Limiting fragmentation from scale-invariant merging of fast partons

Exploiting the idea that the fast partons of an energetic projectile can be treated as sources of color radiation interpreted as wee partons, it is shown that the recently observed property of extended limiting fragmentation implies a scaling law for the rapidity distribution of fast partons. This leads to a picture of a self-similar process where, for fixed total rapidity Y, the sources merge with probability varying as 1/y.


1.
It was shown recently [1] that single particle (pseudo)rapidity distributions in p − p, d − Au, Cu − Cu and Au − Au collisions at 200 GeV c.m. energy [2] can be described by various superpositions of contributions from the "wounded" constituents of the nucleon: a quark and a diquark. The form of the contribution from one wounded constituent W (η) ≈ W (y) was determined. Its characteristic feature is that, although peaked in the direction of the constituent, it is not restricted to one hemisphere but extends over almost full available phase-space [3].
Apart from details inessential for our study, this observation shows that particle production can be discussed independently for the projectile and for the target. This certainly radically simplifies the problem which, originally, may seem to be hopelessly complicated. Moreover, it suggests that the main effect of the target on the emission of particles from the projectile is a passive one: apparently the role of the target is to define the phase-space available for the radiation from the projectile 1 .
The analysis of [1] was performed at one single energy. It is interesting to investigate if (and how) the resulting picture can accommodate the recently discovered property of extended limiting fragmentation [4,5] of the (pseudo)rapidity spectra of particles produced in high-energy collisions [5,6]. This asymptotic property says that when plotted vs. ξ = Y −y where y is the rapidity of produced particle and Y is the rapidity of the projectile, data at increasing energies approach a common curve (a straight line to a good approximation), until a certain value of ξ which increases with increasing Y.
In the present paper we discuss this problem in some detail and formulate a model of particle emission from a strongly interacting composite projectile which satisfies extended limiting fragmentation of the spectra and is able to lead to a satisfactory description of data. The model exploits the idea of distinguishing between the fast partons treated as the sources of radiation and the wee partons considered as a colour field emitted from the sources [7,8]. The emission is described by the bremsstrahlung mechanism where the infrared cut-off, necessary to give a physical meaning to the bremsstrahlung radiation, is treated as a random variable.
By demanding that the observed spectra satisfy extended limiting fragmentation, one can show that the distribution of sources must obey a scaling law, which expresses the explicit rule how the distribution of sources varies as a function of their rapidity y for a given value of Y. Indeed, the smaller is y the less is the number of sources. As a consequence of scaling, the same mechanism describes how the distribution evolves at fixed y for varying energy of the collision. The scaling law also suggests that the mechanism responsible for this structure is a self-similar process in which the sources merge when their rapidity decreases.
In the next section the mechanism we propose is explained. The scaling law following from extended limiting fragmentation is discussed in Section 3. The self-similar merging picture is described in Section 4. Our results are summarized and commented in the last section.
2. Starting with the idea formulated in [7], we assume that the wounded constituent represents a collection of partonic sources, each one emitting soft partons which may be represented by a colour field. Denoting the distribution of sources by R(y + ; Y ), where y + is the rapidity of the source 2 and Y is the rapidity of the projectile, we obtain for the parton distribution where H(y; y + ) is the distribution of all partons coming from a source located at y + . The final distribution dN /dy is obtained from the sum of contributions of the wounded constituents from both particle and target sides. The details of H(y; y + ) depend on the mechanism of radiation. Here, for simplicity, we take the bremsstrahlung or Weizsaecker-Williams distribution in its simplest form: i.e. a flat distribution in rapidity 3 , extending from y + till some (in principle arbitrary) infra-red cut-off value y − . P (y − ; y + ) is the distribution of probability that, for a source located at y + , the radiation is cut at y − . Physically, the very existence of the cut-off is the consequence of the interaction with the target. Since, however, the target is a rather complicated object, one cannot hope to evaluate explicitely the actual distribution of y − . As a statistical approximation we shall assume that it can take any value within the allowed kinematic limits with equal probability 4 . This idea leads to the explicit formula for the distribution P (y − ; y + ): 2 Throughout this paper we use the target rest frame. 3 The general formula for the distribution of partons following from the bremsstrahlung mechanism at high energy is a(1 − x) a + ax(1 − x) a−1 with x = e y−y+ [9]. The first term represents the radiated partons. The second term describes the radiating (leading) one. For a = 1 we obtain the flat distribution (2). 4 We shall discuss some generalization at the end of the paper.
where the factor in front provides the proper normalization of the probability: All in all, we finally obtain This formula allows one to determine the distribution of produced partons from that of the sources contained in the wounded constituent.
3. Let us now introduce a convenient formulation of the constraints imposed by the property of extended limiting fragmentation. Starting with the distribution W (y, Y ) and its derivatives ∂ k W (y,Y ) ∂y k in the target fragmentation region y ∼ Y, it requires on quite general grounds when Y becomes large. Indeed, this reflects the fact that at increasing energies, the rapidity distribution approaches a common curve which is a straight line. We note that from analyticity the higher derivatives cannot be exactly zero 5 , but their decrease with Y as expressed by the constraint (6) implies that limiting fragmentation region extends further in rapidity Y − y with increasing total rapidity. A more precise mathematical formulation of these constraints which reproduces the experimental features described in the introduction can be summarized in the formula with Φ(s = 1) = 0. One easily sees that W (y, Y ) given by (7) satisfies (6). Moreover, one sees that W (y; Y ) scales towards a straight line limit at fixed rapidity y when Y → ∞. More precisely, the slope of the distribution W (y; Y ) in the limiting fragmentation region is approximately constant (given by 5 The extended limiting fragmentation is an asymptotic property valid at large Y. In particular it does not mean that the spectrum depends only on ξ = Y−y even in a restricted region in ξ. In other terms, one has to decribe not only the limiting fragmentation, but also its breaking at some rapidity increasing with Y. Φ(s = 1)), with significant deviations showing up at some finite s 0 and thus at a value of rapidity Y −y ∼ s 0 · Y increasing with total rapidity. Using our formula (5), relating the particle distribution W (y, Y ) to the fast parton distribution R(y; Y ), we shall now show that (7) implies that R(y; Y ) obeys the scaling law Indeed, (5) implies so that (8) is satisfied. The scaling law (8) determines the energy dependence of the distribution of sources. It shows that the separation between the sources and the field scales with the total rapidity. It allows to express the distribution at the total rapidity ωY by the distribution at the total rapidity Y , i.e. the evolution of the distribution of sources: This shows explicitely how the idea of [7] is realized in our mechanism. The extended limiting fragmentation realized by this mechanism is shown in Fig.1, where the particle rapidity spectra at various values of Y are plotted versus Y −y. One sees that extended limiting fragmentation is indeed well satisfied.
4. The scaling law (8) suggests that the source distribution is formed in a self-similar cascade. Since R is expected to decrease with decreasing s + , this process must correspond to merging of fast sources into slower ones.
Indeed, introducing new variables t + = log y + , T = log Y , we see that (8) implies the relation and thus an increment of T is equivalent to a negative shift in t + . This is typical of the cascade with equal steps and a constant merging probability in t + = log y + . In order to make the self-similar structure more explicit, let us give a practical example which in fact will also be physically meaningful. We consider a self-similar system of fast partons evolving as a function of the variable t with constant merging probability. For completion, we also consider the possibility that the decrease at high density has to be somewhat tamed if the sources screen each other. Hence, it is possible to write a simple equation for their distribution, namely where b is the merging probability and a the recombination factor. It is easy to write the scaling solution of (12) as This solution has all desired features. Indeed the source distribution goes to 0 at rapidity y = 0, decreases by merging when y decreases at fixed Y while, at a fixed y, it decreases with increasing Y . In fact this is the solution which was used in Fig.1 and it is compatible with the phenomenological information from data, as a first approximation.
Finally, let us note that such a cascade is equivalent to a cascade with equal steps in rapidity but with the merging probability varying as It is interesting to observe that this result can be obtained if one takes the merging probability in the form where d 0 is the average (longitudinal) distance between partons in the projectile rest frame and d(y) is their average distance at the rapidity y. Indeed, since the longitudinal density is basically determined by the Lorentz factor implied by the boost, one may expect that d(y) = e −y d 0 , leading to α(y) ∝ 1/y. One sees that (15) resembles the well-known perturbative formula for the dependence of the effective colour charge on distance. It should be emphasized, however, that there is also an important difference between them: while d(y) and d 0 refer to longitudinal distances, in the corresponding perturbative expression the four-dimensional distances are involved. The relation between these two situations remains an interesting problem which, however, goes beyond the scope of this paper.

5.
Our results can be summarized as follows.
(i) We investigated the possibility that the particle production can be described by two components: one attached to the projectile and another one to the target both extending through most of the available phase-space in rapidity. This idea is suggested by the success of the phenomenological analysis based on the wounded constituent and wounded nucleon models [10] which showed that such an approach can describe the rapidity spectra of various processes involving hadrons and nuclei [1,3] (ii) Considering one of these components (attached, say, to the projectile) we considered the mechanism of particle production suggested in [7] and [8]. The partons present already in the rest frame of the projectile are treated as sources of the bremsstrahlung radiation producing the colour field. The distribution of sources is determined by two effects: (a) The boost from the rest to the laboratory frame implying a substantial increase of the parton density in longitudinal direction, and (b) the process of recombination accompanied by a decrease of momentum and of the density.
(iii) The meaningful physical interpretation of bremsstrahlung radiation asks for the introduction of an infrared cut-off. To implement the idea that the details of the target should not play an important role, we assume that the cut-off is randomly distributed through the rapidity space available for a given source. We hope that this may be a way to take into account the complexity and multiplicity of elementary contributions to the soft interaction with the target by using statistical arguments.
(iv) It turns out that the property of extended limiting fragmentation, observed in all high-energy processes involving strongly interacting particles, implies a scaling law 7 for the rapidity distribution of the sources. The scaling law states that this distribution depends only on the ratio y/Y and not on y and Y separately. This means that the separation between the sources and the field is not fully defined by rapidity (as is the case in [7]) but their proportion varies with the ratio y/Y .
(v) The scaling law (8) suggests that the recombination process, responsible for the distribution of sources, is a self similar cascade. This self-similar structure differs from the cascade considered in, e.g., [13] in few important aspects. First, it is a merging process instead of a splitting one. Second, it is self-similar in log y rather than in y. Third, the transverse momentum of partons (which in [13] corresponds to the inverse dipole size) does not need to be large.
Following comments are in order. (a) As seen from (7), extended limiting fragmentation is an asymptotic (Y → ∞) property. The rate at which it is realized depends on the shape of the function One sees that for R(s + ) varying slowly [8] near s + = 1 (corresponding to recombination with sizeable value of a in (12) for the fast partons) one obtains a rather fast approach to the limiting line. This is illustrated in Fig.1. Since the data indicate that limiting fragmentation is reached already at relatively low energies, one may infer that indeed recombination is at work. (b) It is worth noticing that the specific structure of the distribution of infra-red cut-offs as given by (3) is not necessary to obtain the scaling law (8). Indeed, one can check that any distribution of cut-offs satisfying the scaling condition also leads to (8).
(c) Our analysis concentrated on the longitudinal properties of the spectra. It would be very interesting, of course, to elucidate also the role of transverse momenta 8 . The key problem should be a better understanding of the process of recombination of sources which may or may not affect their transverse momentum distribution. Unfortunately, our present approach, being purely phenomenological, does not allow to undertake this task.
d) It should be emphasized that, despite the similarity of certain aspects of our approach with perturbative physics, the phenomena we are describing are basically non-perturbative. In particular, the scaling property (11) may only imply a small longitudinal distance d(y), see (15), but not necessary a small transverse distance (i.e. large transverse momentum) as for saturationlike models ( [14]).
A final comment concerns the theoretical implications of the mechanism we propose 9 . Indeed, this mechanism may give some interesting guidelines for an exploration of non perturbative properties. One aspect is a theoretical hint for the "superposition" property of target and projectile components (see e.g. [11]). Another one is the observation that a self-similar interaction is at the origin the longitudinal structure of hadrons boosted to very high momenta, which may represent an important step in understanding soft QCD at high energies.