Pure-mode correlation functions for cosmic shear and application to
  KiDS-1000
  One probe for systematic effects in gravitational lensing surveys is the
presence of so-called B-modes in the cosmic shear two-point correlation
functions \xi_\pm(\vt), since lensing is expected to produce only E-mode shear.
Furthermore, there exist ambiguous modes which can not uniquely be assigned to
either E- or B-mode shear. We derive explicit equations for the pure-mode shear
correlation functions \xi_\pm^E/B(\vt) and their ambiguous components
\xi_\pm^amb(\vt), that can be derived from the measured \xi_\pm(\vt) on a
finite angular interval \tmin\le\vt\le\tmax, such that the latter can be
decomposed uniquely into pure-mode functions as \xi_+=\xi_+^E+\xi_+^B+\xi_+^amb
and \xi_-=\xi_-^E-\xi_-^B+\xi_-^amb. The derivation is obtained by defining a
new set of COSEBIs, for which explicit relations are obtained, and which yields
a smaller covariance between COSEBI modes. We derive the relation between
\xi_\pm^E/B/amb and the underlying E-/B-mode power spectra. The pure-mode
correlation functions can provide a diagnostics of systematics in configuration
space. We then apply our results to SLICS simulations and the KiDS-1000 cosmic
shear data, calculate the new COSEBIs and the pure-mode correlation functions,
as well as the corresponding covariances, and show that the new statistics fit
equally well to the best-fitting cosmological model as the previous KiDS-1000
analysis and recovers the same level of (insignificant) B-modes. We also
consider in some detail the ambiguous modes at first- and second-order level,
finding some surprising results; for example, the shear field of a point mass,
when cut along a line through the center, can not be ascribed uniquely to an
E-mode shear and is thus ambiguous, and the shear correlation functions
resulting from a random ensemble of point masses, when measured over a finite
angular range, correspond to an ambiguous mode.
