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%% section 5 $1/Q$ IR sensitivity of thrust [slacpub7176005 in slacpub7176005: slacpub7176005u1]
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\section{\usemenu{slacpub7176::context::slacpub7176005}{$1/Q$ IR sensitivity of thrust}}\label{section::slacpub7176005}
The DrellYan cross section appears to have no $1/Q$ power
corrections. This is not generally the case for any quantity.
There are good reasons to suspect the existence of $1/Q$
nonperturbative hadronization corrections to event shapes
observables in $e^+e^$ annihilation \cite{17}. Unlike the
DrellYan cross section, these observables
cannot be expressed directly in terms of Feynman diagrams,
and are obtained by
integrating the QCD amplitudes with certain weight functions such as to
emphasize a particular final state configuration.
These weight functions generally destroy the
balance of gluon emission at small and large angles,
and make these observables sensitive to nonperturbative momentum
flow at large angles. As a consequence $1/Q$ corrections are
invariably expected for event shapes.
For example, the average thrust $\langle 1T \rangle$
of the final state
is computed to leading order in
$\alpha_s$ by inserting the factor
\be
1T = (k_0\sqrt{k_0^2k_\perp^2})/Q
\ee
into the phasespace integral for gluon emission, which has a
structure identical to the DrellYan cross section
in moment space, see (\docLink{slacpub7176002.tcx}[DYphasespace]{5}).
The above factor suppresses smallangle emission but causes
$1/Q$ IR sensititvity.
An interesting speculation is whether
$1/Q$ corrections to event shapes are universal in the sense that
they can be related to a single nonperturbative parameter
\cite{18,11,19}. Although, due to importance of large angle
emission, this parameter would not be related to the universal
cusp anomalous dimension, the hypothesis makes sense
so long as the underlying physical process is the same
for all event shapes. Strictly speaking, the answer seems to be
negative, since $1/Q$ corrections also occur outside the twojet
region\cite{20} and higherorder corrections are not suppressed,
because $\alpha_s$ is evaluated at low scale, so that it is not counted.
One may still argue in favour
of {\em approximate} universality \cite{4,21},
if the coupling stays finite and reasonably small in the infrared.
This purely phenomenological hypothesis could in principle
be subjected to experimental tests. This, in fact, seems very hard
in practice, because of poor control of higher orders of the perturbative
series. One may suspect that the largest part of the hadronization correction
to event shapes estimated by Monte Carlo event generators is in fact
related to the perturbative parton cascade which can indeed be
universal to the extent that the event shape variable
is dominated by the twojet kinematics.
To illustrate the difficulty in testing
universality consider the average thrust
$\langle 1T \rangle$. The existing experimental data are well described
by \cite{17}
\be
\langle 1T \rangle(Q) = 0.335\,\alpha_s(Q) +1.02\, \alpha_s(Q)^2 +
\frac{1 {GeV}}{Q}
\ee
where the first two terms give the QCD calculation (to $O(\alpha_s^2$)
accuracy). The power correction is needed to gain agreement
with the data (over the entire range of $Q$).
The second order perturbative correction has a large coefficient,
indicating that the adopted scale $Q$ is in fact inadequate for this
process. The scale setting problem for event shapes is difficult. However,
as a natural first guess one can try to take $\alpha_s$ at a scale
of order of the jet mass $M$, which is related to thrust in the
twojet limit by $M^2 = (1T)\,Q^2/2$ [We take the scale $Q_*^2=(1T)\,
Q^2/4$, since for fixed $T$
this is the upper limit on the gluon transverse momentum.].
Taking $\alpha_s(m_Z)=0.12$ and using the fixedorder QCD result
$\langle 1T \rangle\sim 0.07$, we get $Q_*\sim 0.13 Q$. Fitting
again a $1/Q$ correction to the same data, we get
\be
\langle 1T \rangle(Q) = 0.335\,\alpha_s(Q_*) +
0.19\, \alpha_s(Q_*)^2 +
\frac{0.4 {GeV}}{Q}
\ee
The secondorder coefficient in the perturbative series has
become much smaller and the size of the required hadronization
correction has also decreased.
One might actually think of writing the power
correction as $0.05 GeV/Q_*$. Viewed this way, the issue of universality
becomes inseparable from the problem of determining the most appropriate
scale for the process.
%
%Obtaining theoretical control over $1/Q$ corrections will be very
%interesting. But understanding scale setting and
%higherorder perturbative effects in these observables
%might be even more important phenomenologically and is necessary
%to test universality.
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