified for each l, and a universal slope can be determined giving ~ The average multiplicities at the various Q considered differ somewhat from model to model. For instance, in the gluon model they are 16, 50, 93, 162 and 232.
Volume 236, number 4
O 0.5 2
PHYSICS LETTERS B
S
f.F~
1 March 1990
-
~....*.
4
a, 4
..~~"°
•2
2
4
6
Fig. 2. Slopes of the asymptotes in the log-log plots for the three models: twin (I), ¢~3 (II), and gluon (Ill). The shaded region in I represents a range of possible values of at for each/. I
I
Fig. 1. In Ft in the gluon model (III) plotted against In [ ( Y- Y,) / 6] for various values of Q in TeV. The points are the results of the MC calculation; the straight lines are drawn as best estimates of the universal asymptotes.
scaling indices at. Thus, the situations in models I and II are similar to that shown in fig. 1, except that the deviations from the c o m m o n lines occur sooner, since the linear portions are shorter. We s u m m a r i z e the indices al for all three models in fig. 2. Model I refers to the twin model; the shaded region represents a range o f a/values, since the slopes are m a d e somewhat ambiguous by the shortness o f the linear portions. Evidently, the slopes become greater, the more P(z) is peaked at z = ½. This can be u n d e r s t o o d by recognizing that in the twin model ( I ) all cousins o f the same generation have essentially the same m o m e n t u m fraction. The only fluctuation in x arises from the interaction p r o b a b i l i t y in ( 5 ) , which
affects the q2 values o f the partons and hence the path lengths in the tree before reaching qo2. Thus there are m a n y possible paths that can lead to the same region o f phase space, and therefore the fluctuation in multiplicity depends more strongly on the bin size in rapidity. Conversely, in the gluon m o d e l ( I I I ) the two daughters at a vertex tend to have a large and a small z. Because o f the diverging behavior o f (3) near z-- 0, the y distribution o f the final particles is sharply peaked at y = 0. However, since the m o m e n t u m fraction o f a particle is rather well correlated with the n u m b e r o f vertices involving small-z daughters along a particular path o f a tree, the fluctuation o f possible paths leading to a particular bin is small. Hence, the corresponding intermittency b e h a v i o r is weak. We have considered very high values o f Q, not because o f any delusions that they m a y be physically relevant for a future accelerator, but because we want to learn about the limits o f energy dependence, or independence. We have indeed found that in all three models, even when Q is increased to a value as high as 20 TeV, a universal a s y m p t o t e for each l can be found, thereby defining a Q-independent slope. This 469
Volume 236, number 4
PHYSICS LETTERS B
is therefore a feature which we m a y conclude as characteristic of branching models, irrespective o f the details that differentiate them. In the case o f the gluon model we have arbitrarily chosen an infrared cut-off at Zo=0.0001. To see the dependence on that value, we repeated the calculation with Zo=0.005, and found no essential change in the qualitative features o f the results. Thus the conclusion about the model is unaffected, and that is: the intermittency index at o f the power-law b e h a v i o r for the gluon model is at least an order o f magnitude smaller than that o f either the twin or 03 models. Indeed, the degree o f intermittency is very weak. While at is consistent with becoming zero as l--, 1, its increase with l, for l/> 3, can be well a p p r o x i m a t e d by a/~ 0.04(l-2). The experimental data on e+e - annihilation [ 13 ] reveal power-law behavior with indices that are in the range o f 0.1 for l = 3. Although we do not emphasize any strong connection between our model consideration and the physical process o f particle production, it is nevertheless natural to notice that among the three models considered it is the gluon model that comes closest to a p p r o x i m a t i n g the intermittency indices. F o r hadronic and nuclear collisions the observed power-law behavior is even weaker [ 1 ]. There are at least two factors that further separate the d a t a from our model calculations: one is i m p a c t - p a r a m e ter averaging and the other horizontal averaging. These are some of the issues that we shall investigate in the future, apart from the essential complication that no large virtualities are involved in soft hadronic and nuclear collisions. Nevertheless, a branching model o f the cluster-decay type m a y still be relevant. Indeed, it may be inescapable if the power-law behavior found in the hadronic d a t a persists at higher energy and smaller c~. In conclusion we have studied the intermittency patterns o f three branching models; the results are s u m m a r i z e d in fig. 2, showing the l dependences o f
470
1 March 1990
the intermittency indices. We have found that when Q is higher than some threshold, those indices are Q independent. This is an i m p o r t a n t phase o f our exploration o f the possible d y n a m i c s that can be responsible for the self-similarity seen in the data. As is well known, there is no c o m m o n l y accepted theory for multiparticle production in soft hadronic collisions. The observation o f intermittency in such experiments has caused some excitement in the subject because it suggests a possible signature o f unconventional dynamics. The search for the a p p r o p r i a t e dynamics is an inductive process, in which this work serves as a beginning. This work was supported in part by the US Dep a r t m e n t o f Energy under grant Nos. D E - F G 0 5 85ER-40200 and DE-FG06-85ER-4022.
References
[ 1] W. Kittel, in: Proc. 24th Intern. Conf. on High energy physics (Munich, 1988), eds. R. Kotthaus and J.H. Kuhn (Springer, Berlin, 1989 ) p. 625; invited talk Intern. Europhysics Conf. on High energy physics (Madrid, September 1989 ). [2] See experimental papers in: Leon Van Hove Festschrift (World Scientific, Singapore, 1989). [3] See A. Bialas, and other theoretical papers, in: Leon Van Hove Festschrift (World Scientific, Singapore, 1989). [4] R.C. Hwa, in: Relativistic heavy-ion collisions, eds. R.C. Hwa, C.S. Gao and M.H. Ye (Gordon and Breach, London, 1989). [ 5 ] A. Bialas and R. Peschanski, Nucl. Phys. B 273 ( 1986 ) 703. [ 6 ] A. Capella, K. Fialkowski and A. Krzywicki, BNL preprint (1989). [7] P. Carruthers and I. Sarcevic, preprint AZPH-TH/89-30 (1989). [8] W. Ochs and J. Wosiek, Phys. Lett. B 214 (1988) 617. [ 9 ] B. Andersson, P. Dahlqvist and G. Gustafson, Phys. Lett. B 214 (1988) 604. [10] P. Dahlqvist, B. Andersson and G. Gustafson, preprint LU TP 89-5 (1989). [ 11 ] R.C. Hwa, Nucl. Phys. B 328 (1989) 59. [ 12 ] R. Odorico, Nucl. Phys. B 172 (1980) 157. [ 13 ] B. Buschbeck, R. Lipa and R. Peschanski, Phys. Lett. B 215 (1988) 788.
PHYSICS LETTERS B
1 March 1990
I N T E R M I T T E N C Y IN B R A N C H I N G M O D E L S Charles B. C H I U Centerfor Particle Theory and Department of Physics, Universityof Texas, Austin, TX 78712, USA
and Rudolph C. HWA Institute of Theoretical Science and Department of Physics, Universityof Oregon. Eugene, OR 97403-5203, USA
Received 7 November 1989
The intermittency properties of three branching models have been investigated. The factorial moments show power-lawbehavior as function of small rapidity width. The slopes and energy dependences reveal different characteristics of the models. The gluon model has the weakest intermittency.
Recent experiments on multiparticle production in various collision processes [ 1,2 ] have stirred up considerable interest in the possibility that the underlying dynamics gives rise to the property of self-similarity under the scale transformation of measurement resolution [ 3,4]. It were Bialas and Peschanski [ 5 ] who first suggested the study of factorial moments of multiplicity distributions in decreasing widths of rapidity windows; they referred to a power-law behavior as intermittency, in analogy to similar behavior in turbulent flow and other nonlinear dynamics. There is currently some controversy on whether the observed power-law behavior is indeed the manifestation of a new dynamical mechanism beyond the conventional physics of 1ow-pT particle production. It has recently been shown [6,7] that the present data on factorial moments for hadronic collisions can be fitted by calculations using only the usual short-range rapidity correlations. However, if the power-law behavior is to persist at even narrower width, 5y, of the rapidity window, then the conventional approach would fail and a more specific dynamical mechanism must be invoked. To explore such a mechanism for intermittency in this paper we investigate various branching models for particle production and show the relationship between the nature of branching and the intermittency indices. 466
A cascade model of massive clusters decaying into two smaller clusters at each branching has been shown to exhibit power-law behavior [8 ]. At the partonic level a cascade model [9] based on the dipole approximation of multigluon emission shows hints of some sort of self-similarity, but the power-law behavior does not refer to the rapidities of the final-state hadrons. By means of a multifractal analysis it is shown that the model possesses an unusual scaling behavior, which is a property of the Q C D cascade [10]. In a loose usage of the term intermittency as described in ref. [ 5 ], these models can be said to exhibit intermittent behavior. However, it is not transparent what aspects of the models are responsible for the various properties of intermittency, such as the degree of the power behavior, the range of~y in which there is linearity in the log-log plot, the dependence on energy, etc. In order to understand the origin of those properties, we have made a systematic study of branching models with a range of characteristics of the splitting function P ( z ) , where z is the momentum fraction of a daughter. An analytical description o f branching in the ~3 theory in six dimensions was given earlier [ 11 ]; however, since the coupled integral equations in the formulation were difficult to solve, the existence of intermittency was not proven rigorously, although it was shown to be a possible self-
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
Volume 236, number 4
PHYSICS LETTERSB
consistent property. Here we take the Monte Carlo approach to the problem and generalize P(z) to other possibilities in exploring their consequences on the nature of intermittency. Our purpose in this paper is not to confront data but to examine theoretical issues as a prelude to studying the fractal properties of multiparticle production. For hadronic collisions there is the complication associated with impact-parameter fluctuation, which is outside our scope here. For e+e - annihilation, present data in the 30 GeV range can justify neither the leading log approximation nor our focus on the tree diagrams only. Nevertheless, we believe that it is important to investigate branching models because they serve as a key link between the mathematical problem of fractal structures of geometrical objects and the physical problem of particle production in high-energy collisions. For convenience, we refer to the lines in a tree diagram as partons. We consider only trilinear coupling, so a (mother) parton branches into only two (daughter) partons, all partons being of one species. We work in the infinite-momentum frame with longitudinal momentum of the initial parton being 1, and its daughters having momentum fractions z and 1 - z. The probability for a daughter to have z is specified by the splitting function P(z), which characterizes the branching model. We shall consider three types of P(z): (I) Twin model:
P(z) = ~ ( z - ½),
(1)
(II) 03 model:
P(z)=6z(l-z),
(2)
(III) Gluon model:
P(z)=c
(zl____z+ ~ z
+z(1-z)
).
(3)
In these models the daughter partons at a vertex become increasingly dissimilar in their partition of the mother's momentum, starting from the twin model in which both daughters have identical z, ending with the gluon model in which there is divergence at z = 0 and 1. Eq. (2) is the splitting function for the ~3 theory in six dimensions. We shall use q2 as a generic notation for the vir-
1 March 1990
tuality (four-momentum squared) of an arbitrary parton in the tree, and use q Z and q~ to denote the virtualities of its daughters. The evolution parameter of the branching process is
{ln(q2/AZ)~ t=A In \ln(q2/A2) j ,
(4)
where A is a constant, depending on the model. The branching process continues until the virtuality of a parton reaches q2< q2, at which point the parton is identified as a final-state particle. Given the q2 of the mother parton, the maximum virtualities of the daughters are qZm==Zq2 and q2m.x = ( 1 --z)q 2. Subject to these bounds, kinematically q2 and q2 can have any values, although strictly they are not totally independent of z [ 12 ]. The kinematical boundary near z = 0 and 1 is troublesome to handle, especially in the case of the gluon model, for which ( 3 ) diverges there. Since we want to treat the three models on equal footing using the same algorithm for branching, and since we do not intend to treat QCD branching seriously, we choose to modify (3) slightly by imposing an infrared cut-off, thereby availing ourselves to a very simple procedure for implementing the branching that is common for all three models. For O<~z<~zo and 1-Zo~
(5)
We have now completed the definition of our three 467
Volume 236, number 4
PHYSICS LETTERS B
branching models, except for the specification of A, A and qo in (4). Since our interest is in comparing the three models in their implications on intermittency, we shall adjust those parameters so that all three models have (essentially) the same average multiplicity at the end of each branching process. The values of A, A and qo have no intrinsic significance in our model analyses here; similarly, the value of the initial maximum virtuality, Q, should not be identified seriously with a corresponding quantity in a physical process, such as the CM energy in e+e - annihilation. After all, there is no physical model with twin daughters at each vertex. However, it will be instructive to compare the consequences of these models, if we demand that in all models the extents of branching are the same, at least at one value of Q, leading to the same ( n ) . The values of Q are therefore only convenient labels to denote the extents of branching. We shall also consider the Q dependence of these models, to which we shall loosely refer as the "energy" dependence. Following ref. [ 5], we calculate the factorial moments of multiplicity distributions in narrow rapidity windows, whose width 8y shall be abbreviated by 6. Define Ft(6) =
(k(k-l)...(k-l+l)) (k)t
,
(6)
where k is the number of particles in 6. In this theoretical calculation we can afford the luxury of not making horizontal averages (i.e. averaging over many bins with the same 6), since we are not limited by experimental statistics. Thus we focus our attention at one bin of width 6 situated at y = 0 , and average over all events. This is the (vertical) averaging process indicated in (6), which differs from those considered in ref. [5] and in subsequent experimental analyses [1,2]. Since the three models give widely different y distributions, horizontal averaging combined with vertical averaging will confuse the issues. The subject of horizontal averaging at various scales (coarse and fine) is interesting in its own right, and will be discussed at length elsewhere. In our calculation we work with the m o m e n t u m fraction variables until the branching process is complete. Then for each particle with m o m e n t u m fraction x, the corresponding rapidity is determined by y= sinh-l(xQ/mT). We take the transverse mass to 468
1 March 1990
be rnT=qo/2. Thus for small 6 at y = 0 , we have 6= (2Q/qo)Xo, where Xo is an incremental x interval at x = 0 . A power-law behavior in Xo is therefore also a power-law behavior in 6. We have chosen qo = 2 GeV and adjusted A so that at Q = 5 0 GeV the calculated ( n ) is about 15. For the models I, II, and III, we use A = 0.5, 0.5, and 0.15, and A = 5, 5, and 11.94, respectively, the last number being determined from the usual Q C D formula for four flavors. We have done the calculations for Q = 0 . 0 5 , 0.5, 2, 8, and 20 TeV ~. In each case a power-law behavior
Ft( Q )ac6 -a'~Q)
(7)
has been found. Specifically, for model III a wide range oflinearity exists in In Fl versus In 6 - ~for each of the moments l = 2, ..., 6. In order to establish some relationship among the various results for different Q values, we have made a series of shifts of the horizontal axis and found that all the moments overlap (except the lowest Q case), exhibiting energy independence in the slopes aj. Furthermore, the amount of shift is found to be proportional to the excess of the total rapidity Y over some threshold value lit, where Y = c o s h -~ Q/mT. That is, when l n F t is plotted against In [ ( Y - Yt)/6], as shown in fig. 1, the results for all Q >t 0.5 TeV fall on a universal straight line for each l. The universality is achieved for Yt=6.7. Deviation from the c o m m o n asymptote is expected when 6 gets to be too small, since for every fixed Q the multiplicity k in a bin of width 6 must decrease as 6 is decreased; when k gets below the value of l, Ft must vanish according to (6), resulting in a precipitous drop away from the solid straight line in fig. 1, when 6 is small enough. Nevertheless, before such an effect sets in, the scaling property of the intermittent behavior is striking. In models I and II, the linear portions in the loglog plots are not as extensive as in fig. 1. However, by plotting In Ft versus In [ ( Y - Yt)/6 ] again, but with Yt = 5.4 and 4.5 for models I and II, respectively, the overlap can be achieved in such a manner that a common straight asymptote can still be identified for each l, and a universal slope can be determined giving ~ The average multiplicities at the various Q considered differ somewhat from model to model. For instance, in the gluon model they are 16, 50, 93, 162 and 232.
Volume 236, number 4
O 0.5 2
PHYSICS LETTERS B
S
f.F~
1 March 1990
-
~....*.
4
a, 4
..~~"°
•2
2
4
6
Fig. 2. Slopes of the asymptotes in the log-log plots for the three models: twin (I), ¢~3 (II), and gluon (Ill). The shaded region in I represents a range of possible values of at for each/. I
I
Fig. 1. In Ft in the gluon model (III) plotted against In [ ( Y- Y,) / 6] for various values of Q in TeV. The points are the results of the MC calculation; the straight lines are drawn as best estimates of the universal asymptotes.
scaling indices at. Thus, the situations in models I and II are similar to that shown in fig. 1, except that the deviations from the c o m m o n lines occur sooner, since the linear portions are shorter. We s u m m a r i z e the indices al for all three models in fig. 2. Model I refers to the twin model; the shaded region represents a range o f a/values, since the slopes are m a d e somewhat ambiguous by the shortness o f the linear portions. Evidently, the slopes become greater, the more P(z) is peaked at z = ½. This can be u n d e r s t o o d by recognizing that in the twin model ( I ) all cousins o f the same generation have essentially the same m o m e n t u m fraction. The only fluctuation in x arises from the interaction p r o b a b i l i t y in ( 5 ) , which
affects the q2 values o f the partons and hence the path lengths in the tree before reaching qo2. Thus there are m a n y possible paths that can lead to the same region o f phase space, and therefore the fluctuation in multiplicity depends more strongly on the bin size in rapidity. Conversely, in the gluon m o d e l ( I I I ) the two daughters at a vertex tend to have a large and a small z. Because o f the diverging behavior o f (3) near z-- 0, the y distribution o f the final particles is sharply peaked at y = 0. However, since the m o m e n t u m fraction o f a particle is rather well correlated with the n u m b e r o f vertices involving small-z daughters along a particular path o f a tree, the fluctuation o f possible paths leading to a particular bin is small. Hence, the corresponding intermittency b e h a v i o r is weak. We have considered very high values o f Q, not because o f any delusions that they m a y be physically relevant for a future accelerator, but because we want to learn about the limits o f energy dependence, or independence. We have indeed found that in all three models, even when Q is increased to a value as high as 20 TeV, a universal a s y m p t o t e for each l can be found, thereby defining a Q-independent slope. This 469
Volume 236, number 4
PHYSICS LETTERS B
is therefore a feature which we m a y conclude as characteristic of branching models, irrespective o f the details that differentiate them. In the case o f the gluon model we have arbitrarily chosen an infrared cut-off at Zo=0.0001. To see the dependence on that value, we repeated the calculation with Zo=0.005, and found no essential change in the qualitative features o f the results. Thus the conclusion about the model is unaffected, and that is: the intermittency index at o f the power-law b e h a v i o r for the gluon model is at least an order o f magnitude smaller than that o f either the twin or 03 models. Indeed, the degree o f intermittency is very weak. While at is consistent with becoming zero as l--, 1, its increase with l, for l/> 3, can be well a p p r o x i m a t e d by a/~ 0.04(l-2). The experimental data on e+e - annihilation [ 13 ] reveal power-law behavior with indices that are in the range o f 0.1 for l = 3. Although we do not emphasize any strong connection between our model consideration and the physical process o f particle production, it is nevertheless natural to notice that among the three models considered it is the gluon model that comes closest to a p p r o x i m a t i n g the intermittency indices. F o r hadronic and nuclear collisions the observed power-law behavior is even weaker [ 1 ]. There are at least two factors that further separate the d a t a from our model calculations: one is i m p a c t - p a r a m e ter averaging and the other horizontal averaging. These are some of the issues that we shall investigate in the future, apart from the essential complication that no large virtualities are involved in soft hadronic and nuclear collisions. Nevertheless, a branching model o f the cluster-decay type m a y still be relevant. Indeed, it may be inescapable if the power-law behavior found in the hadronic d a t a persists at higher energy and smaller c~. In conclusion we have studied the intermittency patterns o f three branching models; the results are s u m m a r i z e d in fig. 2, showing the l dependences o f
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the intermittency indices. We have found that when Q is higher than some threshold, those indices are Q independent. This is an i m p o r t a n t phase o f our exploration o f the possible d y n a m i c s that can be responsible for the self-similarity seen in the data. As is well known, there is no c o m m o n l y accepted theory for multiparticle production in soft hadronic collisions. The observation o f intermittency in such experiments has caused some excitement in the subject because it suggests a possible signature o f unconventional dynamics. The search for the a p p r o p r i a t e dynamics is an inductive process, in which this work serves as a beginning. This work was supported in part by the US Dep a r t m e n t o f Energy under grant Nos. D E - F G 0 5 85ER-40200 and DE-FG06-85ER-4022.
References
[ 1] W. Kittel, in: Proc. 24th Intern. Conf. on High energy physics (Munich, 1988), eds. R. Kotthaus and J.H. Kuhn (Springer, Berlin, 1989 ) p. 625; invited talk Intern. Europhysics Conf. on High energy physics (Madrid, September 1989 ). [2] See experimental papers in: Leon Van Hove Festschrift (World Scientific, Singapore, 1989). [3] See A. Bialas, and other theoretical papers, in: Leon Van Hove Festschrift (World Scientific, Singapore, 1989). [4] R.C. Hwa, in: Relativistic heavy-ion collisions, eds. R.C. Hwa, C.S. Gao and M.H. Ye (Gordon and Breach, London, 1989). [ 5 ] A. Bialas and R. Peschanski, Nucl. Phys. B 273 ( 1986 ) 703. [ 6 ] A. Capella, K. Fialkowski and A. Krzywicki, BNL preprint (1989). [7] P. Carruthers and I. Sarcevic, preprint AZPH-TH/89-30 (1989). [8] W. Ochs and J. Wosiek, Phys. Lett. B 214 (1988) 617. [ 9 ] B. Andersson, P. Dahlqvist and G. Gustafson, Phys. Lett. B 214 (1988) 604. [10] P. Dahlqvist, B. Andersson and G. Gustafson, preprint LU TP 89-5 (1989). [ 11 ] R.C. Hwa, Nucl. Phys. B 328 (1989) 59. [ 12 ] R. Odorico, Nucl. Phys. B 172 (1980) 157. [ 13 ] B. Buschbeck, R. Lipa and R. Peschanski, Phys. Lett. B 215 (1988) 788.