  The anomalous scaling exponents $\zeta_{n}$ of the longitudinal structure
functions $S_{n}$ for homogeneous isotropic turbulence are derived from the
Navier-Stokes equations by using field theoretic methods to develop a low
energy approximation in which the Kolmogorov theory is shown to act effectively
as a mean field theory. The corrections to the Kolmogorov exponents are
expressed in terms of the anomalous dimensions of the composite operators which
occur in the definition of $S_{n}$. These are calculated from the anomalous
scaling of the appropriate class of nonlinear Green's function, using an $uv$
fixed point of the renormalisation group, which thereby establishes the
connection with the dynamics of the turbulence. The main result is an algebraic
expression for $\zeta_{n}$, which contains no adjustable constants.
  It is valid at orders $n$ below $% g_{\ast}^{-1}$, where $g_{\ast}$ is the
fixed point coupling constant. This expression is used to calculate $\zeta
_{n}$ for orders in the range $% n=2$ to 10, and the results are shown to be in
good agreement with experimental data, key examples being $\zeta_{2}=0.7$,
$\zeta_{3}=1$ and $% \zeta_{6}=1.8$.
