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\begin {flushright}
ITP-SB-95-45
\end {flushright} 
\vspace{3mm}
\begin{center}
{\Large \bf Bottom quark production cross section at HERA-B} 
\end{center}
\vspace{2mm}
\begin{center}
N. Kidonakis and J. Smith  \\
\vspace{2mm}
{\it Institute for Theoretical Physics, 
State University of New York at Stony Brook, 
Stony Brook, NY 11794-3840, USA} \\  
\end{center}

\begin{abstract}
The cross section for bottom quark production is
calculated for the HERA-B experiment.
We consider both the 
order $\alpha_s^3$ cross section and the resummation of 
soft gluon corrections in all orders of QCD perturbation theory. 
\end{abstract}

\pagebreak
%------------------This is Section 1---------------------------------
\mysection{Introduction}
%----------------------------------------------------------
The calculation of production cross sections for heavy particles 
in QCD is made by invoking the factorization theorem \cite{css} and 
expanding the contributions to the amplitude in powers of the coupling 
constant $\alpha_s(\mu^2)$. Recent investigations have
shown that near threshold there can be large logarithms in the perturbation
expansion which have to be resummed to make more reliable theoretical
predictions. The application of these ideas to fixed-target 
Drell-Yan production has been the subject of many papers over
the past few years \cite{gs}. 
The same ideas on resummation were applied to the calculation of 
the top-quark cross section and differential distributions at the 
Fermilab Tevatron in [3-5]. 
What is relevant in these reactions 
is the existence of a class of logarithms of the type
$(\ln(1-z))^{i}/(1-z)$, where $i$ is the order of the
perturbation expansion, and where one must integrate over the variable
$z$ up to a limit $z=1$. These
terms are not actually singular at $z = 1$ due to the presence
of terms in $\delta(1-z)$. However the remainder
can be quite large. In general one writes such terms 
as ``plus'' distributions, which are then convoluted with
regular test functions (the parton densities).

In this paper we examine the production of $b$-quarks 
in a situation where the presence of these large logarithms is 
of importance, namely in a fixed-target experiment to be performed
in the HERA ring at DESY. This actual experiment has the name HERA-B 
\cite{herab1,herab2}
and involves colliding the circulating proton beam against a stationary
copper wire in the beam pipe. The nominal beam energy of the protons
is 820 GeV, so that the square root of the center-of-mass (c.m.) energy
is $\sqrt{S} = 39.2 $ GeV. Taking the $b$-quark mass as
$m_b = 4.75\: {\rm GeV}/c^2$ then the ratio of $m_b/\sqrt{S} \approx 1/8$. 
If we choose the renormalization scale in the running coupling constant
as $m_b$ then $\alpha_s(m_b^2) \approx 0.2$ so 
$\alpha_s(m_b^2) \ln(\sqrt{S}/m_b) \approx  0.4$, indicating that
perturbation theory should be reliable. 

In perturbation theory with a hard scale we can use the
standard expression for the order-by order cross section
in QCD, namely
\begin{equation}
\sigma(S,m^2) = \int_{\frac{4m^2}{S}}^1 dx_1 
\int_{\frac{4m^2}{Sx_1}}^1 dx_2 \sum_{ij}
f_i(x_1,\mu^2) f_j(x_2,\mu^2) \sigma_{ij}(s = x_1 x_2 S, m^2,\mu^2),  
\end{equation}
where the $f_i(x,\mu^2)$ are the parton densities at the mass factorization
scale $\mu^2$ and the $\sigma_{ij}$ are the partonic cross sections. 
The numerical results for the hadronic cross sections
depend on  the
parton densities, which involve the choice of
$\mu^2$; the choice of the running coupling constant, which
involves the renormalization scale (also normally chosen to be $\mu^2$); 
and the choice for the actual mass of the $b$-quark. 
In lowest order (LO) or Born approximation the
actual numbers for the cross section show a large sensitivity to
these parameters. 
The next-to-leading order (NLO) results follow from the work of the 
two groups \cite{nde1} and \cite{{betal},{bnmss}}.  
However, even the NLO
results do not completely fix the cross section. There is still a sensitivity
to our lack of knowledge of even higher terms in the QCD expansion, which
can be demonstrated by varying the scale choice up and down by factors
of two. In general it is impossible to make more precise
predictions for heavy quark production given the absence of a 
calculation in next-to-next-to-leading order (NNLO).

In the threshold region one can improve on the NLO results.
In this region one finds that there are large logarithms 
of the type mentioned above which arise from
the soft-plus-virtual (S+V) terms in the perturbation expansion.
These logarithms can be resummed to all orders in perturbation theory.
We will see later that the gluon-gluon channel 
is the dominant channel for the production
of $b$-quarks near threshold in a fixed-target $pp$ experiment.  
In the next section we give some results for the parton-parton cross section. 
Section 3 contains the analysis of the hadron-hadron cross section
which is relevant for the HERA-B experiment. We give
results in NLO and after resummation.
Finally in Section 4 we give our conclusions.
  

%------------------This is Section 2---------------------------------
\mysection{Results for parton-parton reactions}
%------------------------------------------------------------------
The partonic processes that we examine are
\begin{equation}
i(k_1) + j(k_2) \rightarrow Q(p_1) + \bar Q(p_2),
\end{equation}
where $i,j = g,g$ or $i,j= q, \bar q$ and $Q, \bar Q$ are heavy quarks
$(c,b,t)$.
The square of the parton-parton c.m. energy is 
$s=(k_1+k_2)^2$. 

We begin with heavy quark production in the $q \bar q$ channel.
The Born cross section in this channel is given by
\begin{equation}
\sigma_{q \bar q}^{(0)}(s,m^2)=\frac{2 \pi}{3} \alpha_s^2(\mu^2)
 K_{q \bar q} N C_F \frac{1}{s} \beta \left(1+\frac{2m^2}{s}\right),
\end{equation}
where $C_F=(N^2-1)/(2N)$ is the Casimir invariant for the fundamental
representation of $SU(N)$,  
$K_{q \bar q}=N^{-2}$ is a color average factor, 
$m$ is the heavy quark mass, $\mu$ denotes the 
renormalization scale, and $\beta=\sqrt{1-4m^2/s}$.
Also $N=3$ for the $SU(3)$ color group in QCD.
The threshold behavior ($s \rightarrow 4m^2$) of this expression is given by
\begin{equation}
\sigma_{q \bar q ,\, \rm thres}^{(0)}(s,m^2)=\pi\alpha_s^2(\mu^2) 
K_{q \bar q} N C_F \frac{1}{s} \beta.
\end{equation}
Complete analytic results are not available for the NLO cross section
as some integrals are too complicated to do by hand.
However in \cite{bnmss} analytic results are given for
the soft-plus-virtual contributions to the cross section, and for the
approximation to the cross section near threshold. 
Simple formulae which yield reasonable approximations to the exact 
$O(\alpha_s^3)$ results have been constructed in \cite{MSSN}. 

The analysis of the contributions to the gluon-gluon channel in NLO
is much more complicated. There are three Born diagrams
and different color structures. 
The exact Born term in the $gg$ channel is 
\begin{eqnarray}
\sigma_{gg}^{(0)}(s,m^2)&=&4\pi\alpha_s^2(\mu^2) K_{gg}N C_F \frac{1}{s}
\left\{C_F\left[-\left(1+\frac{4m^2}{s}\right)\beta\right.\right.
  \nonumber \\ && \quad \quad \quad \quad
+\left. \left(1+\frac{4m^2}{s}-\frac{8m^4}{s^2}\right)
 \ln\frac{1+\beta}{1-\beta}\right] 
%
\nonumber \\ &&  
%
+\left.C_A\left[-\left(\frac{1}{3}+\frac{5}{3}\frac{m^2}{s}\right)\beta
+\frac{4m^4}{s^2}\ln\frac{1+\beta}{1-\beta}\right]\right\},
\end{eqnarray}
where $K_{gg}=(N^2-1)^{-2}$ is a color average factor and $C_A=N$ is
the Casimir invariant for the adjoint representation of $SU(N)$.
The threshold behavior ($s \rightarrow 4m^2$) of this expression is given by
\begin{equation}
\sigma_{gg,\, \rm thres}^{(0)}(s,m^2)= \pi \alpha_s^2(\mu^2) K_{gg}
\frac{1}{s} N C_F [4 C_F-C_A] \beta.
\end{equation}
Again, the complete NLO expression for the cross section in the $gg$ channel 
is unavailable but analytic results are given for the S+V terms in 
\cite{betal}. These were used in \cite{MSSN} to analyze the magnitude of the
cross section near threshold. 

In \cite{LSN} an approximation was given for the 
NLO soft-plus-virtual (S+V) contributions
and the analogy with the Drell-Yan process was exploited
to resum them to all orders of perturbation theory.
We are discussing partonic reactions of the type 
$i(k_1)+j(k_2) \rightarrow Q(p_1) + \bar Q(p_2)+g(k_3)$,
and we introduce the kinematic variables $t_1=(k_2-p_2)^2-m^2$, 
$u_1=(k_1-p_2)^2-m^2$, and $s_4 = s+t_1+u_1$. The variable $s_4$ depends
on the four-momentum of the extra partons emitted in the reaction
and is zero for elastic scattering.   
The first-order S+V result in the $\overline{\rm MS}$ scheme is 
\begin{eqnarray}
s^2\frac{d^2\sigma^{(1)}_{ij}(s,t_1,u_1)}{dt_1 du_1}&=&
\sigma_{ij}^B(s,t_1,u_1)\frac{2C_{ij}}{\pi} \alpha_s(\mu^2)
\nonumber \\ &&
\times
\left[\frac{1}{s_4}\left(2\ln\frac{s_4}{m^2}+\ln\frac{m^2}{\mu^2}
\right)\theta(s_4-\Delta) \right. \nonumber \\ &&
\left. +\left(\ln^2\frac{\Delta}{m^2}
+\ln\frac{\Delta}{m^2}
\ln\frac{m^2}{\mu^2}\right) \delta(s_4)\right] \, \, \, \, 
\end{eqnarray} 
where $\Delta$ is a small parameter used to distinguish between
the soft $(s_4<\Delta)$ and the hard $(s_4>\Delta)$ regions in phase space.
Here we define $C_{q \bar q}=C_F$,
$C_{gg}=C_A$,
\begin{equation}
\sigma^B_{q\bar q}(s,t_1,u_1) = \pi \alpha_s^2(\mu^2) K_{q\bar{q}}
NC_F \Big[ \frac{t_1^2 + u_1^2}{s^2} + \frac{2m^2}{s}\Big]\,,
\end{equation}
and
\begin{eqnarray}
\sigma^B_{gg}(s,t_1,u_1)& =&  2\pi \alpha_s^2(\mu^2) K_{gg}
NC_F \Big[C_F - C_A \frac{t_1u_1}{s^2}\Big] \nonumber \\ && 
\times\Big[ \frac{t_1}{u_1} + \frac{u_1}{t_1} + \frac{4m^2s}{t_1u_1}
\Big(1 - \frac{m^2s}{t_1u_1}\Big) \Big] \,.
\end{eqnarray}
We can define an analogous result for the $q \bar q$ channel in
the DIS scheme (but not for the $gg$ channel) which is
\begin{eqnarray}
s^2\frac{d^2\sigma^{(1)}_{q \bar q}}{dt_1 du_1}(s,t_1,u_1)&=&
\sigma_{q \bar q}^B(s,t_1,u_1)
\frac{2 C_F}{\pi}\alpha_s(\mu^2)
 \nonumber \\ &&
\times
\left[\frac{1}{s_4}\left(\ln\frac{s_4}{m^2}+\ln\frac{m^2}{\mu^2}
\right)\theta(s_4-\Delta) \right. \nonumber \\ &&
\left.+\left(\frac{1}{2}\ln^2\frac{\Delta}{m^2}+\ln\frac{\Delta}{m^2}
\ln\frac{m^2}{\mu^2}\right) \delta(s_4)\right].
\end{eqnarray} 

The resummation of the leading S+V terms has been given in \cite{LSN}.
The result is 
\begin{eqnarray}
s^2\frac{d^2\sigma^{\rm res}_{ij}
(s,t_1,u_1)}{dt_1 du_1}&=&\sigma_{ij}^B(s,t_1,u_1)
\left[\frac{df(s_4/m^2,m^2/\mu^2)}{ds_4}\theta(s_4-\Delta)\right.
\nonumber \\ && \quad \quad \quad \quad \quad \quad 
+\left.f(\frac{\Delta}{m^2},\frac{m^2}{\mu^2})\delta(s_4)\right],
\end{eqnarray} 
where 
\begin{equation}
f\left(\frac{s_4}{m^2},\frac{m^2}{\mu^2}\right)=
\exp\left[A\frac{C_{ij}}{\pi}\bar\alpha_s\left(\frac{s_4}{m^2},m^2\right)
\ln^2\frac{s_4}{m^2}\right]
\frac{[s_4/m^2]^{\eta}}{\Gamma(1+\eta)}\exp(-\eta \gamma_E).
\end{equation}
Expressions for $A$, $\bar\alpha_s$, $\eta$, and $\gamma_E$ are
given in \cite{LSN}.

%------------------This is Section 3---------------------------------
\mysection{Results for bottom quark production at \newline
HERA-B}
%------------------------------------------------------------------
In this section we discuss $b$-quark production at HERA-B, and we examine 
the effects of the
resummation procedure discussed in the previous section.
Following the notation in \cite{LSN} the total hadron-hadron
cross section in order $\alpha_s^{k}$ is 
%--(4.1)
\begin{equation}
\sigma^{(k)}_H(S,m^2) = \sum_{ij}\int_{4m^2/S}^1
%
\,d\tau \,\Phi_{ij}(\tau,\mu^2)\, \sigma_{ij}^{(k)}(\tau S,m^2,\mu^2)\,,
\end{equation}
where $S$ is the square of the hadron-hadron c.m. energy and
$i,j$ run over $q,\bar q$ and $g$.
The parton flux $\Phi_{ij}(\tau,\mu^2)$ is defined via
%--(4.2)
\begin{equation}
\Phi_{ij}(\tau,\mu^2) = \int_{\tau}^1\, \frac{dx}{x} 
%
H_{ij}(x,\frac{\tau}{x},\mu^2) \,,
\end{equation}
and $H_{ij}$ is a product of the scale-dependent parton distribution
functions $f^h_i(x,\mu^2)$, where $h$ stands for the hadron which is
the source of the parton $i$
%--(4.3)
\begin{equation}
H_{ij}(x_1, x_2, \mu^2) = f_i^{h_1}(x_1, \mu^2) f_j^{h_2}(x_2,\mu^2)\,.
\end{equation}
The mass factorization scale $\mu$ is chosen to be identical with
the renormalization scale in the running coupling constant.

In the case of the all-order 
resummed expression the lower boundary in (3.1)  
has to be modified according to the condition
$s_0 < s - 2ms^{1/2}$, where $s_0$ is defined below (see \cite{LSN}).
Resumming the soft gluon contributions to all orders we obtain
%--(4.4)
\begin{equation}
\sigma^{\rm res }_H(S,m^2) = \sum_{ij}\int_{\tau_0}^1
%
\,d\tau \,\Phi_{ij}(\tau,\mu^2)\, \sigma_{ij}(\tau S,m^2,\mu^2)\,,
\end{equation}
where $\sigma_{ij}$ is given in (3.24) of \cite{LSN} and
%--(4.6)
\begin{equation}
\tau_0 = \frac{[m+(m^2+s_0)^{1/2}]^2}{S}\,,
\end{equation}
with $s_0=m^2(\mu_0^2/\mu^2)^{3/2}$ ($\overline{\rm MS}$ scheme) or
$s_0=m^2(\mu_0^2/\mu^2)$  (DIS scheme). Here $\mu_0$ is the non-perturbative
parameter used in \cite{LSN}. It is used to cut off the resummation
since the resummed corrections diverge for small $\mu_0$.

We now specialize to bottom quark production at HERA-B
where $\sqrt{S}=39.2$ GeV.
In the presentation of our results for the exact, approximate,
and resummed hadronic cross sections 
we use the MRSD$\_ ' \:$ parametrization for the parton distributions
\cite{mrs}.
Note that the hadronic results only involve partonic distribution
functions at moderate and large $x$, where there is little difference
between the various sets of parton densities.
We have used the MRSD$\_ '\:$ set 34 as given in PDFLIB \cite{PDFLIB} in the 
DIS scheme with the number of active light flavors $n_f=4$ and the QCD
scale $\Lambda_5 = 0.1559$ GeV. We have used the two-loop corrected 
running coupling constant as given by PDFLIB.    
Note that we have checked scheme differences in the $q \bar q$
channel by using the MRSD$\_ '\:$ 
set 31 as given in PDFLIB \cite{PDFLIB} in the 
$\overline{\rm MS}$ scheme. For the $gg$ channel there is no DIS scheme
so we always use the $\overline{\rm MS}$ scheme. 

First, we discuss the NLO contributions to bottom quark production at HERA-B
using the results in [8-10].
Except when explicitly stated otherwise we will take the factorization
scale $\mu=m_b$ where $m_b$ is the $b$-quark mass.
Also, throughout the rest of this paper, we will use $m$ and $m_b$ 
interchangeably.
In fig. 1 we show the relative contributions of the $q \bar q$ channel in
the DIS scheme and the $gg$ channel as a function of the bottom quark mass. 
We see  that the $gg$ contribution is the dominant
one, lying between 70\% and 80\% of the total NLO cross section 
for the range of bottom mass values given.
The $q \bar q$ contribution is smaller and makes up most of the remaining 
cross section.
The relative contributions of the $g q$ and the $g \bar q$ channels in the 
DIS scheme are negative and very small.
The situation here is the reverse of what is known about top quark production
at the Fermilab Tevatron where $q \bar q$ is the dominant channel 
with $gg$ making up
the remainder of the cross section, and $g q$ and  $g \bar q$ making an even
smaller relative contribution than is the case for bottom quark production
at HERA-B. The reason for this difference between top quark and bottom quark 
production is that the Tevatron is a $p \bar p$ collider while HERA-B is a
fixed-target $pp$ experiment. Thus, the parton densities involved 
are different and since
sea quark densities are much smaller than valence quark densities, the
$q \bar q$ contribution to the hadronic cross section diminishes for a 
fixed-target $pp$ experiment 
relative to a $p \bar p$ collider for the same partonic cross section.

In fig. 2 we show the $K$ factors for the $q \bar q$ and $gg$ channels and for
their sum as a function of bottom quark mass.
The $K$ factor is defined by $K=(\sigma^{(0)}
+\sigma^{(1)}\mid _{\rm exact})/\sigma^{(0)}$, where $\sigma^{(0)}$ is the Born
term and $\sigma^{(1)}\mid _{\rm exact}$ is the exact first order correction.
We notice that all $K$ factors are large. This is to be expected due 
to the new dynamical mechanisms which arise in NLO. The figure 
shows that higher order effects are more important for the $gg$ channel
than for $q \bar q$. 
The $K$ factor for the sum of the two channels is also quite large.
However, the $K$ factor for the total is slightly lower since
we also include the
negative contributions of the $qg$ and $\bar q g$ channels.   

These large corrections come predominantly from the threshold region for 
bottom quark production where it has been shown that initial state gluon 
bremsstrahlung (ISGB) is responsible for the large cross section at 
NLO \cite{MSSN}.
This can easily be seen in fig. 3 where the Born term and the
$O(\alpha_s^3)$ cross section are plotted as functions of
$\eta_{\rm cut}$ for the 
$q \bar q$ and $gg$ channels, where
$\eta=(s -4m^2)/4m^2$ is the variable into which we have incorporated
a cut in our programs for the cross sections. 
The cross sections rise sharply for increasing values of
$\eta_{\rm cut}$ between 0.1 and 1 and they reach a plateau at higher values 
of $\eta_{\rm cut}$. This indicates that the threshold region is very important
and that the region where $s>> 4m^2$ only makes a small contribution 
to the cross sections. 
Note that in the last figure as well as throughout the 
rest of this paper we are assuming that the bottom quark mass is
$m_b=4.75$ GeV$/c^2$.

Next, we discuss the scale dependence of our NLO results. In fig. 4 
we show the total
$O(\alpha_s^3)$ cross section as a function of the 
factorization scale for the $q \bar q$ and $gg$ channels.
We see that as the scale decreases,   
the NLO cross sections peak at a scale close to half the mass of the
bottom quark and then decrease for smaller values of the scale. For the
$q \bar q$ channel the NLO cross section is relatively flat. The situation
is much worse for the $gg$ channel, however, since the peak is very sharp and
the scale dependence is much greater. Since the $gg$ channel dominates, this
large scale dependence is also reflected in the total cross section. 
Thus the variation in
the NLO cross section for scales between $m/2$ and $2m$ is large.

In fig. 5 we examine the $\mu_0$ dependence of the 
resummed cross sections for $b$-quark production at HERA-B
for the $gg$ and $q \bar q$ channels (in the $\overline{\rm MS}$
and DIS schemes).
We also show, for comparison, the $\mu_0$ dependence of
$\sigma^{(0)}+\sigma^{(1)}\mid _{\rm app}+\sigma^{(2)}\mid _{\rm app}$,
where $\sigma^{(1)}\mid _{\rm app}$
and $\sigma^{(2)}\mid _{\rm app}$ denote the approximate first and second
order corrections, respectively, where only soft gluon contributions
are taken into account via the expansion of (2.11). 
Note that we have imposed the same cut on the phase space of $s_4$ ($s_4>s_0$)
as for the resummed cross section. 
The effect of the resummation shows in the difference between the two curves
for each channel.
At small $\mu_0$, $\sigma^{\rm res}$ diverges signalling the 
divergence of the running coupling constant. This is not physical
and should be cancelled by an unknown non-perturbative term. 
There is a region for each channel where the higher-order terms are numerically
important.
At large values of $\mu_0$ the two lines for each channel 
are practically the same.
For the $q \bar q$ channel in the DIS scheme the resummation is successful
in the sense that there is a relatively large region of $\mu_0$ where 
resummation is well behaved before we encounter the divergence. 
This region is reduced for the $q\bar q$ channel in the $\overline{\rm MS}$
scheme reflecting the differences between (2.6) and (2.9).
For the $gg$ channel, however, this region is even smaller.

From these curves we choose what we think are reasonable values for $\mu_0$.
We choose $\mu_0=0.6$ GeV for the $q \bar q$ channel in the DIS scheme
($\mu_0/m \approx 13 \%$) and $\mu_0=1.7$ GeV for the $gg$ channel
($\mu_0/m \approx  36 \%$). The values we chose 
for the $q \bar q$ and $gg$ channels are such that 
the resummed cross sections are slightly larger than the
sums $\sigma^{(0)}+\sigma^{(1)}\mid _{\rm app}+\sigma^{(2)}\mid _{\rm app}$.
Note that these $\mu_0$ values are not exactly
the same as those used in ref. 4,
where $\mu_0/m = 10 \%$ and $\mu_0/m = 25 \%$
for the $q \bar q$ and $gg$ channels respectively, 
which predicted the mass dependence of the top quark cross section. 
In this reference the $\mu_0$ parameters were again
chosen via the criterion that the higher order terms in the perturbation theory
should not be too large. 

It is illuminating to compare fig. 5 with a 
corresponding plot for the top quark case 
(for instance with figs. 12, 13, and 14 in \cite{LSN}, where the top quark 
mass was taken as 100 GeV).
There one can infer that if we take the slightly larger $\mu_0$ values
given above there is very little change in the top quark cross section.
The reason is that in this case the $gg$ channel makes only a small 
contribution and the $\mu_0$ dependence in the $q\bar q$ channel reflects the 
small variation of the running coupling constant at a scale $\mu = 100$ GeV.
As the running coupling constant varies more rapidly at a scale
$\mu = 4.75$ GeV the $\mu_0$ parameters should be taken from 
measurements at the lower scale and then used in the prediction
of the top quark cross section. This emphasizes the importance of 
the proposed measurement at HERA-B. It is clear from fig. 5 that we
cannot choose $\mu_0/m = 25 \%$ for the $gg$ channel for 
bottom quark production
but we can choose $\mu_0/m = 36 \%$ for the $gg$ channel for
top quark production, with very little change in the value of the
top quark cross section.
Both sets of parameters yield cross sections which are within the error bars 
of the recent CDF \cite{CDF} and D0 \cite{D0}
experimental results for the top quark cross section.
Therefore our cut off parameters do have experimental justification.
We would also like to point out that an application of the principal value 
resummation method has been recently completed by Berger and Contopanagos
\cite{BC} leading to 
essentially the same mass dependence of the top cross section as reported
in ref. 4, which again justifies our choice for $\mu_0$.
Finally note that we could just as easily have chosen to work in the
$\overline{\rm MS}$ scheme for both channels by changing
$\mu_0$ in the $q \bar q$ channel
to $\mu_0 \approx 1.3 $ GeV. The reason the DIS scheme is preferred
is simply because it has a larger radius of convergence. 

In fig. 6 we plot the  
NLO cross section for $\mu=m/2$, $m$, and $2m$, and the resummed 
cross section with the values of $\mu_0$ that we chose from fig. 5,
as a function of the beam momentum for $b$-quark production at fixed-target
$pp$ experiments.
The width of the NLO
band reflects the large scale dependence that we discussed above.  
The total NLO cross section for $b$-quark production at HERA-B 
(beam energy 820 GeV) is 28.8 nb for
$\mu=m/2$; 9.6 nb for $\mu=m$; and 4.2 nb for $\mu=2m$. 
The resummed cross section is 18 nb. This value 
was calculated with the cut $s_4>s_0$ while no
such cut was imposed on the NLO result.
As we know the exact $O(\alpha_s^3)$ result, we can make an even 
better estimate using
the perturbation theory improved cross section
defined by
\begin {equation}
\sigma_H^{\rm imp}=\sigma_H^{\rm res}
+\sigma_H^{(1)}\mid _{\rm exact}
-\sigma_H^{(1)}\mid _{\rm app}\,,
\end{equation}
to exploit the fact that $\sigma_H^{(1)}\mid _{\rm exact}$ 
is known and $\sigma_H^{(1)}\mid _{\rm app}$ 
is included in 
$\sigma_H^{\rm res}$.
Therefore we also plot the improved total cross section versus
beam momentum in fig. 6 (where we have
taken into account the small negative contributions of the $qg$ and
$\bar q g$ channels). 
The improved total cross section for 
$b$-quark production at HERA-B is 19.4 nb.

It is interesting to make an estimate of the theoretical uncertainty
in the resummed cross section. For this one would have to determine
the non-leading logarithmic factors in the order-by-order cross
section. This is a difficult problem because the $gg$ channel dominates so
our knowledge of the order-by-order Drell-Yan cross section is not
relevant. The problem is presently under investigation \cite{KS}. 
In \cite{LSN} the logarithms
involving the scale parameter $\mu$ were resummed into the factor 
$\exp(-\eta \gamma_E)$ in (2.11).
This involves the assumption that these factors exponentiate, which
may not be true. However we will use (2.11) as an estimate. The
scale dependence enters into the running coupling constant, the
parton densities, as well as the factors involving
$\eta$ in (2.11). To get some idea of the $\mu$ dependence
of the resummed cross section in (2.11) we have rerun our programs with
$\mu = 4m/5$ and $\mu = 5m/4 $ and plotted the results in fig. 6.
We see that the cross section is still sensitive to the choice of
$\mu$, mainly through the exponential in (2.11). As noted above
this is probably misleading and the investigation in \cite{KS} should 
yield a more reliable result.



%------------------This is Section 4---------------------------------
\mysection{Conclusions}
%------------------------------------------------------------------
We have presented NLO and resummed results for the cross section 
for bottom quark production at HERA-B.
It has been shown that the $gg$ channel is dominant and
that the threshold region gives the main contribution
to the NLO cross section.   
The resummation of the  S+V logarithms 
produces an enhancement of the NLO results and yields the value
of 20 nb at $\sqrt{S}=39.2$ GeV for $m_b=4.75$ GeV/$c^2$. The
theoretical error on this number is under investigation.

{\bf ACKNOWLEDGEMENTS}

The work in this paper was supported in part under the
contract NSF 93-09888.

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\pagebreak
%------------------Figures---------------------------------
\centerline{\Large \bf List of Figures}
\vspace{3mm}

Fig. 1. Fractional contributions of the $gg$ ($\overline{\rm MS}$ scheme,
short-dashed line), $q \bar q$ (DIS scheme, long-dashed line), 
$qg$ (DIS scheme, lower dotted line), and $\bar q g$ (DIS scheme,
upper dotted line) channels to the total $O(\alpha_s^3)$
$b$-quark production cross section at HERA-B as a function of $b$-quark mass. 


Fig. 2. The K factors as a function of $b$-quark mass 
for $b$-quark production at HERA-B for the 
$gg$ channel($\overline{\rm MS}$ scheme,
short-dashed line), the $q \bar q$ channel (DIS scheme, long-dashed line),
the sum of the $gg$ and $q \bar q$ channels (dotted line), and the sum of all
channels (solid line).


Fig. 3. Cross sections for $b$-quark production at HERA-B 
versus $\eta_{\rm cut}$ with $m_b=4.75$ GeV$/c^2$ for the $q\bar q$ channel
in the DIS scheme and the $gg$ channel in the $\overline{\rm MS}$ scheme.
Plotted are the Born term ($gg$, upper solid line at high $\eta_{\rm cut}$;
$q\bar q$, lower solid line at high $\eta_{\rm cut}$)
and the $O(\alpha_s^3)$ cross section ($gg$, upper dashed line;  
$q\bar q$, lower dashed line).


Fig. 4. The scale dependence of the NLO cross section for $b$-quark production
at HERA-B with $m_b=4.75$ GeV$/c^2$ for the $q \bar q$ channel 
in the DIS scheme (lower line) and for the $gg$ channel in the 
$\overline{\rm MS}$ scheme (upper line). 

Fig. 5. The $\mu_0$ dependence of the resummed cross section for 
$b$-quark production at HERA-B with $m_b=4.75$ GeV$/c^2$
for the $q \bar q$ channel in the DIS scheme, in the $\overline{\rm MS}$
scheme, and for the $gg$ channel in the $\overline{\rm MS}$ scheme.
Plotted are $\sigma_{q\bar q}^{\rm res}$ (lower solid line DIS scheme,
middle solid line $\overline{\rm MS}$ scheme) and $\sigma_{gg}^{\rm res}$ 
(upper solid line).  Also we plot the sum 
$\sigma^{(0)}+\sigma^{(1)}\mid _{\rm app}+\sigma^{(2)}\mid _{\rm app}$
(lower dotted line for $q \bar q$ in the DIS scheme,
middle dotted line for $q \bar q$ in the $\overline{\rm MS}$ scheme,
and upper dotted line for $gg$).

Fig. 6. Resummed, improved, and NLO cross sections versus beam momentum
for $b$-quark production in fixed-target $pp$ experiments for $m_b=4.75$
GeV$/c^2$. Plotted are the total resummed cross section  
(long-dashed line), the total improved cross section (short-dashed line),
and the total $O(\alpha_s^3)$) cross section ($\mu=m$ solid line, 
$\mu=m/2$ upper dotted line, $\mu=2m$ lower dotted line).
The scale variation in the resummed cross section
with $\mu = 5m/4$ is plotted as the upper dot-dashed
line and that with $\mu = 4m/5$ as the lower dot-dashed line.  


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