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%% section 2 NEXTTOLEADING ORDER CALCULATIONS [slacpub7191002 in slacpub7191002: slacpub7191003]
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\section{\usemenu{slacpub7191::context::slacpub7191002}{NEXTTOLEADING ORDER CALCULATIONS}}\label{section::slacpub7191002}
While leadingorder calculations often reproduce the shapes of
distributions well, they suffer from several practical and conceptual
problems whose resolution requires the use of nexttoleading order
calculations. These problems are tied to the various logarithms
that can arise in perturbation theory.
The first of these logarithms are `UV' ones, connected with the
renormalization scale. We are forced to introduce a renormalization scale
$\mu$ in order to define the coupling, $\alpha_s(\mu)$; but physical
quantities, such as cross sections or differential cross sections,
should be independent of $\mu$. When we compute such a quantity
in perturbation theory, however, we necessarily truncate its expansion
in $\alpha_s$, and this introduces a spurious dependence on $\mu$. This
dependence is significant in realworld applications of perturbative
QCD because the coupling is not that small, because it runs relatively
quickly, and because we are interested in processes with a relatively
large number of colored `final'state partons. Together, these effects
can lead to anywhere from a 30\% to a factor of 23 normalization uncertainty
in predictions of experimentallymeasured distributions.
At leading order, the only dependence on $\mu$ comes from the resummation
of logarithms in the running coupling $\alpha_s(\mu)$, and the scale choice
is arbitrary. At nexttoleading order, however, the virtual corrections
to the matrix element introduce another dependence on $\mu$. This dependence
can  and in practice, often does  reduce the overall sensitivity
of a prediction to variations in $\mu$. (I should stress, however, that
while varying $\mu$ by some preset amount, say a factor of two up and
down from a typical scale, gives an indication of the sensitivity of or
uncertainty in the calculation, it does {\it not\/} give an estimate of
the error involved; for that one needs a nexttonexttoleading order
calculation.)
The other logarithms are the `IR' ones, connected with the presence
of soft and collinear radiation. Jets in a detector are not infinitely
narrow pencil beams; they consist of a spray of hadrons spread over
a finite segment. Experimental measurements of jet distributions and
the like depend on resolution parameters, such as the jet cone size and
minimum transverse energy. In a leadingorder calculation, jets are
modelled by lone partons. As a result, these predictions don't depend
on these parameters (or have an incorrect dependence on them).
In addition, the internal structure of a jet cannot be predicted at all.
At nexttoleading order, one necessarily includes contributions with
additional real radiation, which either shows up inside one of the jets,
or as soft radiation in the event. This introduces, for
example, the required logarithmic dependence on the cone size $\Delta R$.
At a more conceptual level, we must remember that jet differential
cross sections are multiscale quantities, involving not only a
hard scale (say a jet transverse energy $E_T$) but also scales
characterizing the resolution
defining a jet (for example, $E_T \Delta R$ or $E_{T{\rm min}}$).
As a result, the
perturbative expansion is not one in $\alpha_s$ alone, but contains
logarithms and logarithmssquared of ratios of scales. These logarithms,
which arise from the infrared structure of gauge theories, might
spoil the applicability of perturbation theory if they grow too large.
Only a nexttoleading order calculation can tell us if we are safe.
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