  $\gamma$-ray spectra for positron annihilation with the core and valence
electrons of the noble gas atoms Ar, Kr and Xe is calculated within the
framework of diagrammatic many-body theory. The effect of positron-atom and
short-range positron-electron correlations on the annihilation process is
examined in detail. Short-range correlations, which are described through
non-local corrections to the vertex of the annihilation amplitude, are found to
significantly enhance the spectra for annihilation on the core orbitals. For
Ar, Kr and Xe, the core contributions to the annihilation rate are found to be
0.55\%, 1.5\% and 2.2\% respectively, their small values reflecting the
difficulty for the positron to probe distances close to the nucleus.
Importantly however, the core subshells have a broad momentum distribution and
markedly contribute to the annihilation spectra at Doppler energy shifts
$\gtrsim3$\,keV, and even dominate the spectra of Kr and Xe at shifts
$\gtrsim5$\,keV. Their inclusion brings the theoretical spectra into excellent
agreement with the experimental $\gamma$-spectra across the full range of
Doppler energy shifts. Additionally, the theory enables the calculation of the
`exact' vertex enhancement factors $\bar{\gamma}_{n\ell}$ for individual core
and valence subshells $n\ell$. They are found to follow a simple and physically
motivated scaling with the subshell ionization energy $I_{n\ell}$:
$\bar{\gamma}_{n\ell}=1+\sqrt{A/I_{n\ell}}+(B/I_{n\ell})^{\beta}$, where $A$,
$B$ and $\beta$ are positive constants. It is demonstrated that these factors
can be incorporated in simple independent-particle-model calculations to
successfully reconstruct the true many-body annihilation $\gamma$-spectra. This
formula can be used to determine the enhancement factors for positron
annihilation with core electrons of atoms across the periodic table and
condensed matter systems.
