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%%%%%%%%%%%%%%%    COMANDI PERSONALI    %%%%%%%%%%%%%%%%
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\newcommand{\pp}{(p-p) }
\newcommand{\ud}{\mathrm{d}}
%\newcommand{\ee}{\textrm{\textbf{($\mathbf{e^+e^-}$) }}}
\newcommand{\ee}{$(e^+e^-)$ }
\newcommand{\dis}{\text{DIS }}
\newcommand{\non}{non }
\newcommand{\lead}{$\mathsf{L}$}   %%%%%% ATTENZ. ci sono le $...$: 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% non usare in modalita' math (19/02/01) %%%
\newcommand{\effl}{Leading Effect}
%\newcommand{\ma}{ma }
\newcommand{\atot}{$A_{\text{TOT}}$ }
\newcommand{\mx}{$M_{\text{X}}$ }
\newcommand{\meff}{$M_{\text{eff}}$ }
\newcommand{\rads}{$\sqrt{s}$ }
\newcommand{\ltot}{$\mathcal{L}_{\text{TOT}}$ }
\newcommand{\qht}{$M_{\text{X}}\equiv \sqrt{(q_{\textsc{tot}}^{\textsc{had}})^2}$}
\newcommand{\eneff}{Effective Energy}
\newcommand{\cms}{\textsc{cms}}
\newcommand{\nch}{$\big< n_{ch} \big>$ }
\newcommand{\aqs}{Available Quadri-Scalar}
\newcommand{\eht}{$E^{\textsc{had}}_{\textsc{tot}}$ }
\newcommand{\ml}{$M_{\text{L}}$ }
\newcommand{\qlt}{$M_{\text{L}}\equiv \sqrt{(q_{\textsc{tot}}^{\textsc{lead}})^2}$}
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\author{Corrado Appignani\\[1mm] \emph{Dipartimento di Fisica - 
Universit\`a di Bologna - I-40126 - Italy}}
\title{Introduction to the Total Available Quadri-Scalar:\,\atot}
\begin{document}
\maketitle
\begin{abstract}
A new variable, the ``Total Available Quadri-Scalar'' \atot\!\!, is defined and its
identification with the invariant quantity effectively available to the multihadronic
final states production is proposed. Its definition is justified by means of simple
kinematical considerations and its consistency with the so-called \eneff\ is proved in
the specific situations where the latter works. Using \atot to re-plot existing data, the
quantity \nch is shown to be a ``Universality Feature'' (independent from the process
under exam, the collider nominal energy, and the hadronic invariant mass) as required by
QCD Universality.
\end{abstract}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    INTRODUZIONE     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Basically speaking, QCD shows three main features. One of them is perturbative in nature
and is Asymptotic Freedom (due to a negative $\beta$ function), while the other two
are non--perturbative. These are: Confinement, that is quarks and the gluons
cannot exist in a free state but only inside the hadrons, and \eneff. 

The latter is a very important but forgotten property of QCD and is the least studied
even if it could shed some light on and provide an alternative approach to the question
of how the hadronization  mechanism works. The fundamental idea that lies behind the
\eneff\ approach is that of distinguishing two main phases inside any given interaction,
namely, the Quantum Number Conservation (or Flow), and the Hadronization in the proper
sense of the word, that is the process by which quarks and gluons ``hadronize'' and
become observable matter (hadrons). 

Some papers published in the early '80s \cite{evid,nc2,nc3,nc4} showed that, whatever the
invariant quantity available to the hadronization process is, it should equate the total
hadronic energy of the whole system evaluated in the CMS (from where the name \eneff).
Despite the great importance of such result, up to now, neither the correct invariant
representation has been identified, nor a satisfactory justification for the choice of the
quantities adopted in the past to represent it has been given. 

In fact, the quantity commonly used to plot data and study the world-widely collected
experimental results is \qht\  but, as showed below, this quantity cannot work if we
insist in separating the aforesaid phases of the interaction. Furthermore, as some recent
papers show, (see e.g.~\cite{osaka}) \mx\ is unable to reveal the Universality Features
when also DIS processes are taken into consideration.

Purposes of this paper are to introduce, by means of simple kinematical considerations,
what is believed to be the correct Lorentz--invariant representation of the quantity from
now on called the ``\aqs'' and to show how it is possible, by correcting the above
mentioned result, to let the Universality Features be ``revealed again''.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%       STORIA        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{History} The fact that the total energy available to particles production in a
given type of interaction is not in general the nominal energy of the reaction but
another quantity that takes into account the \effl, is one of the most important
discoveries made in the early '80s \cite{creation}. Before that time, all the measured
quantities were analysed in terms of \rads\ and this brought to different results in
different experiments. This was commonly accepted even if it was in a flagrant contrast
with the QCD Universality.

In fact, at a fundamental level, the final state of any interaction should depend only on
some (Lorentz-invariant) scalar variable believed to be available to the hadronization
mechanism, but not on how that quantity has been put together. This means that, had this
quantity been \rads\  (as thought before), then the results of any analysis should have been
the same, independently on the kind of reaction under exam (e.g. \!\ee\!\!, \pp\!\!,
DIS). Instead, the results were all different and there were no explanation for this
situation, to which people referred as ``the hidden side of QCD''.

As mentioned, in the early '80s it was pointed out \cite{evid} that QCD Universality could
have been made explicit if the quantity \eht\!, that is, the total hadronic energy of the
reaction given by subtracting the energy of the leading particle(s) from the nominal
energy of the reaction (\rads), was used to plot data instead of \rads\!\!. Within few
years, this effect, called the \effl, would have been shown to be universal: in fact, no
matter if the interaction studied was strong, EM or weak, the \effl\ was always present
\cite{nc10,hadmes,creation}.

\eht was soon after called the ``Effective Energy'', the name claiming for \eht
to be the portion of energy effectively available to the production of the multihadronic
systems present in the final state, after that the energy of the leading particle had been
subtracted.

The leading particle was defined to be the particle leaving the interaction vertex with
the highest longitudinal momentum \cite{evid}. The role of this particle was to carry,
totally or partially, the Quantum Numbers (as $J^{PC}$ or flavour) from the initial to
the final state. The transfer of these quantum numbers from the reacting particle(s) to
the leading particle was called the Quantum Number Flow (QNF).

From 1980 to 1984 a series of experiments were conducted at the ISR (CERN) to establish
if \eht was the Effective Energy. All these experiments proved that there were no 
differences between \pp and \ee collisions results if \eht was used to perform the analyses. The
quantities measured in this experiments were called the ``Universality Features'' as they
showed the same behaviour whatever the kind of experiment and the nominal energy of the
collider. Some of them are:
\begin{enumerate} 
\renewcommand{\labelenumi}{ \arabic{enumi} ) \ }
\item ${\displaystyle \big< n_{ch}\big>\ =\ }$ Mean charged particle 
multiplicity ~\cite{nc3}\\[-2mm]
\item ${\displaystyle \frac{\ud\sigma}{\ud x_{\textsc{r}}}}$ = 
\ Fractional energy distribution ~\cite{evid,nc2}\\[-2mm]
\item ${\displaystyle \frac{\ud\sigma}{\ud p^2_t}}$ = \ Transverse squared momentum
distribution ~\cite{nc9}
\item ${\displaystyle \frac{\ud\sigma}{\ud(\frac{p_t}{<p_t>})}}$ = \ Reduced transverse
momentum distribution ~\cite{nc8}
\item ${\displaystyle \alpha\equiv\frac{E_{ch}}{E_{\textsc{had}}}}$ = \ ``Charged'' energy
~\cite{nc5}
\item ${\displaystyle \frac{\ud\sigma}{\ud \big< p_T^2 \big>_{\textsc{in,out}}}}$ = \ 
Event planarity ~\cite{nc4}
\item $P(n_{ch})$ = \ Charged particle multiplicity distribution ~\cite{nc15} \\[-3mm]
\item $N^o$ of propagating quarks vs \lead ~\cite{nc10}
\end{enumerate}

Some studies regarding DIS processes were also made, but only using DIS variables to
plot \pp data, being impossible to re-analyse DIS data in terms of \eht\! 
\cite{nc11,nc14}.

They showed some matching between the two curves but were almost useless to decide whether
\eht worked in DIS as well as it did with other processes, as the variable used was $W^2$
that does not take into account the \effl.

In 1984 the ISR closed and no further  intensive experimental work was planned in this
important field of non-perturbative QCD. The last relevant result was obtained in 1997 at
LEP (CERN), when the meson $\eta'$, very  rich in gluon content, was seen to be produced
in gluon induced jets as a leading particle, that is, having an anomalous high
longitudinal momentum \cite{etalead}. This completed the series of the experiments aimed
to establish the universality of the \effl\  in \pp and \ee processes but nothing of
conclusive had been still obtained in relation with the DIS processes.

As invariant representation of \eht was chosen \qht\: \cite{nc11,nc15,usomx,creation} in
consistency with the previous use of \rads: again a total invariant mass had been chosen
to be the fundamental quantity from which the hadronization should have been depending.
Indeed, \mx\ reduces to \eht when evaluated in the CMS and, as all the experiments were
made in balanced colliders ({\sc cms=lab}), we have
(\eht)$_{\text{CMS\!}}=\;$(\eht)$_{\text{LAB}}$, so the latter was in effect the right
quantity to  be measured.

Nowadays, because of an unjustified extention of the use of \mx to DIS processes,
the almost totality of the experimental works use \mx to study the behaviour of
a given quantity, and the invariant mass is generally but wrongly believed to be the
correct representation for the \eneff. Its use is going on even when a fundamental
quantity as \nch has been shown to be no more universal if also DIS data are considered
\cite{osaka}.

From the viewpoint adopted here, that paper proved that \mx cannot represent the
invariant quantity effectively available for particle production and here is a first
indication that should have long since been considered: the logical step that
brought to the introduction of the \eneff\ was to separate the interaction in the two
sub-interactions yet mentioned. Now, if we insist on associating the total invariant mass
to any given particle system, we should also make the association:\\
\vspace{-5mm}
\begin{center}
\effl \ \ $\longleftrightarrow$ \ \qlt
\end{center}
But, as it is easily seen, an invariant-mass-type quantity cannot be considered as the
variable to be associated to any sub-system into which the whole system is being divided
as
\begin{equation}
\sqrt{s} \; \neq \; M_ {\text{X}}\; + \; M_{\text{L}} 
\end{equation}

So we have to make our choice: either we separate the final state particles into
leading and hadronic, or we use invariant masses, but not both.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%       INTRODUZIONE DI ATOT       %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction of {\boldmath \atot}}
Firstly let us reconsider the ``leading approach''. As recalled more than once, the
interaction is divided into two processes:
\begin{tabbing} 
Leading Effect ~~\= $\equiv$~~ \=the quantum number conservation mechanism \\[1.5mm]
Hadronization      \>$\equiv$ \>the mechanism that transforms some available quantity \\ 
                                \>~~ \>          into particles mass  
\end{tabbing}

The quantum numbers flow from the incident particle to the leading particle (usually but
not always identifiable with the so-called remnant) \cite{nc10}. This leading particle
and its 4-momentum are no more considered. They are not part of the hadronic system
produced. What is left is what is finally studied and analysed.

And here we introduce our doubt:
is it after all correct to ignore the whole leading particle?
The answer is: No. In fact the leading particle leaves the vertex following a trajectory that
is not fully constrained by the particle whose Quantum Numbers it is carrying. For
example, if the leading particle is the proton remnant, it is indeed true that it will
have a large longitudinal momentum, that is, it will be strongly aligned with the incident
particle axis, but not exactly nor completely.

Keeping in mind that there are some exceptions to the situation we are going to describe,
let us try to show the problem by schematizing the incident and the leading particles in a
general situation, see figure \ref{collis}.

\begin{figure}[!h] 
\begin{center} \vspace*{-3.5cm} 
\includegraphics[width=0.87\textwidth]{collis.eps}
\label{collis}
\end{center} \vspace*{-2.5cm}
\caption{Simplified sketch of a collision: in the initial state (top) are represented the
two incident particles, while in the final state (bottom) only the two leading particles
are shown. The transverse contribution to their 4-momenta cannot originate from the
4-momenta  of the incident quarks and must be a consequence of the collision, that is, of
the quark(s) loss/exchange: for this reason it cannot be associated to QNF and must not be
subtracted to \rads\, (see text).}
\end{figure}

This treatment is by now 3-D but we will soon turn to the 4-D approach by simply
extending the formalism, that is, decomponing the energy into its longitudinal and
transverse component.

As we are talking about the QNF, and as these quantum numbers ought to be carried by the
leading particle, it is correct to refer to the quantum numbers trajectory visualizing
that of the leading particles.

Now, if the longitudinal component of the leading particle can be safely associated with
that of the quantum numbers propagating from the initial state, this cannot be the same for
the transverse component. The latter must come from a quark exchange with the other incident
particle, when the leading particle is the same as the incident particle, or from the
quark(s) loss/exchange, in the case where the leading particle is different from the incident
one.

This simple kinematical remark immediately suggests the following hypothesis that
in turn brings to the introduction and allows the successive derivation of \atot\!.
Turning to the 4-D treatment and identifying any particle with its 4-momentum, it is
conjectured that:
\begin{quote} \vspace{-1mm}
The component of the $q^{\textsc{lead}}_{\textsc{tot}}$ that is transverse to the QNF 
is not leading. As such, it must not be subtracted from \rads\ to get the Total \aqs.
\footnote{Here a short note on the name chosen for this new invariant is appropriate: it
is simply the most compact name possible among the correct names. In fact, it would be
wrong to use the word ``energy'', as well as it is not correct to use the term ``mass''
as \atot is not an invariant-mass-like quantity. So, the good practice to call things
with a name that correctly describes their nature has been adopted.}
\end{quote}

The longitudinal component of $q^{\textsc{lead}}_{\textsc{tot}}$ will be called \ltot and
represents the cost to be paid in a single event to force the quantum numbers
conservation. What is left from \rads\,is the invariant to be associated to the
multihadronic final state and that is available to its formation. 
This is the Lorentz-invariant quantity that should be used to analyse
every experimental result. Summarizing we have:
\begin{equation*}
\sqrt{s} \ \ \longrightarrow \ \ 
\begin{cases}
\;\mathcal{L}_{\text{TOT}} &\text{goes to the \effl}\\[3mm]
\;A_{\text{TOT}} &\text{goes to the Hadronization}
\end{cases} 
\end{equation*}

The indices ``\te{tot}'' stays for ``\te{total}'' and reminds us that the whole product of
the interaction is being considered. Anyway, in principle, nothing would prevent us from
considering one hemisphere only by defining $A_1$ (for hemisphere 1) and $A_2$ (for
hemisphere 2). It is easy, in such case, to change the definition of \atot given below: it
suffices to consider the leading (or hadronic) quadrimomentum measured in one
hemisphere instead that looking at both. This situation will not be analysed here and when
we talk about the \aqs\ we are actually talking about the Total \aqs.

It is important at this point to distinguish two kinds of \effl\ that will be
called here direct and indirect. The first type is observed in all the collisions but the
annihilation processes. The second one regards the annihilation processes only.

But how is it possible to have a \effl\ if the incident particles annihilates?  Actually,
it is still possible to talk about a \effl\ but it is no more linked to the propagation
of the quantum numbers of the incident particles. Rather, it is referred to the quantum
numbers of the particles formed by pair production from the $\gamma/Z^0$ produced at
the annihilation vertex.

This second type of \effl\ has been observed in \ee annihilations (it is present at a 1\%
rate) and the most important studies regarded the production of the charmed meson $D^*$
\cite{creation} and that of the ``gluonic'' meson $\eta'$ \cite{etalead}.

As for the $D^*$, the propagating quantum numbers are those of a $c$ or $\bar{c}$ quark
while, when the $\eta'$ is concerned, its strong gluonic component tells us that it is
carrying the quantum numbers of the gluon: in fact, the $\eta'$ appears to be leading
only when emerging from a gluon induced jet (3 or 4-jet events).

Why this distinction? Apart from the conceptual, there is a practical reason to insist on
it. In fact, what we are going to do below, cannot be applied to the indirect \effl,  and
the given definition of \atot itself ought to be changed. The appropriate formulation in
this case is also given below but the experimental evidence has not been pursued yet.

Is anyway evident that, even in such situation, some projection must be done. We can guess
that the axis on which $q^{\textsc{lead}}$ should be projected is the axis given by the mean
momentum of the jet containing the leading particle, namely,  the ``charmed'' jet or the
``gluonic'' jet respectively. 

Keeping in mind this distinction between these two different leading effects, it is now
possible to find a mathematical expression for the quadri-scalar available to the
energy-into-mass transformation.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%    FORMAL DEFINITION OF \LTOT AND \ATOT    %%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Formal definition of {\boldmath \ltot and \atot}}
Recalling what affirmed in the previous section, and using the Minkowskyan metric
instead of the Euclidean, it is immediate to write the expression for \ltot\!:
\begin{equation}
\mathcal{L}_{\text{TOT}} \:\equiv  \: q^{\te{lead}}_{\te{tot}}\cdot
\frac{q^{\te{inc}}_{\te{tot}}}{\sqrt{s}} 
\end{equation}

According to our point of view, this represents the cost of the quantum numbers
conservation in a given event.
What is left is what the physics assign to particle production:
\begin{equation}
%\begin{split}
A_{\text{TOT}} \ \equiv \ \sqrt{(q^{\te{inc}}_{\te{tot}})^2}-q^{\te{lead}}_{\te{tot}}\cdot 
\frac{q^{\te{inc}}_{\te{tot}}}{\sqrt{s}}
\label{eq:at}
%\end{split}
\end{equation}
or, in compact notation:
\begin{equation}
A_{\text{TOT}} \equiv \sqrt{s}- \mathcal{L}_{\text{TOT}}
\label{eq:comp}
\end{equation}
This is the correct formal definition of \atot but we should now find a manageable expression
to be used below. A possible one is given by the expansion:
\begin{equation}
\sqrt{(q^{\te{inc}}_{\te{tot}})^2}=\frac{(q^{\te{inc}}_{\te{tot}})^2}
{\sqrt{(q^{\te{inc}}_{\te{tot}})^2}}
\end{equation}
Substituting the previous in (\ref{eq:at}) we get:
\begin{equation}
A_{\text{TOT}}=q^{\te{had}}_{\te{tot}}\cdot \frac{q^{\te{inc}}_{\te{tot}}}{\sqrt{s}}
\end{equation}
that is, \atot is the projection of the sum of all the 4-momenta of the produced hadrons
(excluding of course the leading particles) on the axis given by the incident particles.

It must be recalled again that this result is only valid for the direct \effl. When we
have to do with the indirect, we should write:
\begin{equation}
\mathcal{L}_{\text{TOT}}(\textsf{ind}) \:\equiv  \: q^{\te{lead}}_{\te{tot}}\cdot
\frac{q^{\te{jet}}_{\te{tot}}}{\sqrt{(q^{\te{jet}}_{\te{tot}})^2}} 
\end{equation}
and coherently modify the expression for \atot\!. The last is valid provided that there is
no further \effl\ observed in the other jets (in which case the definition of \ltot
should be changed accordingly).

Here, the measurement of $q^{\te{jet}}_{\te{tot}}$ ought to be done in addition to that
of $q^{\te{lead}}_{\te{tot}}$: the situation would be only a little bit more complicated.

First of all, the consistency with the previous works must be tested. What these works
showed is: if the variable 
\begin{equation}
E^{\textsc{had}}_{\textsc{tot}} =
E^{\textsc{inc}}_{\textsc{tot}} - E^{\textsc{lead}}_{\textsc{tot}} 
\end{equation}
is used to perform data analysis, you find the Universality Features: a whole set of measured
quantities shows the same behaviour independently of everything but \mbox{\eht\!.} So the 
value of \atot in terms of \eht must be estimated.

The colliders where the \effl\ was studied (ISR, LEP, HERA$(e^+e^-)$) were all balanced, so
that 
\begin{equation}
A_{\text{TOT}} = (E^{\te{had}}_{\te{tot}})_{\te{lab}}= 
(E^{\te{had}}_{\te{tot}})_{\cms}~~~~~\text{(balanced colliders)}
\end{equation}
must hold. In fact, what we get
if we specialize to the CMS is:
\begin{equation} 
\begin{split}
A_{\text{TOT}} \ & =\  \:q^{\te{had}}_{\te{tot}}\cdot
\frac{q^{\te{inc}}_{\te{tot}}}{\sqrt{s}}\: =\:
(E^{\te{had}}_{\te{tot}};\vec{p}\,^{\te{had}}_{\te{tot}})_{\te{cms}}\cdot 
\frac{(E^{\te{inc}}_{\te{tot}};\vec{0})_{\te{cms}}}{\sqrt{s}} \ =\\[4mm]  & =\ \bigg( 
\frac{E^{\te{had}}_{\te{tot}}\cdot E^{\te{inc}}_{\te{tot}}}{E^{\te{inc}}_{\te{tot}}}
\bigg)_{\te{cms}} = \ (E^{\te{had}}_{\te{tot}})_{\te{cms}}
\label{eq:aeht}
\end{split} 
\end{equation}
and there are no consistency troubles as we recover the variable used in that early works
performed in the simple CMS limit.
The previous equation also shows that:
\begin{equation}
A_{\text{TOT}} \neq (E^{\textsc{had}}_{\textsc{tot}})_{\te{non cms}}
\end{equation}
so that, in general, \eht is not the \eneff. In other words, at an unbalanced collider, the
direct use of \eht will not work and is not correct.

This means that the Universality Features were discovered thanks to the fact that the
studies were automatically conducted in the system where the \aqs\ matches the energy of
the hadrons produced (excluding the leading particles). This is also the reason that
induced the discoverers to think that a non-invariant quantity like the hadronic energy,
could have had such a fundamental role. Probably, had those colliders been unbalanced (like
DIS studies at HERA are nowadays), the discovery of the Universality Features would not have
been made.

In order to be exhaustive, it should be said that some late works made at the ISR were
performed using \mx instead of \eht \cite{nc11,nc15,usomx,creation} but the Universality
Features were evident anyway, even if, in general, \mx$\neq$ \eht\!.
How is it possible? Later on it will be showed how
the reason for \mx worked so well in those occasions, lies in the particular cuts made on
the data set at that time.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    RELAZIONE  FRA   MX  E  ATOT    %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Relation between {\boldmath \mx and \atot}}
As mentioned, in general, \atot$\neq$ \mx\!. Can we get an useful expression for their
difference? The best way to get it, is to estimate the quantity
$(A_{\text{TOT}}^2-M_{\text{X}}^2)$:
\begin{align}
&A^2_{\text{TOT}}=(\sqrt{s}-q^{\te{lead}}_{\te{tot}}\cdot \frac{q^{\te{inc}}_{\te{tot}}}
{\sqrt{s}})^2 = s-2q^{\te{lead}}_{\te{tot}}\,q^{\te{inc}}_{\te{tot}}+
\frac{(q^{\te{lead}}_{\te{tot}}\,q^{\te{inc}}_{\te{tot}})^2}{s}
\label{eq:a}\\[4mm]
&M^2_{\text{X}}=(q^{\te{inc}}_{\te{tot}}-q^{\te{lead}}_{\te{tot}})^2= 
s-2q^{\te{inc}}_{\te{tot}}\,q^{\te{lead}}_{\te{tot}}+(q^{\te{lead}}_{\te{tot}})^2 
\label{eq:m}
\end{align} 

\noindent so that the difference is all in the last terms:
\begin{equation}
(A_{\text{TOT}}^2-M_{\text{X}}^2)=
\Bigg[ \frac{(q^{\te{lead}}_{\te{tot}}q^{\te{inc}}_{\te{tot}})^2}{s} - 
(q^{\te{lead}}_{\te{tot}})^2 \Bigg]
\end{equation}

This is the correct invariant expression for the difference we are estimating but we also 
need a manageable expression that is function of some measured quantities in order to
perform the calculations that follow. 

To simplify the notation we introduce the following abbreviations for the indices Total,
Incident, Hadronic, Leading:
\begin{eqnarray*} \vspace*{-6mm}
\te{inc} &\longleftrightarrow &\te{i}\\[-2mm]
\te{lead} &\longleftrightarrow &\te{l}\\[-2mm]
\te{had} &\longleftrightarrow &\te{h}\\[-2mm]
\te{tot} &\longleftrightarrow &\te{t}
\end{eqnarray*}
Choosing again the CMS we have
\begin{equation} 
\begin{split}
\Big[ A^2_{\text{TOT}}-M^2_{\text{X}}  \Big] &=\bigg[ \frac{(q^L_Tq^I_T)^2}{s}-(q^L_T)^2 \bigg]
=(E^L_T)^2_{\cms}-(q^L_T)^2=\\[4mm]
&=(E^L_T)^2_{\cms}-[(E^L_T)^2_{\cms}%
-(\vec{p}\,^L_T)^2_{\cms}]=(\vec{p}\,^L_T)^2_{\cms}=(\vec{p}\,^H_T)^2_{\cms}\\[-2mm] 
\label{eq:am}
\end{split} 
\end{equation}
as is also deducible by observing that:
\begin{equation}
\begin{cases}
A^2_{\text{TOT}}=(E^H_T)_{\cms}^2\\[2mm]
M^2_{\text{X}}=(E^H_T)_{\cms}^2 - (\vec{p}\,^H_T)_{\cms}^2
\end{cases}
\end{equation}
so that
\begin{equation}
\Big[ A^2_{\text{TOT}}-M^2_{\text{X}}  \Big] \ = \ (\vec{p}\,^H_T)_{\cms}^2
\tag{\ref{eq:am}\,$'$}
\label{eq:amam}
\end{equation}

\noindent The importance of the equation (\ref{eq:amam}) is threefold:
\begin{enumerate}
\item It shows that \atot $\ge$ \mx that is, the hadronic system has more 4-scalar at
disposal for particles production than what was thought before;
\item It gives a very simple way to re-analyse data that were wrongly plotted vs \mx\!\!:
it is in fact sufficient to substitute (event by event) \mx with the sum in quadrature of
\mx and  $(\vec{p}\,^L_T)^2_{\cms}$:
\begin{align}
&M_{\text{X}} \longrightarrow \sqrt{M^2_{\text{X}}+(\vec{p}\,^L_T)^2_{\cms}}
~~~~~~~~~~~~~~~\text{or} \label{eq:unom}\\[4mm]
&A_{\text{TOT}}\;=\;\sqrt{M^2_{\text{X}}+(\vec{p}\,^L_T)^2_{\cms}} 
\tag{\ref{eq:unom}\,$'$}
\label{eq:amxpl}
\end{align}
here the knowledge of $(\vec{p}\,^L_T)^2_{\cms}$ is necessary; otherwise the previous
formula becomes useless and the method showed later on must be employed;
\item It highlights the importance of the unbalancing in $(\vec{p}\,^L_T)^2_{\cms}$ between
the two hemispheres: what makes the difference between \atot and \mx is not linked to the
absolute value of the \effl\ but to its unbalancing: the more this unbalancing is, the
less \mx is able to highlight the Universality Features.
In other words, you can have a strong, a medium, or a small \effl: if the momentum of the
leading particles in one hemisphere is equal to that of the other, you will always find
that \atot= \mx, regardless of their absolute value.
\begin{equation}
A_{\text{TOT}}= M_{\text{X}}= (E^{H}_{T})_{\cms} \hspace{1.2cm} 
\big( \text{ if~~~} \ (\vec{p}\,^H_1+\vec{p}\,^H_2)_{\cms}=0\ \big)
\end{equation}
\end{enumerate}

In view of the application of \atot to compare \pp and \ee data, another
fact must be discussed here, not only for future use, but also to check the previous
results compatibility.

When the confrontation between \pp and \ee data were made, the latter was plotted vs
\rads\!. When we do not have a \effl\ in \ee annihilation, the relation \rads=\mx holds. 
But, as stated before, a strong \effl\ in \ee processes is only present at a 1\% rate. 
In the other events you have such a low \effl\ that it can be neglected.

Besides, when an average among all the events is performed, this 1\% showing a large
\effl\ is easily overcome by the remaining 99\%, in the sense that the difference between
\mx and \rads is far below the experimental uncertainties. This means that, for those
samples of \ee data, you have 
\[ \sqrt{s} = M_{\text{X}} \]
and, as \ee colliders always present the feature {\sc lab=cms}, the relation 
\[  M_{\text{X}} = E^{\textsc{had}}_{\textsc{tot}} \]
holds too.
\noindent In other words, for our purposes, in \ee processes:
\begin{equation}
A_{\text{TOT}}= M_{\text{X}}= \sqrt{s}= E^{\textsc{had}}_{\textsc{tot}}
\text{~~~~~~~~~~~~~~~~(\ee ~processes)}
\end{equation}
and we have not to worry about which variable was used. This will turn out to be
important later on, where an experimental evidence of what stated up to
now is provided.

Before turning to check the \aqs\ existence conjecture, another important fact needs to be
highlighted. The theoretically (or empirically) derived conservation laws at vertices are
always expressed in terms of 4-vectorial quantities (like 4-momenta, tensors \ldots). This
is true even in the field of the \effl, where the conservation law at the interaction
vertex reads:
\begin{equation}
q^{\te{had}}_{\te{tot}} = q^{\te{inc}}_{\te{tot}} - q^{\te{lead}}_{\te{tot}}
\label{eq:qmc} 
\end{equation}

On the contrary, when the experimental results are analysed, you always have to do with
some quadri-scalar quantity like \rads or \mx (as the final product of any analysis is almost 
always some scalar function of 1 variable).

The approach pursued here provides a consistent jump between the former and the latter as
it allows to get from the (\ref{eq:qmc}) a quadri-scalar conservation law:
\begin{equation}
\sqrt{s} = A_{\text{TOT}} + \mathcal{L}_{\text{TOT}}
\tag{\ref{eq:comp}\,$'$}
\end{equation}

It seems to be a trivial step but, as previously noted, the world-wide use of \mx proves
that it often happens that we do not recognize simple facts as a conservation law
violation: the same use of \mx violates the well established conservation of 4-momentum
at interaction vertices and of the whole interaction itself.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%    EVIDENCE OF ATOT UNIVERSALITY   %%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Evidence of {\boldmath \nch vs \atot universality}}
Here the legitimacy of the \aqs\ existence conjecture is proved. The proof is obtained by
rescaling the result an already published paper \cite{osaka}. \footnote{Presently, I have 
no access to row data}

It is appropriate here to briefly summarize the results of that paper: it is a study of
the quantity \nch in DIS processes. The main result is that \nch\!$_{\!\te{dis}}$ is more
than 15\% higher than \nch measured for \pp and \ee processes. This is well above the
experimental uncertainties, so the conclusion was: DIS processes disagree with the \effl
phenomenology and ``violates'' QCD universality. Figure \ref{osaka} shows the variable
\nch as measured in \cite{osaka}.
\begin{figure}[!h]
\begin{center}
\hspace*{-7mm} 
\includegraphics[width=1.1\textwidth]{multipdis_4.ps}
\caption{The mean charged particle multiplicity in DIS events plotted as a function of 
\meff and confronted with the same quantity measured for other processes \cite{osaka}.
The inner bars show the statistical errors, the outer ones show the statistical and 
systematic errors added in quadrature.}
\label{osaka}
\end{center}
\end{figure}

The possible explanation given by the authors deals with a supposed high contribution
from events with real gluon exchange ($t$ channel) and a resulting production of
($q\bar{q}$)  pairs in a colour-octet state. It is indeed true that this contribution
cannot be present in \ee events, but it should be in \pp events too; instead, \pp and \ee
data perfectly match each other. Furthermore, it has not been proved that this
contribution brings to a higher particle/mass ratio and honestly, there is no reason
for that to be so, as what is transformed in a real gluon will not be transformed in a
$\gamma$ or $Z^0$ and the contribution of the gluon is not ``more electrically
charging'' than the latter ones (the gluon having zero electrical charge) so that,
in the end, the particle/mass ratio should not change.
%In other words, the real gluon cannot come from the nothing and its production subtracts
%energy from other gauge bosons production.

The study were conducted in a reduced phase space. The charged tracks and the energies
were only measured in the polar angle interval $\theta_{\te{lab}}\! \in[20^0;160^0]$, 
see Figure \ref{intpol}. 
\begin{figure}[!h]
\begin{center}
\includegraphics[width=\textwidth]{multipdis_2.ps}
\caption{Typical DIS event in the ZEUS detector. The shaded areas are those not considered
in the analysis performed in \cite{osaka}.}
\label{intpol}
\end{center}
\end{figure}

This cut is justified by a large decrease in the experimental uncertainties and, as proved
in the same paper by means of MC simulation, this does not affect the result in anyway.
The other cuts are not relevant and are those normally employed to obtain an almost pure
collection of DIS events. More details may be found in \cite{osaka} or below.

Accordingly to the polar angle cut, the variable \mx is substituted with
\mbox{\meff\!:} it is again a hadronic invariant mass but this time it is not the
total one, but that measured in the $\Delta \theta$ employed. Again, MC simulation proved
that the dependence between \nch and \mx (\meff\!) is not affected by the phase space
reduction.

From now on, unless otherwise stated, all variables will be referred to the reduced
phase space without introducing any new notation to recall this fact. The only variable
whose notation will be changed is \mx to \meff in order to help any comparison with
\cite{osaka}.

The correction factor will be of course the ratio between \atot and \mx\! and will be
called $a$. This factor will be applied to the abscissa in the plot showed in Figure
\ref{osaka}, and will be obtained with a high energy approximation (h.E.): this will cause
a light mismatch at low energies even when \atot is used for re-plotting. 
This approximation could be avoided by re-analysing the data in terms of \mbox{\atot\!\!.}

To give an estimation of the correction factor $a$ in our particular situation, the
general result (\ref{eq:am}) is useless as $\vec{p}^{\te{\:lead}}_{\:\te{tot}}$ was not
measured and also because this result is strictly valid only in the full phase
space. Thus the correction has been estimated by averaging over the two variables from
which the said ratio depends, namely $\theta$ and $y$ (as will be soon showed).
The latter is a new variable that will be defined in the next section and that expresses
the unbalancing in hadronic energy between the two hemispheres in a way suitable for our
calculation. This will be done taking into account every relevant cut used in the
original paper.

\noindent This time it is convenient to estimate the quantity 
$s \cdot (A_{\text{TOT}}^2-M^2_{\text{eff}})$:
\vspace{3mm}
\begin{equation}
\begin{split}
s\cdot[A_{\text{TOT}}^2-M_{\text{eff}}^2]
&=[E^HE^I-\vec{p}\,^H\vec{p}\,^I]^2-[(E^I)^2-(\vec{p}\,^I)^2][(E^H)^2-(\vec{p}\,^H)^2]=\\[4mm]
&=(E^HE^I)^2+(p^Hp^I\cos \theta)^2-2E^HE^Ip^Hp^I\cos \theta-\\[2mm]
&\qquad-(E^HE^I)^2-(p^Ip^H)^2+(E^Ip^H)^2+(E^Hp^I)^2=\\[4mm]
&=(p^Hp^I)^2(\cos^2\theta-1)-2(E^HE^Ip^Hp^I)\cos \theta+\\[2mm]
&\qquad+(E^Ip^H)^2+(E^Hp^I)^2=\ldots  \label{eq:diff}  
\end{split} \vspace*{-3mm}
\end{equation} 
where: \vspace*{-2mm}
\begin{itemize}
\item[-] the index \te{tot} has been suppressed; 
\item[-] the absolute value of 3-vectors is denoted by suppressing the arrow above
the vectors. 
\end{itemize}
Of course $\theta$ is the angle between $\vec{p}\,^I$ and $\vec{p}\,^H$ that is, considering
the HERA kinematics, between $\vec{p}\,^H$ and the $z$ axis:
\begin{equation}
\theta =  \angle \:(\vec{p}\,^H_T\, ; \, \hat{z})
\end{equation}
This means that $\theta$ is the polar angle, and will be from now on identified with it.
Introducing the high energies approximation:
\begin{equation*}
\begin{cases} 
p^H\!\simeq E^H\\  p^{\,I}\simeq E^I
\end{cases}
\end{equation*}
and substituting in (\ref{eq:diff}) we get:
\begin{equation}
\begin{split}
\ldots&=(E^HE^I)^2(\cos^2 \theta-1)-2(E^HE^I)^2\cos \theta+2(E^HE^I)^2=\\[4mm]
&=(E^HE^I)^2(1-2\cos \theta+\cos^2\theta)=\\[4mm]
&=[E^HE^I(1-\cos \theta)]^2 
\end{split} \tag{\ref{eq:diff}\,$'$}
\end{equation}

\noindent To sum up, we have:
\begin{equation}
s\cdot[A_{TOT}^2-M_{\textrm{eff}}^2]\ \simeq\  [E^HE^I(1-\cos \theta)]^2\ 
\ \ \ \ \quad (E^{H,I} \simeq p^{H,I})
\end{equation}

In order to further simplify the last expression we are now forced to choose a reference
frame. Again the best choice is the CMS, both to simplify $s$ and for sake of
visualization. Besides, it is in general the most ``balanced'' system, and this helps in
avoiding any trouble with the high energies approximation. The only complication induced by this
choice is that it forces to perform the needed transformations on data
collected in the \te{lab} system.

\noindent Specializing ourselves to the \te{cms} we get:
\begin{equation}
s\cdot[A_{\text{TOT}}^2-M_{\text{eff}}^2]\ \simeq\ 
(E^H)^2_{\cms}(E^I)^2_{\cms}(1-\cos \theta)^2_{\cms}
\end{equation}
and using 
\begin{equation}
s=(E^{\te{inc}}_{\te{tot}})^2_{\cms}
\end{equation}
we obtain
\begin{equation}
[A_{\text{TOT}}^2-M_{\text{eff}}^2]\simeq (E^H)^2_{\cms}(1-\cos \theta)^2_{\cms}
\label{eq:atmxosa}
\end{equation}

As already mentioned, the difference (and thus the ratio) between \atot and \mx is a
function of two variables: $\Delta \theta$ and $y$. This means that, after having
imposed the cuts used in the paper to be corrected, considering the collision parameter
at HERA, and finally averaging over these two variables, we are left with a constant
dependence of \atot from \meff\!:
\begin{equation}
A_{\text{TOT}}\ \propto\ M_{\text{eff}} \qquad \text{(at fixed kinematical conditions)} 
\end{equation}

\noindent Thus, if in general we have:
\begin{equation}
\frac{A_{\text{TOT}}}{M_{\text{eff}}} \equiv a(\Delta \theta , y) \label{eq:aa}
\end{equation}
the previous remark allows the calculation of the constant 
$a=\big< a(\Delta \theta , y) \big>_{(\Delta \theta , y)}$ 
that expresses the ratio between \atot and \mx at fixed kinematical conditions:
\begin{equation}
a \; \equiv \; \big< a(\Delta \theta , y) \big>_{(\Delta \theta , y)} \qquad 
\text{(at fixed kinematical conditions)}
\end{equation}
Note that $a$ is invariant being the ratio between two invariants.

The dependence of $a$ from $\Delta \theta$ and $y$ will be used later on to show how it
is possible, by tuning these parameters that gives $a$ a particular value, to obtain a
different value for $a$ (up to a factor 2). This fact supports the correctness of the use
of \atot\!, as we obtain a value that allows DIS data to match the other processes curves
only if the cuts used in \cite{osaka} are taken into account.

\noindent Using the notation introduced in (\ref{eq:aa}), the (\ref{eq:atmxosa}) can be
rewritten as:
\begin{equation}
[a^2(\Delta \theta , y)-1] \cdot M^2_{\textrm{eff}} \; \simeq \; 
(E^H)^2_{\cms}(1-\cos \theta)^2_{\cms}
\tag{\ref{eq:atmxosa}\,$'$}
\end{equation}
or, isolating $a$,
\begin{equation}
a^2(\Delta \theta , y) \; \simeq \; 
1+\frac{(E^H)^2_{\cms}}{M^2_{\textrm{eff}}}\cdot(1-\cos \theta)^2_{\cms}
\label{eq:ageneral}
\end{equation}

\noindent The relevant cuts and numerical values are:
\begin{itemize}
\item  $\theta_{\textsc{lab}}\in [20^0;160^0]$: the already mentioned polar angle cut
\cite{osaka}. 
It must be re-evaluated in the \cms;
\item  $(E'_e)^{\textsc{lab}}_{\textsc{min}}=8\ \text{Gev}$, where the accent refers to
the final state and the ``$e\,$'' indicates the final state positron (this cut is
employed to guarantee an almost pure level of DIS events). Even this value  must be
re-evaluated in the \cms;
\item $\beta_{\textsc{boost}}$\,=\,0.93524 is the value of the boost between the \te{lab}
and the \cms\ at HERA.
\end{itemize}
and we are now ready to estimate the correction factor we are seeking,
that is, the mean of $a$ with respect to the variables from which it depends:
\begin{equation}
a \; \equiv \; \sqrt{1+\frac{\big< (1-\cos \theta)^2_{\cms} \big>\phantom{2}}%
{\big< M_{\textrm{eff}}/(E^H_T)_{\cms} \big>^2}}
\label{eq:val}
\end{equation}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   MEDIA SU  COS THETA   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Average over the polar angle}
In this section an accent is used to indicate \cms\ variables while, if no indices are
present, the variables are referred to the \te{lab}.
Writing the appropriate transformations for the 4-momenta
\begin{equation}
\begin{cases}
E'=\gamma(E-\beta p_z)\\[2mm]
p'_z=\gamma(p_z-\beta E)
\end{cases}
\end{equation}
and using again the high energies approximation in the form
\begin{equation}
\begin{cases}
p_z=p \cos \theta \simeq E \cos \theta\\[2mm]
p'_z=p' \cos \theta' \simeq E' \cos \theta'
\end{cases}
\end{equation}
we get
\begin{equation}
\begin{cases}
E'=\gamma(E-\beta E \cos \theta)\\[2mm]
E' \cos \theta'=\gamma(E \cos \theta-\beta E)
\end{cases}
\end{equation}
so that
\begin{equation}
\cos \theta' \simeq \frac{\gamma E(\cos \theta-\beta)}{\gamma E(1- \beta \cos \theta)}=
 \frac{\cos \theta-\beta}{1- \beta \cos \theta}
\end{equation}
that is the formula for beams collimation. The use of the last result is again
cause of some overestimation of $a$ at low energies.

Working in the \cms\ allows to perform the mean over the polar angle by simply averaging
over the distribution obtained in (\ref{eq:ageneral}) as, if the sample is large enough, 
$\big< \theta \big>_{\cms} \simeq \, \pi/2$\,.

Using the interval given in \cite{osaka}, \!$\theta \in[20^0;160^0]$, and performing the
needed transformations we get
\begin{equation}
\begin{cases}
20^0 &\longrightarrow ~87.895^0~ ~\:=~ 1.5361~ \text{rad}\\[2mm]
160^0 &\longrightarrow ~176.305^0~ =~ 3.0771~ \text{rad}
\end{cases}
\end{equation}
that represents the extremes of integration to be used. The integral yields
\begin{equation}
\big< (1-\cos \theta)^2 \big>_{\cms} \: =\: \frac{{\displaystyle \int_{1.5361}^{3.0771} 
(1-\cos \theta)^2 \ud \theta}}{\Delta\theta} \: \simeq \: 2.679
\end{equation}
so that
\begin{equation}
a(y) \: \equiv\:  \big<\, a(\Delta \theta,y)\, \big>_{(\Delta \theta)} \: \simeq \:
\sqrt{1+\frac{2.679}{(M_{\text{eff}}/E^H)^2}}
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%       MEDIA SU  y y y     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Average over the unbalancing variable \,{\boldmath ``$y$''}}
First of all, it is necessary to define a variable that suitably represents the unbalancing
in hadronic energy between the two hemispheres. This will be called $y$ and is defined as
follows:
\begin{equation} \begin{cases} 
E^H_{1,p}=y\cdot E^H_T\\[2mm] E^H_{2,e}=(1-y)\cdot E^H_T
\label{eq:ydef}
\end{cases} \end{equation}
that is, $y$ is the fraction of hadronic energy measured in hemisphere 1 (that of the
``target region'', where the proton remnant is detected).

By using $y$ we can express \mx as a function of \eht in a very useful way. In fact,
assuming that $\hat{k}\,^H_1 \simeq -\hat{k}\,^H_2$ (where $\hat{k}\,^H$ represents the
hadronic 3-momentum versor) we have
\begin{equation}
\begin{split}
M_{\text{X}} \equiv \sqrt{(q_{\te{tot}}^{\te{had}})^2} &= 
\big[ (q^H_1)^2 + (q^H_2)^2 + 2E^H_1E^H_2 -%
2\vec{p}\,^H_1 \cdot \vec{p}\,^H_2 \big]^{1/2} \simeq (~~\text{h.E.}) \\[2mm]
& \simeq \big[ 2E^H_1E^H_2 + 2E^H_1E^H_2 \big]^{1/2} = 2 \sqrt{E^H_1E^H_2}
\end{split}
\end{equation}
that together with definition (\ref{eq:ydef}) yields
\begin{equation}
M_{\text{X}} \simeq 2 \sqrt{E^H_1E^H_2} = 2E^{\te{had}}_{\te{tot}} \sqrt{y(1-y)}
\label{eq:mxdiy}
\end{equation}

It must be pointed out here, that this method only works if we have 2 hadronic jets. In
case of 3 or more jets, another method should be employed or a direct data analysis must be
performed.

The hadronic energies employed above are ``real'' hadronic energies, in the sense that the
symbols represent what is really produced at the interaction vertex independently from the
cuts applied to perform the analysis.

This does not create problems in case of a phase space reduction as there is no
dependence of \mx(\eht\!\!) from the phase space \cite{osaka}. On the contrary, the cut
in the final positron energy must be considered, as the events that do not meet this
condition do not become part of the statistics. 

As $\big< y \big>_{\cms\,} = 1/2$\, holds, for the same reason exposed in the previous
section, it is sufficient to calculate the mean of the distribution obtained in the
(\ref{eq:mxdiy}). Had there been no cuts in the leading particles energy, this integral
would have yielded $\pi/8$, that is half the area of a circle of radius 1/2. This is not
our case and we will have to take into account the relevant cuts, one of which has been
already mentioned and is:
\[ (E'_e)^{\textsc{lab}}_{\textsc{min}} = 8\: \text{Gev} \]
that, as usual, must be re-evaluated in the \cms :
\begin{equation}
\begin{split}
(E'_e)^{\textsc{cms}}_{\textsc{min}} \: &=\: \gamma \big[
(E'_e)^{\textsc{lab}}_{\textsc{min}}- \vec{\beta}\cdot \vec{p}\,^{\textsc{lab}}_z \big]
\: \simeq \: (E'_e)^{\textsc{lab}}_{\textsc{min}} \cdot \frac{1+\beta}{\sqrt{1-\beta^2}}
\: =\\[2mm] &=(E'_e)^{\textsc{lab}}_{\textsc{min}} \cdot \sqrt{\frac{1+\beta}{1-\beta}}
\: \simeq \: 5.467\: (E'_e)^{\textsc{lab}}_{\textsc{min}} \: \simeq \: 43.7~ \text{Gev}
\end{split}
\end{equation}
where we have:
\begin{itemize}
\item considered that $\hat{\beta}$ e $\hat{z}$ are antiparallel, from where the change in
sign at the second equality
\item used the high energy approximation and the strong collinearity of the 
$e^+_{\textsc{out}}$ (again at the second equality)
\item used the value of $\beta_{\textsc{boost}}$ for $\beta$
\end{itemize}
The highest momentum of the leading particle in the target region will be measured when
this is a proton: as its maximum value will be taken that representing the
limit between the leading physics and the diffractive physics (in the sense described in
\cite{evid} or \cite{creation}), namely:
\begin{equation}
(E^{\te{lead}}_{1,p})_{\te{max}} \: = \: (x_F)_{\te{max}}^{\cms}\cdot E^{\te{inc}}_{1,p}\: 
=\: 0.8\: E^{\te{inc}}_{1,p} \label{eq:xf}
\end{equation}
where $x_F$ is the Feynman variable and is in practice the fractional longitudinal 
momentum of the leading particle.

It should be noted that the relation between energies or momenta expressed in fractional
terms are approximately invariant, in fact
\begin{equation}
\begin{split}
&\begin{cases}
E'_{\te{lead}} = \gamma (E^L-\vec{\beta}\cdot \vec{p}\,^L)\\[4mm]
E'_{\te{inc}} = \gamma (E^I-\vec{\beta}\cdot \vec{p}\,^I)
\end{cases} \longrightarrow\\[4mm]
& \longrightarrow ~~ 
\frac{E'_L}{E'_I} = \frac{\gamma (E^L-\vec{\beta}\cdot \vec{p}\,^L)}{\gamma (E^I-\vec{\beta}\cdot
\vec{p}\,^I)} \simeq \frac{\gamma (1-\beta)}{\gamma (1-\beta)} \cdot \frac{E^L}{E^I} =
\frac{E^L}{E^I}
\end{split}
\end{equation}
so that the energy of a particle expressed with the Feynman variable is about the same in
two different reference systems at high energies and in a collinear approximation.
The leading particle, as such, respects these approximations.

The two values just obtained allow to find the maximum interval in hadronic energy 
compatible with the cuts:
\begin{equation*}
\begin{cases}
(E^H_1)_{\textsc{min}}^{\cms}={\displaystyle \frac{\sqrt{s}}{2}-\frac{4}{5}\cdot 
\frac{\sqrt{s}}{2}}\: \simeq\: 30.0\: \text{Gev} \\[4mm]
(E^H_2)_{\textsc{max}}^{\cms}={\displaystyle 
\frac{\sqrt{s}}{2}-(E'_e)^{\textsc{cms}}_{\textsc{min}}}\: \simeq\: 106.5\: \text{Gev}
\end{cases}
\end{equation*}
from which the minimal value for $y$ immediately follows:
\begin{equation}
y_{\textsc{min}} = 
\frac{(E^H_1)_{\cms}^{\textsc{min}}}{(E^H_1)_{\cms}^{\textsc{min}}+
(E^H_2)_{\cms}^{\textsc{max}}} \simeq 0.220
\end{equation}

In order to estimate $y_{\textsc{max}}$ it is necessary to do some assumptions regarding 
$x_F^{\textsc{min}}$ and  $(E'_e)^{\textsc{cms}}_{\textsc{max}}$\,. The most acceptable
choice, in view of a confrontation with ISR data, is to adopt the same value of 
$x_F^{\textsc{min}}$ chosen at ISR, namely \mbox{0.4\,:}
\[ (x_F)_{\textsc{min}}^{\cms} = 0.4\: E^{\te{inc}}_{1,p} \]
while it will be supposed that the final state positron loses at least the 30\% of its 
energy, value above which the diffractive events begin to prevail over the DIS ones:
\[ (E'_e)^{\textsc{cms}}_{\textsc{max}} = 0.7 {\displaystyle \frac{\sqrt{s}}{2}}\: 
\simeq\: 105\: \text{Gev} \]
From the previous two equations we get
\begin{equation}
\begin{cases}
(E^H_1)_{\cms}^{\textsc{max}}={\displaystyle \frac{3}{5}\cdot \frac{\sqrt{s}}{2}}\:
\simeq\: 90.1\: \text{Gev} \\[4mm]
(E^H_2)_{\cms}^{\textsc{min}}={\displaystyle 
\frac{\sqrt{s}}{2}-(E'_e)^{\textsc{cms}}_{\textsc{max}}}\: 
\simeq\: 45.0\: \text{Gev}
\end{cases}
\end{equation}
that in turn yields
\begin{equation}
y_{\textsc{max}} = 
\frac{(E^H_1)_{\cms}^{\textsc{max}}}
{(E^H_1)_{\cms}^{\textsc{max}}+(E^H_2)_{\cms}^{\textsc{min}}} \simeq 0.667
\end{equation}

\noindent Summarizing:
\begin{equation}
\begin{cases}
y_{\textsc{min}} = {\displaystyle 
\frac{(E^H_1)_{\cms}^{\textsc{min}}}
{(E^H_1)_{\cms}^{\textsc{min}}+(E^H_2)_{\cms}^{\textsc{max}}}} \simeq 0.220
\\[4mm]
y_{\textsc{max}} = {\displaystyle 
\frac{(E^H_1)_{\cms}^{\textsc{max}}}
{(E^H_1)_{\cms}^{\textsc{max}}+(E^H_2)_{\cms}^{\textsc{min}}}} \simeq 0.667
\end{cases}
\end{equation}
and it is now possible to give a numerical estimation for the mean of the distribution in
hadronic energy from which $a$ depends:
\begin{equation}
\big< \sqrt{y(1-y)}\: \big>_{\cms}^{\textsc{hera}} = \:
\frac{{\displaystyle \int_{0.220}^{0.667} \sqrt{y(1-y)} \:\ud y}}{0.667-0.220} \:
\: \simeq \: 0.479
~~\text{(cuts considered)}
\end{equation}
or, in our particular situation,
\begin{equation}
\big< M_{\text{X}} \big>_{\cms}^{\textsc{hera}} \: =\: 2E^{\te{had}}_{\te{tot}}\cdot 
\big< \sqrt{y(1-y)}\: \big>_{\cms}^{\textsc{hera}}\: \simeq \: 0.959 \cdot 
E^{\te{had}}_{\te{tot}}
\end{equation}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%      IL  RISULTATO  FINALE      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Final result}
Having performed all the necessary means, it is now possible to give an estimation for
the correction factor $a$ to be applied in abscissa to the result we are correcting:
\begin{equation}
a \: \equiv \: \sqrt{1+\frac{\big< (1-\cos \theta)^2_{\cms} \big>}%
{\big< M_{\textrm{eff}}/(E^H_T)_{\cms} \big>^2}} \: =%
\: \sqrt{1+\frac{2.679}{(0.959)^2}} \: = \: 1.98
\end{equation}
that is, the plot will be showing the \nch vs \atot dependence if we multiply DIS data
abscissas by a factor $a$= 1.98. It should be recalled that \ee data do not necessitate
any rescaling while for \pp data an estimation of the correction factor will be soon
given, but we may anticipate that it will be not relevant. The result is showed in Figure
\ref{rescaled}.
\begin{figure}[!h] 
\begin{center}
\hspace*{-7mm}
\includegraphics[width=1.1\textwidth]{corrected.ps}
\caption{The same figure as \ref{osaka} but with DIS data (ZEUS 95) rescaled by the factor 
$a \simeq 2$ and plotted in violet: now the X-axis variable is \atot\!\!. The universality
of \nch is evident, particularly at high energies ($a$ being an asymptotic value).}
\label{rescaled}
\end{center}
\end{figure}

It seems clear that, particularly at high energies, DIS data and other processes data 
match each other very well. The quantity \nch is again a Universality Feature.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%          PP  DATA          %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\pp data rescaling}
Recalling that the ISR was balanced and that there were no relevant cuts in the polar
angle, we can conclude that the analyses performed using \eht (the almost totality)
were automatically conducted in terms of \atot (see equation \ref{eq:aeht}). This means
that those works do not necessitate any correction

On the contrary, some other works (among the early works see for example
\cite{nc15,nc11,usomx}) were conducted by using \mx to analyse the collected data: these
ones need to be corrected for the reasons exposed in the introductory section. This may
seem in contrast with the fact that no differences were found at that time when data
were analysed both with \eht and \mx but it is not so, the reason being in the low
precisions and energies then obtainable.

As an example we will examine here the difference between \atot and \mx that could have been
revealed in the early works at ISR where the only condition was the presence of a leading
proton leaving the vertex with 
\begin{equation}
x_{\text{F}} \in [0.4\:,\:0.8]
\end{equation}
from which it may be deduced that
\begin{equation}
E^{\te{had}}_{\te{max}} \:\simeq\: 3\,E^{\te{had}}_{\te{min}}
\end{equation}
and this in turn allows to get the maximum value of $a$ in this situation:
\begin{equation}
\begin{split}
\frac{A_{\text{TOT}}}{M_{\textrm{X}}} &=
\frac{(E^{\te{had}}_{\te{tot}})_{\textsc{cms,lab}}} {M_{\textrm{X}}} \simeq ~~~~~~~
\text{(worst situation)} \\[2mm] & \simeq
\frac{E^{\te{had}}_{\te{max}}+E^{\te{had}}_{\te{min}}}
{2\sqrt{E^{\te{had}}_{\te{max}}E^{\te{had}}_{\te{min}}}}
\simeq  \frac{4\,E^{\te{had}}_{\te{min}}}{2\sqrt{3(E^{\te{had}}_{\te{min}})^2}} =
\frac{2}{\sqrt{3}} \simeq 1.155
\end{split}
\end{equation}
where again the high energies approximation has been used in obtaining the second
equality.  Thus, in a single event and in the worst situation, the difference between
\atot and \mx could had been as large as 15\%, but this is a limiting event; when averaging over
all the events the difference becomes $\ll$ 15\% and consequentely hidden by the uncertainties
on \nch\!\!. It is therefore possible to safely employ ISR \pp data analysed in terms of
\mx to make a comparison with other processes data if a very high precision is not required.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%      DELTA THETA AND YYY DEPENDENCE      %%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Dependence of {\boldmath $a$ from $\Delta \theta\:$ and $\Delta y$}}

In this last section the dependence of $a$ from $\Delta \theta\:$ and $\Delta y$ is
studied with the main purpose of proving that the result here obtained cannot be
fortuitous as changing kinematical conditions means to change the numerical value of $a$:
other values of $\Delta \theta$ and of $\Delta y$ would bring to different values of $a$
and the match between \ee\!\!, \pp and DIS achieved in this paper would have not been
obtained.

From equation (\ref{eq:ageneral}) we see that $a$ increases with $\theta$ and decreases with the
interval in $y$ (centered at 0.5). The situations will be studied when $a$ has its maximum
and minimum value, and when there is no phase space reduction.

It is of course safer to vary $\Delta \theta\:$ than $\Delta y$ as from the latter depends
the purity of the DIS sample. Only safe values for the minimum and the maximum of $y$
will be considered.

A maximum for $2 \big< \sqrt{y(1-y)}\: \big>$ is obtained by selecting a sample of
balanced events:
\begin{equation}
y = \frac{1}{2} ~\longrightarrow  ~2 \big< \sqrt{y(1-y)}\: \big>_{\textsc{max}} \: = \: 1
\end{equation}
while unbalanced events induce a minimum for $a$. If events with a fast leading proton
and a low energy outgoing positron are chosen (event unbalanced in $E^H$ towards
hemisphere 2)
\begin{equation}
\begin{cases}
x_F \in [0.6\, ;\, 0.8]\\[4mm]
(E'_e)^{\textsc{cms}}_{\textsc{min}} \in [43,7\, ;\, 70] ~\text{Gev}
\end{cases}
\end{equation}
then we get, with the same procedure used above:
\begin{equation}
\begin{cases}
y_{\te{min}}=0.220 ~~~~ \text{(already obtained)}\\[2mm]
y_{\te{max}}=0.431
\end{cases}
\end{equation}
from which
\begin{equation}
\big< \sqrt{y(1-y)}\: \big>_{\textsc{min}} \: = \: 0.464
\end{equation}
and this value may be used to re-evaluate $a$ in changed conditions.
As showed in (\ref{eq:amxpl}) the analytical minimum for $a$ is 1:
\begin{equation}
a \equiv \frac{A_{\text{TOT}}}{M_{\text{eff}}} \le 1
\end{equation}
but it is not possible to reach this value in true experiments as measurements should be
performed in very small and collinear intervals in $\theta\:$ (say $\theta_{\textsc{lab}}
\in [0^0;5^0]$).

Perhaps, a realistic possibility could be $\theta_{\cms}\! \in\! [0^0;90^0]$ that is
$\theta_{\textsc{lab}}\! \in\! [0^0;20^0]$. In such a case 
\begin{equation}
\big< (1-\cos \theta)^2_{\cms} \big> \simeq 0.23
\end{equation}
which yields
\begin{equation}
a_{\textsc{min}} \simeq \sqrt{1+ \frac{0.23}{1}} \simeq 1.1
\end{equation}
that is, in such conditions \atot and \mx would be in practice undistinguishable owing to the
already recalled bad resolution. Such a study would have never been able to reveal any
difference between \atot and \meff and the Universality Features would have been evident
even using \meff as it happened in some previous experiments.

In order to get much greater values for $a$ than that obtained in this work, we should go
to the other side of the polar angle range, imposing, for example, 
$\theta_{\cms} \in [150^0;180^0]$. But this would mean to say 
$\theta_{\textsc{lab}} \in [174^0;180^0]$ that is a ridiculous interval.
We are thus forced to assume
\begin{equation}
a_{\textsc{max}} \gtrsim 2 ~~~(\text{using } 
\big< \sqrt{y(1-y)}\: \big>_{\textsc{min}} \: = \: 0.464)
\end{equation}
This shows how this study has already been conducted around the maximum value for $a$.
In the end, it is interesting to see what would have been obtained by working in the full
phase space. Imposing:
\begin{equation}
\Delta \theta_{\textsc{lab}} = \Delta \theta_{\cms} = [0^0;180^0] 
\end{equation}
we get
\begin{equation}
\big< (1-\cos \theta)^2_{\cms} \big> = 1.5
\end{equation}
so that
\begin{equation}
a_{\textsc{(full phase space)}} = \sqrt{1+ \frac{1.5}{(0.959)^2}} \simeq 1.62
\end{equation}
a value that is about 20\% less than what has been found in the reduced phase space. This
result, combined with the worse resolution at the two ends of the ZEUS calorimeter, shows
that a full phase space study would have been loosely able to reveal a smaller difference
between \atot and \mx\!.

What obtained in this section shows how the estimated value for $a$ and the consequent
gathering of data referred to different processes, has a quite pronounced dependence
from the ``contour'' conditions. This means that a possible fortuitousness of the
agreement obtained would be quite unlikely. Anyway, it is surely necessary to perform 
direct analyses to check if \atot works well in different conditions and kinds of
experiments, particularly in those involving the indirect \effl: above all, this could be
the final way to check the geometrical justification for the introduction of \atot\!.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%      CONCLUSIONI        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions} 
After reconsidering the kinematics of the \effl, a new invariant has
been introduced. It is likely to be the universal quantity effectively available to
multiparticle production and, as such, the variable that correctly describes the
hadronization processes in any kind of particle collisions. Consequently, it has been
called the Total \aqs\ and denoted with \mbox{\atot\!.}

\noindent The universality of \nch has been recovered after that other works \cite{osaka} had
pointed out its non-universality when considering DIS processes and analysing in terms of
the invariant hadronic mass \mx\!\!.

\noindent This result suggests that \mx is definitively ruled out and proposes \atot as
its ``successor''.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     BIBLIOGRAFIA        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}

