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\title{QCD analysis of heavy quarks production in hadronic collisions }
\author{A. Mirjalili\address{Physics Department, Yazd
University, Yazd, Iran}\address[MCSD]{Institute for Studies in
Theoretical Physics and Mathematics (IPM)\\P.O.Box 19395-5531,
Tehran, Iran}
\thanks{a.mirjalili@yazduni.ac.ir},
Ali. N. Khorramian\address{Physics Department, Semnan University,
Semnan, Iran}\addressmark[MCSD], S. Atashbar
Tehrani\address{Physics Department, Persian Gulf University,
Boushehr, Iran}\addressmark[MCSD]}
\begin{document}
\begin{abstract}
The problem of renormalization scheme dependence in QCD
perturbation theory remains on obstacle to making precise tests of
the theory. The renormalization scale dependence of dimensionless
physical QCD observable, depending on a single energy scale $Q$,
can be avoided provided that all ultraviolet logarithms which
build the physical energy dependence on $Q$ are resummed. This was
termed complete Renormalization Group improvement(CORGI). This
argument can be extended to processes involving factorization of
operator matrix elements and coefficient functions. We are trying
to employ the idea of CORGI approach on analyzing of heavy quarks
production in hadron collisions. There is a sizable and systematic
discrepancy between experimental data on the $b\bar{b}$ production
in $p\bar{p}$, $\gamma{p}$ and $\gamma{\gamma}$ collisions and
existing theoretical calculations within perturbative QCD. One
suggested way to cope with this discrepancy is to employ the CORGI
approach in which one should perform a resummation to all-orders
of renormalization and factorization group -predictable terms at
each order of perturbation theory. Then the scales dependence will
be avoided and it is expected that the mentioned discrepancy is
reduced significantly. \vspace{1pc}
\end{abstract}
\maketitle
\section{Introduction}
One of the clean test of perturbative QCD is the heavy quark
production in hard collisions of hadrons, leptons and photons. The
results of the exploratory study demonstrate the sensitivity of
fixed order QCD calculation of the total cross section for heavy
pair quark production to the choice of renormalization
$(\mu)$ and factorization ($M$) scales. As emphasized by Politzer
\cite{Chyla-22} in connection to the central aspect of these two
scales, there is no compelling reason for identifying these two
scales. The point is that whereas the renormalization scale
defines, roughly speaking, the lower limit on the virtualities of
loop particles included in the definition of the renormalized
coupling, the factorization scale $M$ specifies the upper limit on
virtualities of partons included in the definition of dressed
parton distribution functions. In other words, the renormalization
scale reflects ambiguity in the treatment of short distances,
whereas the factorization scale comes from similar ambiguity
concerning large distances. It is therefore natural to keep the
$\mu$ and $M$ scales as independent free parameters of any finite
order perturbative approximations. The calculations presented here
demonstrate that the cross section $\sigma_{tot}(M,\mu)$ depends,
indeed, on these two scales in quite different way.\\
The scale dependence of QCD observable can also be argued in a
different view. It was stressed that standard Renormalization
Group (RG)-improvement, as customarily applied with a
$Q$-dependent scale $\mu= xQ$ , omits an infinite subset of all
ultraviolet logarithm terms. One should rather keep $\mu$
independent of $Q$, and then carefully resum to all-orders the
RG-predictable of these terms. In this way all $\mu$-dependence
cancels between the renormalized coupling and the logarithms of
$\mu$ contained in the coefficients, and the correct physical
$Q$-dependence is built. The approach of the ``complete
RG-improvement" (CORGI) is based on this idea. It is possible to
employ this approach to the processes which involves additional
factorization scale \cite{max-mir-00}. DIS phenomena where hard
scattering cross sections are convoluted, is an observable which
we are investigating it from this point view. The CORGI approach
serves to separate the perturbation series into infinite subset of
terms which when summed are renormalization scheme (RS) invariant.
So we
expect to have more consistent result in comparison with
experimental data, specially for total cross section of heavy
quark pair production which they lie systematically by a factor of
about 2-4 above the median of current theoretical calculations
\cite{Chyla-1}.\\
In dealing to Complete RG improvement approach, what we need to do
is to extend first the perturbative expressions of the required
observable to any desired order and then to calculate the
partial derivatives of total cross section $\sigma_{tot}(M,\mu)$ expression
with respect to $\mu$, M and other scheme labelling parameters, using the
QCD-${\beta}$ function and evolution equations of parton
distribution functions (PDF) and finally demanding consistency
principle. This help us to find the strict dependence of coefficient function
to the scales $\mu$ and M and in general case
to other parameters. At this step we can obtain the factorization
and renormalization scheme (FRS) invariants X2, X3,... . To
convert the standard perturbative series into the CORGI approach,
one can use from the scheme invariant property of the resummed
terms at any specified order. In this case all scheme labelling
parameters can be equal to zero and as a result the coupling
constant $a_0$ will be appeared. In fact $a_0=a(\texttt{scheme
labelling paremeters=0}), $ where $a $ is the usual RG-coupling
constant. This new coupling constant can be expressed in terms of
Lambert-W function \cite{max-mir-02}.
\section{Heavy quarks production}
According to the factorization theorem the total cross section of
heavy quark pair production in $p\bar{p}$ collision at the center
of mass energy $\sqrt{s}$ has the form \ba
&&\sigma_{tot}(p\overline{p}\rightarrow{Q\overline{Q}},s)=\int
\int{dx}{dy}\sum_{i,j}D^{\bar{p}}_i(x,M)\nonumber\\
&&{\times}D^{{p}}_j(y,M)
\times\sigma_{ij}(s=xyS,M)\;,\nonumber\\\ea where partonic cross
section $\sigma_{ij}$ as well as PDF of the beam particles depend
on the factorization scale $M$. The above expression is basically
of non perturbative nature. Fixed order perturbation theory enters
if we we insert in above equation the solution of evolution
equations with the splitting function $P_{ij}$ and calculate
$\sigma_{ij}$ as power expansion in the coupling $\alpha_s(\mu)$,
\ba
\sigma_{ij}(S,M)=\alpha^2_s(\mu)\sigma^2_{ij}(s)+\alpha^3_s(\mu)\sigma^3_{ij}(s,M,\mu)+...\hspace{-0.75
cm}
\nonumber\\
\ea The latter dependence is cancelled by that of PDF provided the
splitting function $P_{ij}$ in the evolution equation retaken to
all orders. The splitting functions admit expansion in powers of
$\alpha_s(M)$ \ba &&\hspace{-0.75
cm}P_{ij}(x,M)={\frac{\alpha_s(M)}{2\pi}}P^{(0)}_{ij}(x)+
\left({\frac{\alpha_s(M)}{2\pi}}\right)^2{P^{(1)}_{ij}(x)}\nonumber\\
&&+\;.\;.\;. \ea where $P_{ij}^{(0)}(x)$ are unique, where as all
higher splitting functions $P_{ij}^{(k)}$, $k\;{\ge}\;1$ depend On
the choice of the factorization scheme (FS). Conversely, they can
be taken as defining the FS. \\
At NLO, i.e. taking into accont the first two terms in related
perturbative series for $\sigma_{ij}$ and $P_{ij}$ we get \ba
&&\sigma^{NLO}_{tot}(M,\mu)=\alpha^2_s(\mu)\left\{\int\int{dx}{dy}
\sum_{i=1}^{2n_f}q_i(x,M)\right.
\nonumber\\
&&\times{q_i(y,M)}\times[\sigma^{(2)}_{q\bar{q}}(xy)
+\alpha^2_s(\mu)\sigma^{(3)}_{q\bar{q}}(xy,M,\mu)]+
\nonumber\\
&&2\int\int{dxdy}\Sigma{(x,M)}G(y,M)\alpha_s(\mu)\sigma^{(3)}_{qG}(xy,M)+\nonumber\\\nonumber\\
&&\left.\int\int{dxdy}G{(x,M)}G(y,M)[\sigma^{(2)}_{GG}(xy)\right.\nonumber\\
&&\left.+\alpha_s(\mu)\sigma^{(3)}_{GG}(xy,M,\mu)]\right\}\nonumber
\ea where here the sum is over number of active flavor $n_f$
quarks and antiquarks and the relation between PDF of protons and
antiprotons was taken into account. The PDF for the quark singlet,
non-singlet and Gluon distribution function can be obtained form
some phenomenological model like valon model \cite{Hwa-81,KMA-JHP}.\\
In this stage, one prefers to rewrite the above equation in terms
of rapidity and the minimum transverse momentum ($P_T^{min}$) of
outgoing heavy quarks . The converting relations can be found, for
instance, in Sec.10.3 of \cite{QCD Collider}. This make enable to
compare the theoretical result with available experimental data
from $D{\emptyset}$ collaboration on $b\bar{b}$ production in
$p\bar{p}$ collisions \cite{Chyla-1}. This comparison has been
plotted in Fig. 1. \vspace{-1.2 cm}
\begin{figure}[htb]
\vspace{9pt}
\includegraphics[scale=0.2,width=17pc,height=17pc]{fig1.eps}
\vspace{-1 cm} \caption{$D{\emptyset}$ data on $b\bar{b}$
production which has been compared to the results of MRST2 model
\cite{MRST2}.} \label{fig:largenenough}
\end{figure}
\vspace{-1.1 cm}
\section{CORGI approach}
As we wrote before, in addition to standard approach to
perturbative QCD calculations which are usually is performed in
the $\overline{MS}$ scheme with a physical choice of
renormalization and factorization scales set equal to $Q$,
one can use the
approach of the Complete Renormalization Group Improvement
(CORGI) \cite{max-mir-00}. In this approach, all perturbative
terms are scheme independent, so there is no dependence on
renormalization
scale ($\mu$) or factorization scale ($M$) in these terms.\\
If an observable $R(Q)$ dependent on a single scale has a
perturbative
series
\begin{equation}%=============================================
R(Q)=a+r_1a^2+r_2a^3+\cdots+r_na^{n+1}+\cdots ,\label{obs-snd}
\end{equation}%===============================================
where $a\equiv \alpha_s(\mu)/\pi$ is the RG-improved coupling,
then it can be shown that in CORGI approach, this observable has
the perturbative expansion \cite{max-mir-00} like
\begin{equation}%=============================================
R(Q)=a_0+X_2a_0^3+X_3a_0^4+\cdots+X_na_0^{n+1}+\cdots
.\label{obs-cgi}
\end{equation}%===============================================
In Eq.~(\ref{obs-snd}) all terms including $a,r_1,r_2,\cdots$
depend on renormalization scale ($\mu$), while in
Eq.~(\ref{obs-cgi}), $a_0=a_0(Q)$ (Q is the physical energy scale)
and
$X_2,\; X_3,\cdots$ are constants and scheme invariants.\\
The $a_0$ is defined in terms of the Lambert-W function
\cite{max-mir-02} as
\begin{eqnarray}%=============================================
a_0=-{1\over c[1+W_{-1}(z(Q))]}\ ,\ z(Q)=-{1\over e}\left({Q\over
\Lambda_{\cal R}} \right)^{-b/c}\nonumber\label{lamb}
\end{eqnarray}%===============================================
where $b$ and $c$ are the first two universal coefficients of the
QCD $\beta$-function
\ba%=============================================
b={33-2N_f\over 6}\ ,\ c={153-19N_f\over 12b},\nonumber
\ea%===============================================
with $N_f$ the number of the active flavors. $\Lambda_{\cal {R}}$
depends on the chosen observable and is related to the QCD
dimensional transmutation parameter $\Lambda_{\overline{MS}}$.\\
In order to improve and increase the reliability of the
perturbative QCD calculation, we try to employ this approach in
QCD analysis of total cross section for heavy pair quark
productions. The advantage of CORGI approach is that not only each
term in the related perturbative series is scheme independent but
also because it involves a resummation of RG-predictable terms, it
should yield
more accurate approximations than the standard truncation of Eq.~(\ref{obs-snd}).\\
As a preliminary task to use this approach in the LO
approximation, we need just to change the usual RG-coupling
constant $a$ to the $a_0$ which is appearing in the invariant
scattering matrix element squared of partonic cross section
\cite{QCD Collider} and also in the evolution parameter of the
calculations in moment space. Using the inverse Mellein
technique, we can take the results form the moment space to
x-space and extract parton distribution. The preliminary result
for the total cross section is\hspace{0 cm} indicated in Fig. 2.
\vspace{-2.7 cm}
\begin{figure}[htb]
\vspace{9pt}
\includegraphics[scale=0.2,width=23.8pc,height=28.8pc]{fig2.eps}
\label{fig:largenenough}
\end{figure}
\hspace{0 cm}\vspace{-5.8 cm} \newline Figure 2. Results of
$\sigma_{tot}$ verses $P_T^{min}$ in two different CORGI and
standard approaches.\\\\As it can be seen, employing the CORGI
approach will shift forward the the obtained curve for total cross
section and it is a good sign that employing this approach will
increase the precision of calculation. So the results confirm the
anticipated better agreement of the CORGI approach with the data.
\section{Discussion}
However in previous section we employed the CORGI approach as a
preliminary task and the obtained result indicated that we would
get more precise result if we use this approach in hard
scattering, but we should note that the real situation will be
arising out when we are able to do the resummation on perturbative
terms in $\sigma_{tot}$ in which the $\mu$ and $M$ scales can be
avoided. In this case we will be able to distinguish the real
feature of the related pertrbative series in CORGI approach.
Latter on we should do the numerical calculations of
$\sigma_{tot}$ for heavy quark production. This will be the real
situation for comparison of CORGI result with the standard result
for the hard scattering processes. Of course to achieve to this
situation we need to high mathematical attempts. Primarily
calculations show that due to existence of convolution integral in
process of solving the partial derivative equations of
perturbative terms, resulting from self consistency principal,
their $M$-scale dependence can not be obtained easily. At the
moment this
makes an impediment to find the relation which exist between
perturbative terms. As a results, we are not able to do the resumation on the
perturbatve terms. This demands a big challenging and we hope to
be able to report on it in future.
\section{Acknowledgement}
A.M. Acknowledges from Yazd university to support him financially
to do this research job. A.M. is also grateful to the ``Abdus
Salam International center for theoretical physics (ICTP)" in
Italy for their hospitality whilst part of this research was
performed there.
%We are acknowledge Institute for studies in
%theoretical Physics and Mathematics (IPM) for partial financial
%supporting of this project.
\begin{thebibliography}{9}
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\bibitem{max-mir-00} C.J. Maxwell, A. Mirjalili, Nucl. Phys. B {\bf
577}(2000)209.
\bibitem{Chyla-1} B. Abbott et al. (D$\emptyset$ Collab):, Phys. Rev. Lett {\bf{85}} (2000), 5068.
%\bibitem{Chyla-2} B. Abbott et al. (D$\emptyset$ Collab):, Phys. Rev. D {\bf{65}} (2002), 052005.
\bibitem{max-mir-02} C.J. Maxwell, A. Mirjalili, Nucl. Phys. B {\bf
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\bibitem{Hwa-81} R.C. Hwa and M.S. Zahir, Phys. Rev. D {\bf 23}(1981) 2539.
\bibitem{KMA-JHP} Ali.N. Khorramian, A. Mirjalili, S. Atashbar Tehrani,
JHEP 0410 (2004) 062.
\bibitem{QCD Collider} R.K. Ellis, W.J. Stirling, B.R. Webber,
QCD and Collider Physics, Cambridge University Press, 1996.
\bibitem{MRST2} A.D. Martin, R.G. Roberts, W.J. Stirling and R.S.
Thorne, Phys.Lett.{\bf{B}} 531 (2002) 216.
\end{thebibliography}
\end{document}