  The Burkhardt-Cottingham (BC) sum rule connects the twist-3 light-cone parton
distribution function (PDF) $g_{T}(x)$ to the twist-2 helicity PDF $g_{1}(x)$.
The chiral-odd counterpart of the BC sum rule relates the twist-3 light-cone
PDF $h_{L}(x)$ to the twist-2 transversity PDF $h_{1}(x)$. These BC-type sum
rules can also be derived for the corresponding quasi-PDFs. We perform a
perturbative check of the BC-type sum rules in the quark target model and the
Yukawa model, by going beyond the ultra-violet (UV) divergent terms. We employ
dimensional regularization (DR) and cut-off schemes to regulate UV divergences,
and show that the BC-type sum rules hold for DR, while they are generally
violated when using a cut-off. This violation can be traced back to the
breaking of rotational invariance. We find corresponding results for the sum
rule relating the mass of the target to the twist-3 PDF $e(x)$. Moreover, we
supplement our analytical results with numerical calculations.
