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\begin{document}
\begin{center}
\bf{$\boldsymbol{k_\perp}$ inclusive distribution and two-particle correlation at
MLLA and beyond}\newline
Redamy Perez Ramos
\end{center}

The hadronic $k_\perp$-spectrum inside a high energy
 jet is determined including corrections of relative magnitude
 ${\cal O}(\sqrt{\alpha_s})$ with respect to the Modified Leading Logarithmic
Approximation (MLLA),  in the limiting spectrum approximation
 (assuming an infrared cut-off $Q_0 =\Lambda_{QCD}$) and beyond ($Q_0\ne\Lambda_{QCD}$).
The results in the limiting spectrum approximation are found to be in
impressive agreement with preliminary measurements by the CDF collaboration, unlike
what occurs at MLLA, pointing out small overall
 non-perturbative contributions.
Within the same framework, 2-particle  correlations inside a jet
are also predicted at Next-to-MLLA and compared to previous MLLA calculations.

MLLA corrections, of relative magnitude ${\cal O}(\sqrt{\alpha_s})$ with respect
to the leading double logarithmic approximation (DLA), were shown to be quite
substantial for single-inclusive distributions and 2-particle correlations
~\cite{PerezMachet,RPR2}. Therefore, it appears legitimate
to wonder whether corrections of order ${\cal O}(\sqrt{\alpha_s})$, that is
next-to-next-to-leading or next-to-MLLA (NMLLA), are negligible or not.

The starting point of this analysis is the MLLA evolution equation for
the generating functional of QCD jets \cite{Basics}.
Together with the initial condition at threshold, it determines
jet properties at every energies. At high energies one can represent
the solution as an expansion in $\sqrt{\alpha_s}$. Then,
the leading (DLA) and next-to-leading (MLLA) approximations are complete.
The next terms (NMLLA) are not complete but they include an important
contribution which takes into account energy conservation and an
improved behavior near threshold.

Some results for such NMLLA terms have been
studied previously for global observables and have been found to
better account for recoil effects. They were shown to
drastically affect multiplicities and particle correlations in jets:
this is in particular the case in  \cite{CuypersTesima}, which 
deals with multiplicity correlators of order 2, and in \cite{DokKNO},
where  multiplicity correlators involving a higher number of partons are
studied; in particular, the higher
this number, the larger turn out to be  NMLLA corrections.

The present study makes use of this evolution equation to estimate NMLLA
contributions to our differential observables.
It presents the complete calculations of the single
inclusive $k_\perp$ distribution leading to the results
published in \cite{PRL}, and extends them to 2-particle correlations
inside a high energy jet.

The shown agreement between the CDF results
and the NMLLA distributions over
the whole $k_\perp$-range is excellent.
The NMLLA calculation is in particular able to capture the shape
of CDF spectra at every jet hardness $Q$. Conversely, 
predictions at MLLA prove
only reliable at not too large $k_\perp$.

The domain of validity of the predictions has been enlarged
to larger $k_\perp$ (and thus to larger $x$ since $Y$ is fixed)
computing from MLLA to NMLLA accuracy. This agreement further support
the Local Hadron Parton Duality (LPHD) hypotheses \cite{LPHD}.

The MLLA, NMLLA equations for correlators were analyzed and solved
iteratively \cite{RPR2}. This allowed us to generalize the result previously
obtained by Fong and Webber in \cite{FW} that was valid in the vicinity
of the maximum of the single inclusive parton energy distribution
("hump").

In particular, we have analyzed the regions of moderately small $x$
above which the correlation becomes "negative" (${\cal C}-1<0$).
This happens when suppression because of the limitation of the
phase space takes over the positive correlation due to gluon cascading.
This region turned out to be larger in NMLLA than in MLLA.

Also, the correlation vanishes (${\cal C}\to1$) when one of the partons
becomes very soft ($\ell = \ln1/x \to Y=\ln E\Theta/Q_0$). The reason
for that is dynamical rather than kinematical: radiation of a soft
gluon occurs at {\em large angles}\/ which makes the radiation
coherent and thus insensitive to the internal parton structure of the
jet ensemble.

Qualitatively, our MLLA and NMLLA result agrees better
with available OPAL data
than the Fong--Webber prediction. There remains however a significant
discrepancy, markedly at very small $x$. In this region
non-perturbative effects are likely to be more pronounced. They may
undermine the applicability {\em to particle correlations}\/ of the local
parton--hadron duality considerations that were successful in
translating parton level predictions to hadronic observations in the
case of more inclusive {\em single particle energy spectra} \cite{Basics}.

\begin{thebibliography}{50}

\bibitem{PerezMachet}
R. Perez-Ramos \& B. Machet: ``MLLA inclusive hadronic distributions inside
one jet at high energy colliders'', hep-ph/0512236, JHEP {\bf 04} (2006) 043..

\bibitem{RPR2}
R. Perez-Ramos:
``Two particle correlations inside one jet at Modified Leading
Logarithmic Approximation of Quantum Chromodynamics;
I: Exact solution of the evolution equations at small $x$'',
hep-ph/0605083, JHEP {\bf 06} (2006) 019, and references therein.

%
\bibitem{Basics}
Yu.L. Dokshitzer, V.A. Khoze, A.H. Mueller \& S.I. Troyan: ``Basics of
Perturbative QCD'', (Editions Fronti\`eres, Gif-sur-Yvette, 1991)

\bibitem{CuypersTesima}
F. Cuypers \& K. Tesima: ``Recoil effect on multiplicity correlation''
Z. Phys. {\bf C 54} (1992) 87-93.

\bibitem{DokKNO}
Yu.L. Dokshitzer, {\it Improved QCD treatment of the KNO phenomenon}
Phys. Lett. {\bf B 305} (1993) 295.

\bibitem{PRL}
F. Arleo, R. P\'erez-Ramos \& B. Machet: ``Hadronic single inclusive kt
distributions inside one jet beyond MLLA'',
arXiv:07072391 [hep-ph], Phys. Rev. Lett.100:052002,2008

R. P\'erez-Ramos, F.Arl\'eo \& B. Machet: ``Next-to-MLLA
corrections to single inclusive kt
distributions and two-particle correlations in a jet'',
arXiv:0712.2212 [hep-ph]

\bibitem{LPHD}
Ya.I. Azimov, Yu.L. Dokshitzer, V.A. Khoze \& S.I. Troian:
``Similarity of Parton and Hadron Spectra in QCD Jets'',
Z. Phys {\bf C 27} (1985) 65

Yu.L. Dokshitzer, V.A. Khoze \& S.I. Troian: ``On the concept of local
parton hadron duality'', J. Phys. {\bf G 17} (1991) 1585-1587

\bibitem{FW} C.P. Fong \& B.R. Webber:
``One and two particle distributions
at small $x$ in QCD jets'', Nucl. Phys. {\bf B 355} (1991) 54

C.P. Fong \& B.R. Webber:  ``Two particle correlations
at small $x$ in QCD jets'', Phys. Lett. {\bf B 241} (1990) 255

\end{thebibliography}

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