---
title: |
    Space-Time Evolution of Hadronization in DIS:\
    Semi-Exclusive Processes and Grey Track Production[^1]
---

Introduction {#intro}
============

Nuclear targets serve as a natural and unique analyzer of the space-time development of strong interactions at high energies. Due to Lorentz time dilation, projectile partons may keep their coherence for some time, but once they become incoherent the cross section of final state interaction (FSI) increases. One needs observables sensitive to such modifications of FSI. One way for experimental testing of theoretical models is the measurement of the nuclear modification factor for inclusive production of leading hadrons [@knp], and various theoretical approaches have been advocated [@knph; @ww; @amp; @giessen] to explain recent data from HERMES experiment at HERA [@hermes]. The production of hadrons with large fraction $z_h$ of the initial parton momentum is a rare, nearly exclusive process, which has quite a specific time development. In the main bulk of events the jet energy is shared by many hadrons. It is a difficult task to find observables sensitive to the time development of hadronization in this case, and additional, complementary processes which could be sensitive to the space-time evolution of hadronization should be investigated. It is the aim of this paper to review two approaches which have been recently pursued along this line, namely the Semi- Exclusive DIS (SEDIS) processes $A(e,e'B)X$ [@ck; @ckk], where $B$ is a detected recoiling heavy fragment (e.g a nucleus $A-1$) and $X$ the unobserved jets from hadronization, and the grey track production in DIS off nuclei [@grey].

![Cartoons of the process $A(e,e'B)X$ (with $B=A-1$) (left), and the grey track production (right) in DIS. [*Left*]{}: after $\gamma*$ interaction with a quark, the nucleon debris interacts with the spectator nucleus $A-1$, which coherently recoils and is detected in coincidence with the scattered lepton. [*Right*]{} the nucleon debris breaks apart the spectator nucleus $A-1$, and nucleons with momentum $200-600 \,\,MeV/c$ are emitted and detected as grey tracks.[]{data-label="fig1"}](fig1a.eps "fig:"){height="4.0cm" width="6.0cm"} ![Cartoons of the process $A(e,e'B)X$ (with $B=A-1$) (left), and the grey track production (right) in DIS. [*Left*]{}: after $\gamma*$ interaction with a quark, the nucleon debris interacts with the spectator nucleus $A-1$, which coherently recoils and is detected in coincidence with the scattered lepton. [*Right*]{} the nucleon debris breaks apart the spectator nucleus $A-1$, and nucleons with momentum $200-600 \,\,MeV/c$ are emitted and detected as grey tracks.[]{data-label="fig1"}](fig1b.eps "fig:"){height="4.0cm" width="6.0cm"}

Unlike the rare process of leading hadron production, both processes, which are depicted in Fig. 1, depend upon the bulk of the FSI of the nucleon debris with the nuclear medium, and turn out to be very sensitive to the hadronization mechanism. Their theoretical treatment within a Glauber-like approach of FSI requires an [*effective time-dependent nucleon debris-nucleon cross section*]{}; a recent model of the latter [@ck] will be reviewed in the next Section and its use in the theoretical description of the two processes will be presented in Sections 3 and 4.

The effective debris-nucleon cross section {#debris}
==========================================

The effective debris-nucleon cross section obtained in [@ck], is based upon the hadronization model which combines the soft part of the hadronization dynamics, treated by means of the color string model, with the hard part one, described within perturbative QCD. In this model, which is inspired by Ref. [@knp], the formation of the final hadrons occurs during and after the propagation of the created debris through the nucleus, with a sequence of soft and hard production processes. Introducing a mass scale $\lambda=0.65\,\, GeV$ [@scale], soft production occurs at $Q< \lambda$ and hard production at $Q> \lambda$; in the former case $npQCD$ is taken care of by the color string model, whereas in the latter case pQCD is described within the gluon radiation model [@knp]. The details of the approach are given in Ref. [@ck], here only the basic elements of the model will be recalled. The ingredients of the treatment of hadron (mostly mesons $({\bf M})$) formation from the string $({\bf S})$ decay, are the following ones:

-   the probability $W(t)$ for a string to create no quark pairs since its origin;

-   the time dependent length of the string $L(t)$ with $L_{max}={m_{qq}}/{\kappa}$, where $m_{qq}$ is the mass of the “diquark” and $\kappa \simeq 1 \,GeV fm^{-1}$ the string tension;

-   the creation of a baryon ([**B**]{}) and a shorter string after the first breaking of the latter (within $\Delta t \simeq 1\,fm$), and the sequence of decays according to the scheme

    $\bf S$ $\Rightarrow$ [**B**]{}+$\bf S$ $\Rightarrow$ [**B**]{}+ $\bf S$+ [**M**]{} $\Rightarrow$ [**B**]{}+ $\bf S$+ [**2**]{} [**M**]{} +...

    leading to the following multiplicity of mesons $$n_{M}(t)=
    \frac{{\rm ln}(1+t/\Delta t)}
    {{\rm ln}2}\ ,
    \label{50}$$ with $\Delta t \simeq 1 fm$.

The gluon radiation mechanism is governed by:

-   the coherence time $$t_c =\frac{2\,E_q\,\alpha\,(1-\alpha)}
    {k_T^2}\ ,
    \nonumber$$ which is the time which elapses from the creation of the leading quark and the emission of the gluon which lost coherence with the color field of the quark. Here $\alpha$, $k_T=|{\bf k}_T|$, and $E_q$ are the fraction of the quark light-cone momentum carried by the radiated quantum, its transverse momentum, and the quark energy, respectively;

-   the mean number of radiated gluons, given by $$n_G(t)=
    \int\limits_{\lambda^2}^{Q^2} dk_T^2
    \int\limits_{k_T/E_q}^1  d\alpha\, \frac{dn_G}{dk^2_T\,d\alpha}\,
      \Theta(t- t_c )\ ,
      \nonumber$$ where the number of radiated gluons as a function of $\alpha$ and $\vec k_T$ is [@guber] $$\frac{dn_G}{d\alpha\,dk_T^2} =
    \frac{4\alpha_s(k_T^2)}{3\,\pi}\;\frac{1}{\alpha\,k_T^2}
    \nonumber$$

-   the time dependence of the gluon radiation, controlled by the parameter. $t_0 ={(m_N\,x_{Bj})}^{-1}=0.2 fm /x_{Bj}$

The final result for $n_G$ is , $$n_G(t) = \frac{16}{27}\,\left\{
{\rm ln}\left(\frac{Q}{\lambda}\right)\,+\,
{\rm ln}\left(\frac{t\,\Lambda_{QCD}}{2}
\right)\,{\rm ln}\left[\frac{{\rm ln}(Q/\Lambda_{QCD})}
{{\rm ln}(\lambda/\Lambda_{QCD}}\right]\right\}\ ,
\label{60}$$ for $t < t_0$, and $$\begin{aligned}
n_G(t) &=& \frac{16}{27}\,\left\{
{\rm ln}\left(\frac{Q}{\lambda}
\,\frac{t_0}{t}\right)\,+\,
{\rm ln}\left(\frac{t\,\Lambda_{QCD}}{2}
\right)\,{\rm ln}\left[\frac{{\rm ln}(Q/\Lambda_{QCD}
\sqrt{t_0/t})}
{{\rm ln}(\lambda/\Lambda_{QCD})}\right]
\right.\nonumber\\ &+& \left.
{\rm ln}\left(\frac{Q^2\,t_0}{2\,\Lambda_{QCD}}
\right)\,{\rm ln}\left[\frac{{\rm ln}(Q/\Lambda_{QCD})}
 {{\rm ln}(Q/\Lambda_{QCD}\,\sqrt{t_0/t})}\right]\right\}
\ ,
\label{65}\end{aligned}$$ for $t > t_0$, with saturation at $t > t_0\,Q^2/\lambda^2 = 2\nu/\lambda^2$. Using Eqs. (\[50\]), (\[60\]), and (\[65\]), one obtains the effective debris-nucleon cross section in the following form

$$\sigma_{eff}(t)=\sigma_{tot}^{NN}+
\sigma^{MN}_{tot}\Bigl[n_M(t) +
n_G(t)\Bigr]\ 
\label{70}$$

The time (or $z$, for a particle moving with the speed of light) dependence of $\sigma_{eff}(t)$ is shown in Fig. \[fig2\] for a fixed value of $x_{Bj}$ and various values of $Q^2$.

![The debris-nucleon effective cross section (Eq. (\[70\])) plotted [*vs*]{} the distance $z$ for a fixed value of the Bjorken scaling variable $x \equiv x_{Bj}$ and various values of the four-momentum transfer $Q^2$ (after Ref. [@ck]).[]{data-label="fig2"}](fig2.eps){height="6.0cm" width="6.0cm"}

The Semi-Exclusive DIS Process A(e,e’B)X
========================================

This process has been discussed in [@fs; @cks] within the Plane Wave Impulse Approximation (PWIA). If we consider, for ease of presentation, a deuteron $(D)$ target, the PWIA process consists of the hard interaction of $\gamma*$ with a parton of, e.g. the neutron, the creation of a debris, with the spectator ($s$) proton recoiling and detected in coincidence with the scattered electron (cf. Fig. 1, left, disregarding the wavy lines that represent the FSI between the debris and the spectator nucleons). The PWIA cross sections reads as follows

![The effective semi exclusive deuteron structure function (Eq. (\[feff\])) calculated including (FSI) and omitting (PWIA) the FSI of the debris, compared with preliminary experimental data from Jlab [@kuhn]. $\alpha$ and $p_t$ are the light cone fraction and perpendicular momentum component of the detected proton and $x_A=x_{Bj}/(2-\alpha)$ (after [@claleo]).[]{data-label="fig3"}](fig3.eps){height="6.5cm" width="6.5cm"}

$$\frac{d\sigma}{dx dQ^2\ d\alpha dp_T^2}
=
K(x_{Bj},Q^2,p_s)\, n_D(|{\bf p}_s|) \, F_2^{N/D}(Q^2,x_{Bj},p_s),
\label{crossPWIA}$$

is obtained, where $p_s =(p_{s}^0,{\bf p}_s)$ is the four-momentum of the recoiling detected nucleon , $\alpha = [p_{s}^0 - |{\bf p}_s|\cos \theta_s]/M$ ($\theta_s$ being the nucleon emission angle with respect to the direction of $\bf q$), $K(x_{Bj},Q^2,p_s)$ is a kinematical factor, $F_2^{N/A}(Q^2,x_{Bj},p_s) = 2x_{Bj} F_1^{N/A}(Q^2,x_{Bj},p_s)$ is the DIS structure function of the hit nucleon, and $n_D$ the nucleon momentum distribution, i. e. $$n_D(|{\bf p}|)=\frac13\frac{1}{(2\pi)^3} \sum\limits_{{\cal
M}_D} \left |\int d^3 r 
 \Psi_{{1,\cal
M}_D}( {\bf r})\exp(-i{\bf p r}/2) \right|^2.
\label{dismom}$$ where $\Psi_{{1,\cal
M}_D}( {\bf r})$ is the deuteron wave function. When the FSI of the debris is taken into account, the momentum distribution is replaced by the distorted momentum distribution $$n_D^{FSI}( {\bf p}_s,{\bf q}) =
\frac13\frac{1}{(2\pi)^3} \sum\limits_{{\cal
M}_D} \left | \int\, d  {\bf r} \Psi_{{1,\cal
M}_D}( {\bf r}) S( {\bf r},{\bf q}) \chi_f^+\,\exp (-i
{\bf p}_s {\bf r}) \right |^2,
 \label{dismomfsi}$$ where $\chi_f$ is the spin function of the spectator nucleon, ${\bf q}$ is the three-momentum transfer (oriented along the $z$ axis), and $$S( {\bf r},{\bf q}) = 1-\theta(z)\, \frac{\sigma_{eff}(z,Q^2,x)(1-i\alpha)}{4\pi b_0^2}\,
\exp(-b^2/2b_0^2)
\label{gama}$$ takes care of the final state interaction between the debris and the spectator. The approach can readily be generalized to complex nuclei for which the experiments are difficult to carry out. However, preliminary experimental data for the deuteron have already been obtained at Jlab [@kuhn]. A limited set of these data is shown in Fig. 3, where the quantity $$F_{eff} \left( \frac{x_{Bj}}{2-\alpha},p_T,Q^2 \right)\equiv \frac{\sigma^{exp}}{K\cdot n_D(|{\bf p}|)}
\label{feff}$$ is compared with theoretical calculations [@claleo] which omit (Eq. (\[dismom\])) and include (Eq. (\[dismomfsi\])) the FSI of the debris; it can be seen that the latter is extremely important.

Hadronization and grey track production {#grey}
=======================================

The dominant channels of DIS are the ones in which the recoiling nucleus $B$ breaks apart to fragments; the investigation of the nature and the kinematical dependence of these fragments can shed light on the time evolution of the jet in the nuclear medium. This seems particularly true in the processes producing grey tracks, which are hadrons, predominantly protons, with momenta in the range of few hundred MeV/c. The basic mechanism of grey track production is the inelastic interaction of the jet with the spectator nucleons of the target, which recoil in the given momentum interval and are detected (cf. Fig. 1, right). The $Q^2$ and $x_{Bj}$ dependence of the average number of grey tracks produced in the Fermilab E665 experiment [@e665] ( protons with momentum $200-600\,\, MeV/c$ produced in $\mu-Xe$ and $\mu-D$ DIS at $490\,\, GeV$ beam energy), have been recently analyzed [@grey] within the theoretical framework which employs the effective debris-nucleon cross section of [@ck].

![[*Left*]{}: mean number of grey tracks $< n_g >$ produced in the $\mu - Xe$ DIS experiment [@e665] [*vs*]{} $Q^2$ in the non-shadowing region with fixed $x_{Bj}=0.07$ (dashed). [*Right*]{}: mean number of grey tracks $< n_g >$ [*vs*]{} $x_{Bj}$ with fixed value of $Q^2=14.3\,\, GeV^2$ (dashed). In both Figures the solid curve includes the $Q^2-x_{Bj}$ correlation found in the experiment (after Ref. [@grey]).[]{data-label="fig4"}](fig4a.eps "fig:"){height="5.8cm" width="6.0cm"} ![[*Left*]{}: mean number of grey tracks $< n_g >$ produced in the $\mu - Xe$ DIS experiment [@e665] [*vs*]{} $Q^2$ in the non-shadowing region with fixed $x_{Bj}=0.07$ (dashed). [*Right*]{}: mean number of grey tracks $< n_g >$ [*vs*]{} $x_{Bj}$ with fixed value of $Q^2=14.3\,\, GeV^2$ (dashed). In both Figures the solid curve includes the $Q^2-x_{Bj}$ correlation found in the experiment (after Ref. [@grey]).[]{data-label="fig4"}](fig4b.eps "fig:"){height="5.8cm" width="5.5cm"}

The basic elements of the approach are the following ones:

-   DIS on a bound nucleon occurs at coordinate $(\vec b,z)$ and the debris propagates through the nucleus interacting with spectator nucleons via $\sigma_{eff}(z-z')$. The mean number of collisions (plus the recoiling nucleon formed in the hard $\gamma*-N$ act) is $${\langle \nu_c\rangle = \int d^2b\,
    \int\limits_{-\infty}^\infty dz\,\rho_A(\vec b,z)
    \int\limits_z^\infty dz'\,\rho_A(\vec b,z')\,
    \sigma_{eff}(z-z')\,+\,1\ .}
    \label{medio}$$ where $\rho_A$ is the nuclear density and $\sigma_{eff}(t)=\sigma_{in}^{MN}[n_M(t)+n_G(t)]$ with $\sigma_{in}^{MN}=\sigma_{tot}^{\pi\,N} -\sigma_{el}^{\pi\,N} - \sigma_{dif}^{\pi\,N}=17.7\,mb$.

-   the mean number of collisions has been calculated according to Eq. (\[medio\]), and the mean number of grey tracks $<n_g>$ has been obtained using the empirical relation found in [@e665], [*viz.*]{} $${<n_g> =\frac{\langle {\nu}_c\rangle - (2.08\pm0.13)}{(3.72\pm0.14)}\,}
    \label{empirical}$$

The results of the parameter-free calculations, which are shown in Fig. 4, exhibit a good agreement with the experimental data; worth being mentioned is the $Q^2$ dependence of the data, which is explained by the adopted hadronization model, namely by the gluon radiation mechanism. It turns out, therefore, that the observed $Q^2$ dependence of the average number of grey tracks is a very sensitive tool to discriminate between different models of hadronization.

Conclusions
===========

The main conclusions of my talk can be summarized as follows:

-   a time-dependent debris-nucleon cross section has been obtained; it incorporates both non perturbative (string) and $Q^2$-dependent perturbative (gluon radiation) effects; it accounts for the bulk of the FSI interaction of the debris created by the hard interaction of a bound nucleon with $\gamma*$; it should be used in all kinds of DIS off nuclei to describe the FSI of the hit nucleon debris with the spectator nucleons;

-   the (parameter-free) calculation of the Semi Exclusive DIS off the deuteron $D(e,e'p)X$ exhibits good agreement with preliminary data from Jlab; a systematic investigation of the $x_{Bj}$ and $Q^2$ dependence of these processes at Jlab and HERA energies would shed further light on the hadronization mechanism;

-   the (parameter-free) calculation of the $Q^2$ and $x_{Bj}$ dependence of the average number of grey tracks produced in Deep Inelastic $\mu-Xe$ scattering exhibits a very satisfactory agreement with the experimental data, thanks to the gluon radiation mechanism for hadron production.

Acknowledgments {#acknowledgments .unnumbered}
===============

I am grateful to the Organizers of the Workshop for the invitation and to Leonid Kaptari and Boris Kopeliovich for a fruitful and stimulating collaboration.

[99]{}

B.Z. Kopeliovich, J. Nemchik and E. Predazzi, in [*Future Physics at HERA*]{}, Proceedings of the Workshop 1995/96, edited by G. Ingelmsan, A. De Roeck and R. Klanner, DESY, 1995/1996, vol.2, p. 1038 (nucl-th/9607036);\
in Proceedings of the [*ELFE Summer School on Confinement Physics*]{}, edited by S.D. Bass and P.A.M. Guichon, Editions Frontieres, 1995, p. 391, Gif-sur-Yvette (hep-ph/9511214).

B.Z. Kopeliovich, J. Nemchik, E. Predazzi and A. Hayashigaki, Nucl. Phys. [**A740**]{} (2004)212.

E. Wang and X.-N. Wang, Phys. Rev. Lett. [**89**]{} (2002) 162301. A. Accardi, V. Muccifora and H.J. Pirner, Nucl. Phys. [**A720**]{} (2003) 131.

T. Falter, W. Cassing, K. Gallmeister, U. Mosel, nucl-th/0303011; nucl-th/0406023.

HERMES Collaboration, A. Airapetian et al., Eur. Phys. J. [**C20**]{} (2001) 479; Phys. Lett. [**B577**]{} (2003) 37.

C. Ciofi degli Atti and B.Z. Kopeliovich, Eur. Phys. J. [**A17**]{} (2003) 133.

C. Ciofi degli Atti, L.P. Kaptari, and B.Z. Kopeliovich, Eur. Phys. J. [**A19**]{} (2004) 145.

C. Ciofi degli Atti and B.Z. Kopeliovich, Phys. Lett. [**B606**]{} (2005) 281.

B.Z. Kopeliovich, A. Schäfer and A.V. Tarasov, Phys. Rev. [**D62**]{} (2000) 054022 (hep-ph/9908245).

J.F. Gunion and G. Bertsch, Phys. Rev. [**D25**]{} (1982) 746.

W. Melnitchouk, M. Sargsian, M. I. Strikman, Z. Phys. [**A359**]{} (1997) 99.\
S. Simula, Phys. Lett. [**B387**]{} (1996) 245.

C. Ciofi degli Atti, L.P. Kaptari, and S. Scopetta, Eur. Phys. J. [**A5**]{} (1999) 191.

Jlab Experiment 94-102, S. E. Kuhn, K. A. Griffioen, co-spokespersons, [*Inelastic electron scattering off a moving nucleon in deuterium*]{};\
S. E. Kuhn, [*Private Communication*]{}.

C. Ciofi degli Atti and L. P. Kaptari, [*unpublished*]{}.

E665 Collaboration, M.R. Adams et al., Z. Phys. [ **C65**]{} (1995) 225.

[^1]: Talk given at the Workshop on [*In-Medium Hadron Physics*]{}, Giessen, Nov. 11-13, 2004
