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\begin{document}    
\title{Inclusive Spectra and Quantum Stochastic Processes}           
\author{ A. Krzywicki\footnote{E-mail address: krz@th.u-psud.fr}}           
\address{Laboratoire de Physique Th\'eorique, B\^atiment 210,            
Universit\'e Paris-Sud,           
91405~Orsay, France}            
\maketitle           
\begin{abstract}           
We try to explain the apparently 
"thermal" shape of the inclusive transverse 
momentum spectrum. We conjecture that  
prior to the collision, parton-parton interactions 
generate a kind of a stochastic process driving the 
one-particle spectrum within a hadron towards 
the thermal shape it would have in a classical gas. 
We illustrate this idea,  which might have a more 
general relevance, with a simple, exactly solvable 
model.\par     
\smallskip\noindent{PACS numbers: 13.85Hd , 02.50Ey } \par
\smallskip\noindent{Key words: hadron, inclusive, stochastic}
\end{abstract}     
\par\hfill LPT Orsay 02/31 (April 2002)

\begin{multicols}{2}
This research is partly motivated by
a long standing puzzle associated with
multiparticle production at high energy: in soft
hadronic collisions the inclusive transverse 
momentum spectrum at not too large  $p_\perp$ 
has an approximately boltzmannian shape, 
\begin{equation}
\frac{d\sigma}{d^2p_{\perp}} \sim e^{-E(p_\perp)/T} ,
\label{0}
\end{equation}
as if it resulted from a thermalization process occuring 
during the collision. Here, $E(p_\perp)$  denotes the 
"transverse energy" of a  secondary \cite{foot1} 
and $T$ is a "temperature"  parameter,
roughly independent of the secondary hadron species. 
However, the thermalization in the final
state, debatable but plausible in a heavy-ion collision, is
nearly impossible in a minimum bias nucleon-nucleon collision, 
where (\ref{0}) is the best observed. Hence, one can 
wonder whether the suggestive and universal behavior of the
data does not reflect some simple mechanism operating at 
the level of the wave function describing the incoming hadron's 
internal motion.
\par
As is well known, in the infinite momentum frame $P \to \infty$ 
a hadron can be regarded as a 2D nonrelativistic, coherent 
multiparton state enclosed in a box, with dimensions fixed by 
the confinement scale. The kinetic energy of a parton is 
$p_{\perp}^2/2xP$,  $x$ denoting its longitudinal 
momentum fraction. If the state were belonging to
a classical statistical ensemble, it would thermalize 
because of the residual parton-parton interaction.
Then, one could argue, invoking the parton-hadron duality,  
that after the hadron is broken by the interaction
the inclusive spectrum reflects the   Boltzmann distribution 
holding at the parton level. But the hadron state is a pure 
quantum state. The question is whether the one-particle 
momentum distribution does nevertheless exhibit a kind 
of pseudo-thermalization \cite{foot2}.
\par
Apart from this phenomenological motivation,
it is quite interesting to inquire to what
extent the wave function of a multiparticle quantum
state reflects the features of a corresponding classical 
ensemble, at least at the one-particle distribution 
level. Or else: under what conditions the one-particle
spectrum is expected to approach, for time $ \to + \infty$,
a limit independent of the initial state of the system, 
like in a generic classical Markov process?
\par
It is difficult to approach the problem in its full
generality. Consequently, we adopt the
one-particle approximation: we follow a 
single particle moving in an external mean 
field, whose action represents the multiparticle 
environement. We know, that in a classical gas this motion 
generically leads to thermalization, ie the momentum
spectrum tends to the Boltzmann form. A time arrow 
is attached to the motion of the test particle. This 
time arrow is also present when one goes over to 
the quantum description. This means that the evolution
generator describing the motion of the test particle is
non-hermitian. Hence, it will, in general, have
complex eigenvalues. Consequently, a single
quantum state of the test particle, independent of 
its initial state, will be selected for time 
$\to + \infty$. This is the general idea of this
paper. It will be illustrated by a simple example.
\par 
Consider a gas of classical particles first.
How does the thermalization occur? A given particle of the gas
is slowed down (accelerated)  when it collides with
a particle less (more) energetic than itself.
Eventually, its momentum fluctuates near the
average momentum measured by the gas
temperature. To mock this behavior assume
that the particle gets random kicks, such that 
the transition $p \to p  \pm \delta$ occurs with
probability
\begin{equation}
Prob(p \to p  \pm \delta) \sim 
e^{\Delta^\pm(p)}
\label{2}
\end{equation}
where
\begin{equation}
\Delta^\pm(p) = \frac{1}{2T}[E(p) - E(p\pm\delta)]
\label{2bis}
\end{equation}
Here $E(p)$ is the energy of a parton. We do not 
postulate any particular dispersion formula to 
make it clear that the discussion is quite general. 
The transition probability (\ref{2})
satisfies the detailed balance equation
\begin{equation}
Prob(p) Prob(p  \to p  \pm \delta) = 
Prob(p  \pm \delta)Prob(p  \pm \delta \to p)
\label{3}
\end{equation}
with $Prob(p) \sim \exp{[-E(p)/T]}$, which is therefore 
the limit distribution of the process defined by 
(\ref{2}) \cite{foot3}.   
The process (\ref{2})
violates the time reversal. It must be so, the process has the 
time arrow built into it, since it mocks the effect of 
the interaction of the test particle with its multiparticle
environement. 
\par
In order to simplify writing we assume  first that 
$\delta$ is a constant vector. Thus the motion is
one-dimensional. The extension of the discussion to 
a spectrum of momentum transfers is trivial.
\par
Eq. (\ref{2}) implies the following master equation for the
momentum probability distribution $w(p,t)$:
\begin{eqnarray}
\partial w(p, t) /\partial t =  
w(p + \delta, t) e^{\Delta^-(p + \delta)} \nonumber  \\
+ \; w(p - \delta, t) e^{\Delta^+(p - \delta)}
- w(p, t) [e^{\Delta^-(p)} + e^{\Delta^+(p)}]    
\label{fp}
\end{eqnarray}
where the time $t$ is measured in appropriate 
units, so that the interaction strength does not appear 
explicitly. It can be immediately seen that 
$w(p) = e^{- E(p)/T}$ is a stationary 
solution of (\ref{fp}), as expected. 
\par
We now go over to the quantum world and replace the 
particle by a wave. We assume that the kick transition probabilities 
are the same as before. This fixes the moduli of the corresponding 
amplitudes, the phases remaining free. It is convenient to use
the second quantization formalism and to work in the 
interaction representation. The kick is due to the interaction
\begin{equation}
V =  \sum_p [ \eta^- e^{\Delta^-(p+\delta)/2} a^\dagger_p a_{p+\delta}
+  \eta^+ e^{\Delta^+(p-\delta)/2} a^\dagger_p a_{p-\delta}]
\label{8}
\end{equation}
where $\eta^\pm$ are some  phase factors, scalar functions 
of $p$ and $\delta$, while $a^\dagger_p$ ($a_p$) are 
time independent creation (annihilation) operators.
The interaction operator is not hermitian, the time 
reversal invariance is broken like in the classical model. 
\par
The number of particles is - by assumption - conserved. Hence the 
state at time $t$ has necessarily the form
\begin{equation}
\mid t \rangle = \sum_p \psi (p, t) a^\dagger_p \mid 0 \rangle
\label{9}
\end{equation}
The norm of this state is time dependent, the evolution being 
non-unitary. This does not matter for our purposes, since we are 
interested in the shape of the momentum distribution 
only. The Schroedinger equation describing the evolution 
of the wave packet (\ref{9}) reads
\begin{eqnarray}
i\partial \psi (p,t)/\partial t = 
\eta^- e^{\Delta^-(p+\delta)/2}\psi (p+\delta, t) \nonumber \\ 
+ \; \eta^+ e^{\Delta^+(p-\delta)/2} \psi (p-\delta, t)  
\label{10}
\end{eqnarray}
In order to determine the behavior of the solution we have 
to solve first the eigenvalue equation:
\begin{equation}
 \eta^- e^{\Delta^-(p+\delta)/2} \psi (p+\delta) +  
\eta^+ e^{\Delta^+(p-\delta)/2} \psi (p-\delta) = \lambda \psi (p)
\label{11}
\end{equation}
The transition probabilities are determined by the values of $\Delta^\pm$.
In the absence of a better insight into the problem it is
reasonable to assume that $\eta^\pm$ also depends primarily on this
parameter. In order to obtain an analytically solvable model
we postulate a linear behavior 
\begin{equation}
 \mbox{\rm Arg}(\eta^\pm) = \xi_0 + \xi_1 \Delta^\pm (p\mp \delta)
\label{11bis}
\end{equation}
Set 
$\psi(p) =  e^{-E(p)(1+2i\xi_1)/4T} z(p)$  to get
\begin{equation}
 e^{i\xi_0}[z(p +\delta) + z(p-\delta)] = \lambda z(p)
\label{12}
\end{equation}
From here on the calculation becomes elementary. With
periodic boundary conditions
\begin{equation}
 2 e^{i\xi_0}\;\cos{(x\cdot\delta)}
\;  \tilde{z}(x) =  \lambda \tilde{z}(x)
\label{13}
\end{equation}
where
\begin{equation}
 \tilde{z}(x) = \sum_p z(p) e^{ip\cdot x}
\label{14}
\end{equation}
Hence the sought eigenvalues and eigenvectors  are
\begin{equation}
\lambda_x = 2 e^{i\xi_0}\; \cos{(x\cdot\delta)}
\label{15}
\end{equation}
 and 
\begin{eqnarray}
\psi^{(s)}_x(p) = e^{-E(p)(1+2i\xi_1)/4T} \cos{(p\cdot x)}  \\ 
\psi^{(a)}_x(p) = e^{-E(p)(1+2i\xi_1)/4T} \sin{(p\cdot x)} 
\label{16}
\end{eqnarray}
When one considers a spectrum of momentum transfers, one gets
instead of (\ref{15}):
\begin{equation}
\lambda_x =  e^{i\xi_0} \; 
\sum_\delta \mu(\delta) \cos{(x\cdot\delta)} 
\label{15bis}
\end{equation}
with some $\mu(\delta)$, which we assume to be a positive weight
so that, generically, $x=0$ corresponds to the eigenvalue with
the largest modulus. The general solution of the Schroedinger equation is
\begin{equation}
\psi(p,t) = \sum_x [A_x \psi^{(a)}_x(p) + 
S_x \psi^{(s)}_x(p)] e^{-it\lambda_x}
\label{17}
\end{equation}
Let $0 < \xi_0 < \pi$.
Then only the contribution corresponding to $x=0$ 
survives in the limit $t \to + \infty$, because this is the one
which is "explosing" the most rapidly (remember that the
state is not normalized). Consequently, the asymptotic spectrum 
is $w(p) \sim e^{-E(p)/2T}$. This is the generic behavior, 
which can be avoided   by choosing very unnatural initial 
conditions only. At this point the goal we have set for 
ourselves at the beginning of this paper has been 
achieved: a rather robust quantum model with the desired 
properties has been constructed.
\par
Notice, that the physical contents of the classical and of
the quantum  model are    different. In the classical model 
a particle keeps changing its momentum. In the quantum 
model the components of a wave keep splitting. All
these components are a priori present at any time, only
their amplitudes get modified. However, when the
momentum transfer probabilities are in both models 
the same, the observable one-particle spectrum
has the same "thermal" shape, except that in the 
quantum case the "temperature" is twice larger. Moreover,
both models have the Markov property: the generic
limiting behavior is independent of the initial
state of the system.
\par
As already mentioned, the problem discussed in this
paper is of intrinsic theoretical interest,  independently
of the original phenomenological motivation
of the author. The subsequent discussion 
makes plausible, we believe, the conjecture alluded 
to in the introduction part of this paper: the 
evolution of the wave function of a system 
of very many interacting particles is likely to drive  
 the one-particle spectrum towards a 
"thermal" shape. However, our actual result is 
modest, it is merely a first step 
and much remains to be done. 
\par
As long as one stays within the one-particle framework,
several specific   assumptions of our quantum 
model can be relaxed. Still, even in this framework,
it would be interesting to check how generic is the
result obtained in this paper, viz.  does it hold for
processes which are significantly different from (\ref{2}).
The difficulty is mainly technical, at least if one looks
for an analytic solution.  However, going beyond the one-particle
picture and checking the conjecture in a genuine 
multiparticle context requires a significant conceptual 
advance. 
\par
On the phenomenology side, our exercise 
supports the idea that the "thermal" spectrum
observed in hadron-hadron collisions 
does actually reflect pseudo-thermalization 
affecting the structure of the
multiparton wave functions of the
incoming hadrons. If true, this is to large extent
independent of the details of the
microscopic dynamics - a common
feature of many multiparticle production
phenomena - except that the validity of
the parton picture requires an
asymptotically free theory, of course \cite{foot4}.
\par
{\bf Acknowledgements}: I am indebted to Roger Balian,
Andrzej Bialas, Ian Kogan and Jean-Pierre Leroy for helpful comments
and suggestions. This work was partially supported by
EC IHP grant HPRN-CT-1999-000161. Laboratoire de
Physique Th\'eorique is Unit\'e Mixte du CNRS UMR 8627.

\begin{thebibliography}{99}
\bibitem{foot1} The data
are best fitted with $E(p_\perp)$ set equal to the
transverse mass $\sqrt{p_{\perp}^2 +m^2}$. From
the theory point of view, one would perhaps prefer the
non-relativistic expression $p_\perp^2/2m$, because
the shape of the spectrum is expected to reflect the
galilean invariance of the original parton state. The
two choices are equivalent for soft enough $p_\perp$.
There is plenty of data the reader can consult.
Relatively recent ones can be found, for example,
in \cite{data}.
\bibitem{data} M. Aguilar-Benitez et al., Z. Phys. C50 (1991) 405.
\bibitem{foot2} In an alternative non-thermal
interpretation of (\ref{0}) one refers to the Schwinger mechanism
\cite{schw} of pair production in a uniform (chromo)electric
field \cite{strings}.
\bibitem{schw} J. Schwinger, Phys. Rev. 82 (1951) 664.
\bibitem{strings} E. Brezin and C. Itzykson, Phys. Rev. D2 (1970) 1191;\\
A. Casher, H. Neuberger and S. Nussinov, Phys. Rev. D20 (1979) 179;\\
B. Andersson, G. Gustafson and T. Sjoestrand, Z. Phys. C6 (1980) 235;\\
A. Bialas, Phys. Lett. B466 (1999) 301 .
\bibitem{foot3}  We do not claim that the transition
probabilities are exactly given by (\ref{2}) in real life.
There is an infinity of stochastic processes yielding the
same limit distribution.
\bibitem{foot4} Transverse momentum distributions
of the form (\ref{0}) are also observed in
jets produced in $e^+e^-$ annihilations.
In principle, we do not see any major
obstacle in extending the speculations of
this paper to the evolution of the
chromodynamical state produced there.
After this paper has been completed, our
attention was called to ref. \cite{bhal}
where a surprisingly
good and economic description of structure
functions is presented, using
a statistical ansatz. This might be
another example of pseudo-thermalization,
this time oberved directly at the parton level.
\bibitem{bhal}R.S. Bhalerao, Phys.Lett. B380 (1996) 1;
R.S. Bhalerao, N.G. Kelkar, B. Ram, Phys.Lett. B476 (2000) 285;
R.S. Bhalerao, Phys.Rev. C63 (2001) 025208.
\end{thebibliography}
\end{multicols}
\end{document}






