MPI H-V29-1996

Hadronization in Nuclear
Environment^{*}^{*}*
To appear in the Proceedings of the Workshop on Future Physics at HERA,

DESY, September 25, 1995 – May 31, 1996

Boris Kopeliovich, Jan Nemchik and Enrico Predazzi

Max-Planck Institut für
Kernphysik, Postfach
103980, 69029 Heidelberg, Germany

Joint Institute
for Nuclear Research,
Dubna, 141980
Moscow Region, Russia

Università di Torino
and INFN, Sezione di Torino, I-10125, Torino, Italy

Institute of Experimental Physics SAV,
Solovjevova 47, CS-04353 Kosice, Slovakia

Abstract: We present a space-time description of hadronization of highly virtual quarks originating from a deep-inelastic electron scattering (DIS). Important ingredients of our approach are the time- and energy–dependence of the density of energy loss for gluon radiation, the Sudakov’s suppression of no radiation, and the effect of color transparency, which suppresses final state interaction of the produced colorless wave packet. The model is in a good agreement with available data on leading hadron production off nucleons and nuclei. The optimal energy range for study of the hadronization dynamics with nuclear target is found to be a few tens of GeV, particularly energies available in the experiment HERMES.

A quark originated from a hard process, converts into colorless hadrons owing to confinement. Lorentz dilation stretches considerably the duration of this process. While hadrons carry a little information about the early stage of hadronization, a nuclear target, as a set of scattering centres, allows us to look inside the process at very short times after it starts. The quark-gluon system produced in a hard collision, interacts while passing through the nucleus. This can yield precious information about the structure of this system and the space-time pattern of hadronization.

The modification of the quark fragmentation function in nuclear matter was considered for high- hadron production in [1, 2], for deep-inelastic lepton scattering in [3, 4, 5], and for hadroproduction of leading particles on nuclei in [6, 7]. The data are usually presented in the form,

(1) |

where and are the quark fragmentation function in vacuum and in a nucleus, respectively.

We treat hadronization of a highly virtual quark, perturbatively a gluon bremsstrahlung and the deceleration of the quark as a result of radiative energy loss. We assume that subsequent hadronization of the radiated gluons, which includes the nonperturbative stage, does not affect the energy loss of the quark.

The radiation of a gluon takes the time

(2) |

This expression follows from the form of the energy denominator corresponding to a fluctuation of a quark of energy into a quark and a gluon, having transverse momenta and relative shares of the initial light-cone momentum and , respectively. If one calculates radiated energy taking into account condition (2) one arrives at the density of energy loss per unit of length, which turns out to be energy and time independent [8] like in the string model.

In the case of inclusive production of leading particles at , however, energy conservation forbids the radiation of gluons with energy greater than . Then, the time dependence of the radiative energy loss can be written as

(3) |

where represents the distribution of the number of gluons. The factor .

Although soft hadronization is usually described in terms of the string model, we model it by radiation as well, choosing the bottom limit in (3) small. We fix the QCD running coupling at , in the region which is supposed to be dominated by nonperturbative effects. the parameter is chosen to reproduce the density of energy loss for radiation of soft gluons () corresponding to the string tension, . This value of is consistent with the transverse size of a string corresponding to the gluon-gluon correlation radius, fm suggested by QCD lattice results.

After integrating eq. (3) in the soft radiation approximation, we get

(4) |

Here we have set and , where is the Bjorken variable.

Eq. (4) shows that for , the density of energy loss is constant, , exactly as in the case with no restriction on the radiated energy [4, 5]. At longer time intervals, more energetic gluons can be radiated and the restriction becomes important. As a result, the density of energy loss slows down to which is a new result compared to what was known in the string model. At still longer , no radiation is permitted, but obviously a color charge cannot propagate a long time without radiating which must be suppressed by a Sudakov’s type formfactor. Assuming a Poisson distribution for the number of emitted gluons we get the formfactor, , where is the number of non radiated gluons,

(5) |

In order to calculate a time interval for the leading hadron production (or, better, a colorless ejectile which does not loose energy anymore), one needs a model of hadronization and of color neutralization. In the large limit, each radiated gluon can be replaced by a pair, and the whole system can be treated as a system of color dipoles. It is natural to assume that the leading (fastest) hadron originates from a dipole made of the leading quark and of the antiquark coming from the last emitted gluon. This dipole is to be projected into the hadron wave function, , where and are the relative shares of the light-cone momentum carried by the quarks, and is the relative transverse momentum of the quarks. The result of this projection leads to the fragmentation function of the quark into the hadron, which reads

(6) |

where is a distribution function of the leading hadrons over the production time .

(7) | |||||

Here the quark energy . We have chosen a hadronic wave function in the light-cone representation which satisfies the Regge end-point behaviour, , where is the charge radius of the hadron.

Fig.1 shows function for several values of and exhibits the approximate -scaling of the mean production time, , which depends weakly on and vanishes at .

Our predictions for the fragmentation function depicted in Fig. 2 nicely agrees with the EMC data [9].

The production of the leading colorless wave packet with the desired (detected) momentum completes the process of hadronization. Any subsequent inelastic interaction is forbidden, otherwise a new hadronization process begins and the leading hadron energy falls to lower values. Such a restriction means a nuclear suppression of the production rate.

On the other hand, soft interactions of the leading quark during the hadronization in nuclear matter cannot stop or absorb the leading quark [10]. Although rescatterings of the quark in the nucleus result in an additional induced soft radiation, [10, 11], at the same time the quark looses much more energy due to the hard gluon radiation following the deep-inelastic scattering just as in vacuum. Thus, the induced soft radiation can be treated as a small correction to the energy loss and can be neglected, provided is high enough.

The transverse size of the colorless wave packet produced in a hard reaction can be small, therefore the nuclear suppression is weaker due to color transparency. We take into account the evolution of the wave packet during its propagation through the nucleus using the path integral technique developed in [12]. Figs. 3 - 5 show quite a good agreement of our parameter-free calculations with available data [13, 9] on the -, - and -dependence of nuclear suppression. Unfortunately, there is still no data in the region, .

Note that at high energies many of interesting effects go away or are difficult to observe. Nuclear suppression integrated over vanishes (see Fig. 3). The region of high , where nuclear suppression is expected to be enhanced (see Fig. 4) squeezes at high towards , where the cross section vanishes. There is almost no -dependence at high and moderate values of (see Fig. 5).

We present in Figs. 5 and 6 our predictions for the and dependence of nuclear suppression for the energy range of the HERMES experiment. We expect the onset of nuclear effects at moderate values of (Fig. 5) as well as in the region of (Fig. 6), which can be tested by the HERMES experiment.

To conclude, we have developed a phenomenology of electroproduction of leading hadrons on nuclei which is based on the perturbative QCD. Our parameter-free model is in a good agreement with available data. We stress that the energy range of the HERMES experiment is especially sensitive to the underlying dynamics of hadronization.

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