The zero to four loop contribution to the cross section
Re+e− for
e+e−⟶ hadrons, when combined with the renormalization
group equation, allows for summation of all leading-log (
LL),
next-to-leading-log
(NLL)…N3LL perturbative contributions. It is also
shown how all logarithmic contributions to
Re+e− can be summed and
that
Re+e− can be expressed in terms of the log independent
contributions, and once this is done the running coupling
a is evaluated at a
point independent of the renormalization scale
μ. All explicit dependence
of
Re+e− on
μ cancels against its implicit dependence on
μ
through the running coupling
a so that the ambiguity associated with the
value of
μ is shown to disappear. The renormalization scheme dependency of
the "summed" cross section
Re+e− is examined in three distinct
renormalization schemes. In each case,
Re+e− is expressible in terms
of renormalization scheme independent parameters
τi and is explicitly and
implicitly independent of the renormalization scale
μ. Two of the forms are
then compared graphically both with each other and with the purely perturbative
results and the
RG-summed
N3LL results.