A GEOMETRICAL DETAILS

Scaling Behavior of the Azimuthal and Centrality Dependencies of Jet Production in Heavy-ion Collisions

Abstract

For heavy-ion collisions at RHIC a scaling behavior is found in the dependencies on azimuthal angle and impact parameter for pion production at high essentially independent of the hadronization process. The scaling variable is in terms of a dynamical path length that takes into account detailed properties of geometry, medium density and probability of hard scattering. It is shown in the recombination model how the nuclear modification factor depends on the average . The data for production at 4-5 and 7-8 GeV/c at RHIC are shown to exhibit the same scaling behavior as found in the model calculation. Extension to back-to-back dijet production has been carried out, showing the existence of scaling also in the away-side yield per trigger. At LHC the hard-parton density can be high enough to realize the likelihood of recombination of shower partons arising from neighboring jets. It is shown that such 2-jet recombination can cause strong violation of scaling. Furthermore, the large value of that exceeds 1 can become a striking signature of such a hadronization process at high energy.

pacs:
25.75.-q, 25.75.Gz, 24.85.+p

I Introduction

Recent analysis of the high-statistics data on production in heavy-ion collisions has provided a wealth of information on the nuclear modification factor () as a function of centrality (), azimuthal angle (), and transverse momentum () (1). The complexity revealed is clearly in need of some organizational simplification, if useful insight is to be gained from it to advance a theoretical understanding of the various features. For example, the measured ranges from 1 to 10 GeV/c; the conventional wisdom is to separate the high and low regions and treat them with different dynamics. The dependence is studied by determining the second harmonics (), whose dependencies on and are presented as if they are variables orthogonal to , although it is known that and are correlated by geometry and depends on path length. On the theory side treatments of fundamental issues at the microscopic level cannot incorporate experimental details without sacrificing clarity and precision. In the interest of finding simplifying features of the data we examine in detail the geometrical aspect of the problem associated with the propagation of a hard parton through the medium and explore the possibility of scaling behavior in a variable, , that involves both impact parameter () and . On the basis of that result we then organize the dynamical problem of hadronization in a way that can exhibit the scaling behavior. With that behavior derived on a theoretical basis in certain approximation, the data on can then be shown in the same format to check the relevance of the model-dependent finding. A form of universality does exist in reality.

At low where thermal and semihard partons dominate, a study of the surface factor that relates and has led to the elucidation of the dependencies of and ridge yield (2). At higher shower partons become more important (3). Accordingly, the path length of a hard parton in the dense medium becomes the major factor that affects . That length depends on , and the point of creation of the hard parton, the probability of which depends on the nuclear overlap function. All those quantities are calculable from geometrical considerations. Of course, at some point dynamics enters the picture. Recombination can handle well the late-stage hadronization process at all (3). Our primary objective is to expose the simplifying features of the nuclear complexity that a hard parton experiences at the early stage of the production process. It is complementary to the study of the jet quenching process in QCD that assumes a simple path length (4); (5). The two meet at the point where momentum degradation is expressed in terms of the scaling variable . The connection is complicated by the role that thermal partons play in the hadronization of shower partons at the intermediate range. Nevertheless, a simple behavior of in terms of can be determined theoretically and experimentally.

If indeed a universal behavior of can be found in its dependence on a dynamical path length, then it should be natural to follow up the investigation with a study of the properties of back-to-back dijets, since the dependence of the yield on the momenta of the trigger and away-side particles ( and , respectively) is closely related to the path lengths traveled by the hard parton and its recoil. A detailed study of the dependence on and has been carried out earlier, but with averaged over (6). Now, with specific values taken into account, the joint dependence on and presents complications that must be simplified, if useful comparison between theory and experiment is to be facilitated.

Dijets mentioned above are directed in opposite directions, but two jets going in the same direction would be of tremendous importance when collision energy is very high as at LHC. Pion production arising from 2-jet recombination should overwhelm 1-jet fragmentation when the rate of hard scattering is significantly increased at high initial energy. Our procedure of studying the and dependence can readily be extended to the coalescence of shower partons from parallel jets. Our result shows the possibility of dramatic departure from the usual expectation on the nuclear modification factor.

In this work we shall focus on the production of pions, since the high-statistics data (1) that can check our analysis are for only. The extension to the production of protons and other hadrons can be done in the same framework, but will not be carried out here.

Ii Inclusive pion DISTRIBUTION AT FIXED AZIMUTHAL ANGLE

We shall work in a range where thermal-thermal recombination is a small correction to what we shall calculate. That would be for GeV/c at GeV. It does not mean that thermal partons are unimportant because their recombination with shower partons (the TS component) can be dominant (for production). Both TS and SS components are proportional to the probability of hard-parton production, so we may write the basic structure of the single-particle distribution in nuclear collisions in the form

(1)

where is the probability that a parton of species with momentum at azimulthal angle (relative to the reaction plane, or more precisely relative to the minor axis of the overlap almond in our theoretical consideration) emerges at the surface of medium. We consider only the transverse plane at mid-rapidity. describes the hadronization of parton with momentum to a hadron of whatever type under consideration with momentum . In the recombination model the medium effect on hadronization is taken into account by the TS component in in addition to the fragmentation term that is represented by the SS component (3); (7). Such details are not relevant to our concern here. The main point in Eq. (1) is that the dependence on and are in . The thermal partons can have a minor dependence of those variables, and is an issue that will be discussed later in this paper.

The essence of our problem is the effect of the medium on the hard parton from creation to its emergence on the surface. For every hard parton created with momentum the probability of its emergence with momentum may be denoted by so that we have

(2)

where is the distribution for parton momentum at the creation point and has been parametrized in Ref. (8). The momentum degradation factor strictly should depend on the parton type, but we make the approximation of ignoring that dependence, since our focus is on the path length dependence. To show explicitly how path length enters the problem, we must exhibit the creation point at , which is to be integrated over the region of nuclear overlap, weighted by the probability of a hard collision. That is, we write

(3)

where is the overlap function to be detailed below, and is the momentum degradation factor that changes the parton momentum from to in a length . That length depends on the creation point , the angle that the hard parton is directed, and the exit point for a nuclear collision at impact parameter . Again, we defer the calculation of until later so that our description of the general scheme is not interrupted.

Putting Eqs. (2) and (3) in (1), we can break up the long equation into four parts by introducing , and write

(4)
(5)
(6)
(7)

A parameter is introduced in Eq. (4) to represent the dynamical effect of energy loss on the variable that we call the dynamical path length. If there is no energy loss, then the medium is transparent and whatever the kinematical path length may be. The value of will be determined by phenomenology later, after the form of momentum degradation, specified by , is discussed. The equations from (4) to (7) exhibit the different parts of the production process separately: (4) shows the probability of having a dynamical path length for a parton directed at ; (5) gives the distribution of a parton with momentum at the surface at angle ; (6) is the corresponding distribution after traversing an absorptive distance in the medium; (7) is the final hadron distribution at and . A point to note is that the relationship between and that depends on the initial configuration is completely contained in Eq. (4).

We now define the quantities introduced earlier in Eq. (4) in the framework of the Glauber model for collisions. The thickness function normalized to is

(8)

where is the nuclear density normalized to 1

(9)

with fm and fm for Au. We shall use fm for the effective nuclear radius. For any point in the transverse plane, we have

(10)
(11)

where all lengths are scaled to . The longitudinal lengths of and at are

(12)

where . With being the inelastic nucleon-nucleon cross section, and are related by (10)

(13)

for . The probability for producing a hard parton at is proportional to , where are the Cartesian coordinates of , so the normalized is

(14)

where is the thickness function of nucleus whose center is at an impact parameter on the axis from that of , located at .

The path length from the point of creation to the exit point on the boundary is

(15)

where the integrand is weighted by the local density along the trajectory marked by , since energy loss is proportional to the medium density, and our aim here is to calculate the geometrical part of the factors that contribute to momentum degradation. Apart from an overall normalizaation constant, that density inside the almond-shaped overlap region is given by

(16)

The normalization is not of concern here because appears only in conjunction with in Eq. (4), where is an adjustable parameter to be fixed later in fitting the data.

The exit point can be determined in terms of and . The initial point is inside the overlap region, where is non-vanishing. The exit point is somewhere on the elliptical boundary, since some time elapses before the hard parton reaches the surface. There is no rigorous way to calculate the transition from the almond region to the ellipse, since hydrodynamics is not valid before local equilibrium is established; besides, semihard partons and the ridge formation that they give rise to can significantly change the boundary, unaccounted for by collective flow (2); (9); (10). We determine and the local density along the trajectory in the following way. First, we take the boundary of the ellipse to satisfy

(17)

where

(18)

independent of the transit time from to . We map the density for the almond region to the density of the elliptical region by the identification

(19)

along any radial direction measured from the origin where . The distances from the origin to the almond and elliptical boundaries are denoted by and , respectively, and described in the Appendix. The ratio is very nearly 1 for almost all angles, since and are the - and -axis intercepts of both boundaries. We take the trouble to do the mapping in Eq. (19) for the short duration before thermal equilibrium is established in order to render Eq. (15) well defined, where the upper limit is at the boundary of the ellipse. If the point is far from the boundary, the transit time for the hard parton to arrive at the medium boundary can be large so the ellipse would be larger than that described by Eq. (17). For large transit time the dynamics of medium expansion predominantly in the longitudinal direction should be taken into consideration. In that case we rely on the dynamical scaling behavior found in Ref. (11) to regard our procedure as being insensitive to that expansion. Specifically, it means that if the RHS of Eq. (17) were larger, would be larger, and the rescaling of would redistribute the density accordingly and the net result of the integration in Eq. (15) would not deviate by too much. in Eq. (19) is identified with the density given in (16) for every point inside the almond. By restricting Eq. (15) to only the short time duration when (19) is valid, it is much easier to calculate the exit point , where the hard parton trajectory along intersects the ellipse, i.e.,

(20)

where can be determined by solving Eq. (17), as done in the Appendix. That is what we use as the upper limit of the integration in Eq. (15), thereby giving dependence on , , and to the length . The result gives an estimate of the path length that may possess more general validity than what the static approximation of ignoring explicit transverse expansion may imply. In this way , as defined in Eq. (4), is completely calculable from geometrical considerations.

Iii Scaling Behavior

We now focus on and try to extract as much information as possible about its dependence on and before proceeding to the dynamical problem of momentum degradation and particle production. From Eqs. (4) and (14) is properly normalized as

(21)

so the mean is

(22)

Since centrality is more directly accessible in experiments, we shall freely change from the dependence on to that on , which is our symbol for centrality; for example, stands for 0-10% centrality. The relationship between and is well established, and is tabulated in Ref. (12), for instance.

Using Eq. (4) we obtain

(23)

In the next section we shall show that . For our general discussion here need not be specified, although the numerical value will be used when plots are presented in figures. We show in Fig. 1 the dependence of on for six values of for AuAu collisions at GeV. Except for a shift in the magnitude, that dependence seems to be insensitive to for ranging from 0.05 to 0.55, which corresponds to from 0.48 to 1.58 in units of . It suggests that there is a universality in the dependence for . That universality cannot be exact, since we know that when there can be no dependence on . However, for approximate universality seems valid in Fig. 1. If so, then the question becomes whether the probability distribution that is a function of three variables can more economically be expressed in terms of few variables.

Figure 1: Average dynamical path length vs for six values of centrality.

Our first step in addressing that question is to ask whether, at fixed , the distribution is a scaling function. More specifically, we define

(24)

and ask whether can be written in terms of a scaling function in the form

(25)

such that

(26)

Distributions that have such properties are referred to as satisfying KNO scaling, well-known in multiparticle production (13); (14). We show in Fig. 2 the behavior of for (a) and (b) ; in each case there are six values of : We use for classification, instead of , because the data to be shown in the next section are given in terms of centrality. What is notable is that in each case all dependencies on collapse to one universal curve in terms of , when expressed in the format of Eq. (25). That is scaling in . The next question is whether there is scaling in .

Figure 2: (Color online) Scaling function for six values of at (a) 0-10% centrality and (b) 50-60% centrality. The red lines are fits of the two figures, parametrized by Eq. (28) and (29); the green line in (b) is a reproduction of the red line in (a).

To quantify the dependence on , we fit by a 2-parameter formula

(27)

where is the Beta function. The best fits yield

(28)
(29)

They are shown by the (red) dashed lines in Fig. 2(a) and (b). The (green) dashed line in Fig. 2(b) is a reproduction of the (red) dashed line in Fig. 2(a) for the purpose of comparison. Visually, the two scaling curves do not differ by too much, although numerically the values of and in Eqs. (28) and (29) are noticeably different between the two cases. To a 10% accuracy, one may regard to be universal for ranging from 0.05 to 0.55 and from 0 to . That offers a remarkable degree of simplicity in the complex geometrical problem of nuclear collisions. In words it means that depends only on and , not on and separately.

Iv Nuclear Modification Factor

Having found some general properties of the collision system that are basically geometrical, we now proceed to the problem of hadron production and see how the geometrical insight gained in the preceding section can help us organize the dependence on and . As a consequence we shall find an efficient way to compare our result with the data on .

Let us start by inserting some details that are omitted in Sec. 2. Since the PHENIX data (1) on are for production, we review the formalism for calculating the single-pion inclusive distribution (3). We restrict our consideration to midrapidity and write the invariant distribution in the recombination model for a pion in the direction in the 1-dimensional form as

(30)

where the distribution is in general

(31)

and the recombination function (RF)

(32)

To reproduce the exponential behavior of the observed distribution at low by TT recombination, we have assumed the thermal parton distribution to have the form

(33)

so that the thermal pion distribution is

(34)

For the shower parton distribution we have

(35)

where is the matrix of shower parton determined from the fragmentation functions (7), and is the hard parton distribution at the medium surface before fragmentation. The TS contribution to the final pion distribution is

(36)
(37)

Finally, the SS component that is equivalent to the fragmentation component is

(38)

where is the fragmentation function. When these equations are compared to Eq. (1), it should be clear what in the latter stands for.

The above is a summary of the basic formulas when if averaged over. Now, as we consider and dependencies explicitly, let us revisit the recombination process. A quark and an anitquark with momenta and are unlikely to recombine to form a pion, if and are not approximately parallel, since their relative momentum normal to the pion momentum should not be larger than the inverse size of the pion. Thus we shall continue to use the 1-dimensional description of the RF. However, the distributions of the coalescing partons should be taken into account.

For the thermal partons a description is given in Ref. (2) that can simultaneously reproduce three empirical facts, which are: (a) at GeV/c (1), (b) (1), and (c) ridge yield (15). The physics developed there involves a consideration of ridge formation with or without triggers. We omit a description of that here, since the details are not of crucial importance to our calculation below. We simply adopt the thermal parton distribution in the form

(39)

where a small -dependent term is included in the square brackets that represents the effect of semihard scattering. is as defined in Eq. (16) with being evaluated at 0.17 from the boundary of the almond overlap on the axis. is a surface factor defined by

(40)

where is the elliptic integral of the second kind, and . It causes a small enhancement of the thermal partons due to semihard partons with GeV and in Eq. (39). is given in terms of as

(41)

These complications are included here for completeness, even though their effect on the final result is small.

The shower partons, on the other hand, depend crucially on , as a comparison between Eqs. (7) and (35) reveals the presence of in in the former, which is the crux of the problem at hand. In Eq. (2) the distribution of a parton with momentum at the medium surface is related to the parton distribution at the point of creation whose dependence on is known (8). The relationship between and is complicated in view of Eqs. (2) and (3), but has been simplified to Eq. (6) by the introduction of . In terms of , the degradation of momentum from to can be written as a simple exponential for GeV/c, i.e.,

(42)

whose justification based on an approximation of QCD result is given in Ref. (6). Using this in Eq. (6) yields the simple connection

(43)

With this substituted into Eqs. (36) and (38) in place of , we obtain the TS and SS contributions to the pion distribution as functions of and

(44)

where is as noted after Eq. (38). To determine the dependence on and , it is necessary to make one last transform according to Eq. (5), using what we have learned about . It should now be clear that whereas Eqs. (43) and (44) describe the dynamical processes of hard parton scattering, momentum degradation and hadronization, the geometrical complication is contained in . The validity of this separation remains to be verified by comparison with data in a way that can best reveal the role of .

We calculate the nuclear modification factor

(45)

where is the average number of binary collisions and is the pion distribution in collisions. For AuAu collisions we include the TT component, although it is small compared to the TS and SS components for GeV/c. In Ref. (1) good quality data are given for GeV/c, six bins of and five bins of . At higher the errors are larger. We calculate the same quantities for and 7.5 GeV/c, and for similar ranges of and . An important step we take is to present our results as functions of .

Figure 3: Theoretical result showing scaling behavior of at GeV/c for six values of centrality. The points for different values between 0 and have the same symbol when the centralities are the same, with increasing from left to right with . The solid line is a fit by Eq. (47) and the dashed line is an exponential fit by .

Since for each pair of values of and , we can calculate and , they can therefore be plotted one versus the other, as in Fig. 3. We have done this for six values of at , and six values of at , with fixed at 4.5 GeV/c. Points for various with the same have the same symbol; they are spread out from left to right with increasing values of . All 36 points lie remarkably well on one universal curve. It means that different experimental conditions on and yield the same value of so long as their values of are the same. The nature of the behavior of depends on the details of the dynamics. It can be approximated by an exponential

(46)

as shown by the dashed straight line in Fig. 3, although a better fit is

(47)

shown by the solid line. That is a simple formula for a wide range of and .

Figure 4: Same as for Fig. 3 but for GeV/c. The solid line is a fit by Eq. (48)

Similar calculation can be done for GeV/c. Fragmentation dominates at such a high . We include all contributions to the hadronization with the result shown in Fig. 4. Again, good scaling is found for all and . The solid line shows a fit of the dependence by

(48)

A comparison with Eq. (47) indicates that there is no significant change in as is increased from 4.5 to 7.5 GeV/c. Average values of the two sets of parameters can yield acceptable fits of all points calculated. Thus we learn that there is an approximately exponential suppression of in terms of the path length for all and all centrality , for GeV/c, bearing in mind that is no greater than 0.65. The degree of suppression is no more than a factor of 5, which is a familiar fact. In Figs. 3 and 4, we see clearly how that suppression is related to one another at different and .

To compare our calculations with the data we have one basic parameter to vary, which is in Eq. (4) that relates to , defined in Eq. (15). That integral for is along the path of the hard-parton trajectory weighted by the local density , which is proportional to the longitudinal length at . With the normalization factor being absorbed by , itself has the characteristic of length, and behaves as inverse length, though all length are in units of . The dimensionless is the dynamical path length, in terms of which the momentum degradation can be expressed as . Our formalism does not provide a way to calculate , but it does account for the geometrical details. In QCD energy loss can be calculated, but without the details of the medium. If the scaling behavior that we have found can also be found in the data, then the value of can be extracted, giving justification to our procedure of condensing all the geometrical complications into .

Figure 5: vs for 3 values of at =4-5 GeV/c. The data are from Ref. (1). The lines are from calculation in the recombination model.

In Fig. 5 we show the data from Ref. (1) on vs at GeV/c for 3 values of . We adjust the parameter to fit the data and obtain

(49)

There is another parameter that has not been mentioned so far. It is the lower limit of the integral in Eq. (15). It can differ slightly form to take into account the absence of immediate effect of energy loss at the very beginning of a hard-parton trajectory. A non-zero shortens the effective path length and can increase at small and small . A better fit of the data in that region is achieved by using , which corresponds to less than 0.2 fm, a value small enough so that it does not play a significant role in the general description of the problem. All figures shown above are results of calculations done with and given these values. Another way of presenting the results is to plot versus for different centralities. That is done in Fig. 6, showing good agreement with data (1).

Figure 6: vs for 5 bins of centrality at =4-5 GeV/c. The data are from Ref. (1). The lines are from calculation in the recombination model.

As we can see from these two figures, the data plotted in terms of and vary over wide ranges of values, exhibiting no organized simplicity. It is therefore important to replot versus to check the scaling behavior found in our theoretical study. Fig. 7 shows the data replotted in that format; the solid line is a direct transfer from the solid line in Fig. 3 that fits the theoretical points by Eq. (47). It is a way to show the relationship between the two figures, but in actuality the theoretical and experimental points agree with one another better than how the analytic formula can represent them. Given the experimental errors, the scaling behavior is evidently in the data.

Figure 7: Experimental data on for =4-5 GeV/c from Ref. (1) are plotted against the scaling variable . The values are the same as described in Fig. 3. The solid line is the same as the solid line in Fig. 3 that represents the result from theoretical calculation.
Figure 8: Experimental data on for =7-8 GeV/c from Ref. (1) are plotted against the scaling variable . The values are the same as described in Fig. 3. The solid line is the same as the solid line in Fig. 4 from theoretical calculation.

For in the 7-8 GeV/c range we show the data only in the scaling format, as in Fig. 8. The solid line is a reproduction of the solid line in Fig. 4, described by Eq. (48). Within errors which are large, the agreement between theory and experiment is good in both the magnitude and the dependence. No parameters have been adjusted. Thus we learn not only that scaling is robust, but also that the dependence on changes with mainly because the SS component of hadronization becomes most dominant at higher , since no other factor besides in Eq. (44) depends on . The medium effect is primarily contained in .

It should be mentioned that the path length dependence has also been investigated in Ref. (1). Scaling behavior has been found in different variables for different ranges. Their definition of in fm is closest in spirit to our dimensionless . The linear dependence of on the path length and has also been checked experimentally in Refs. (17); (18), as predicted in QCD (5). Since our is far more detailed than and our range includes the contribution from thermal-shower recombination, our result cannot be related simply to any QCD formula.

It is interesting to show how the dependence can be obtained in a more direct and transparent way, when an approximation is made at high . The hard-parton distribution has been parametrized in Ref. (8) in the form

(50)

where for GeV/c and . For simplicity, we approximate the formula by a simple power law for GeV/c

(51)

where and The advantage is that becomes, from Eq. (43),

(52)

which renders the and dependencies separable. We can then perform the integration in Eq. (44) independent of , and obtain the gluon and u-quark contributions separately as

(53)

The scaling property of , as described by Eq. (25), can now be used to yield

(54)

which can simply be evaluated in conjunction with Eq. (27). The result is shown in Fig. 9 for gluon in (green) solid line and for quark in (red) dashed line, the normalizations for which are freely adjusted. It should be noted that the curve in Fig. 7 is from complete calculation taking the contributions from all partons into account without using explicitly the scaling property of , while the two curves in Fig. 9 are obtained through the approximation of Eq. (51) so as to exhibit directly the roles of (a) the parton distribution function , (b) the exponential relationship between and , and (c) the scaling function . All three of these properties are the main content of the integrand in Eq. (54) that shows clearly the three basic aspects of the physics that give rise to the observed behavior of .

Figure 9: (Color online) The same data as shown in Fig. 8 for =7-8 GeV/c. The (green) solid line is for gluon fragmentation only and the (red) dashed line is for -quark fragmentation; the normalizations for both contributions are arbitrarily adjusted so as to show the dependence compared to data.

It is of interest to point out the implication of the two factors in the integrand of Eq. (53), or (54). The factor is basically an expression of the hard-parton distribution in terms of , indicating that the probability of a hard parton contributing at large to an observed pion is exponentially suppressed. gives the probability of creating a hard parton at from the surface. Their product, expressed as in Eq. (54), shows that, although the mean is 1, the small portion dominates due to higher suppression at large . It means that most of the partons that contribute to pion production are created near the surface. If the result of the integration over can be approximated by with , we see that the dependence of on is roughly the exponential behavior given by Eq. (46).

V Back-to-Back Dijets

Having seen how the dynamical path length plays a crucial role in determining the general behavior of , it is natural to extend the study to consider back-to-back dijets, whose properties clearly depend on the distances that the two generating hard partons traverse in the medium. That problem has been studied previously in Ref. (6) without taking into account azimuthal dependence explicitly. We now can investigate the effect of scaling and find a more transparent way to show the dependence of away-side yield per trigger on and .

Using the same notation as in [4] where and denote the momenta of the trigger and the associated particle on the away side, respectively, at mid-rapidity, we have by simple generalization from Eq. (1)

(55)

where is the parton momentum on the near-side surface, and on the away-side surface. They are related to the momentum at the creation point by Eq. (4) is now generalized to

(56)

where is as defined in Eq. (15), and is defined similarly but in the opposite direction with the upper limit of integration replaced by

(57)

In our calculation below we shall set the lower limit of both integrals at , ignoring the small mentioned after Eq. (49). As in Eqs. (5) and (6) we have

(58)
(59)

Computation of can now be performed from the equations given above without questioning the scaling properties. However, we have learned from the preceding section that considerable clarity can be gained by presenting the result in terms of the scaling variable .

Our next step in that direction is therefore to study the properties of . We define

(60)

where is the same as determined in Sec. 3 and shown in Fig. 1. It should be noted that although Eq. (23) expresses in terms of the near-side path length , it is the same as for the away side, since is integrated over the entire overlap so replacing by does not lead to any difference. We now ask whether has the scaling behavior

(61)

where

(62)

For and , we have and we can plot in the format of vs , as shown in Fig. 10. It is essentially symmetric in and , as it should, and peaks near the boundary of a maximum . The existence of such a boundary is due to the finite size of the medium that puts an upper limit on . Fig. 11 shows the maximum as a function of . Evidently, the symmetry in and suggests the use of the variables

(63)

so that . We show in Fig. 12 the projections of on for 4 values of . The four curves are in essence very similar although in detail they cannot be identical because and must both be positive so has different lower limits depending on .

Figure 10: (Color online) A 3D plot of the scaling function for and .
Figure 11: Maximum vs for and .
Figure 12: The scaling function vs for four values of at and .

Figures 10-12 are for and . To show scaling it is necessary to do the calculation for other values of and and demonstrate universality. We have done that and found plots like Fig. 12 that have almost no perceptible differences among them and are thus not exhibited. We conclude that does satisfy scaling as described by Eq. (61).

We can now return to the dijet spectra and present the results on yield per trigger defined as