  Recently, a novel four-dimensional Einstein-Gauss-Bonnet (EGB) theory was
presented to bypass the Lovelock's theorem and to give nontrivial effects on
the four-dimensional local gravity. The main mechanism is to introduce a
redefinition $\alpha\rightarrow\alpha/(D-4)$ and to take the limit
$D\rightarrow4$. However, this theory does not have standard four-dimensional
field equations. Some regularization procedures are then proposed to address
this problem [arXiv:2003.11552, arXiv:2003.12771, arXiv:2004.08362,
arXiv:2004.09472, arXiv:2004.10716]. The resultant regularized four-dimensional
EGB theory has the same on-shell action as the original theory. Thus it is
expected that the novel four-dimensional EGB theory is equivalent to its
regularized version. However, the equivalence of these two theories is
symmetry-dependent. In this paper, we test the equivalence in a cylindrically
symmetric spacetime. The well-defined field equations of the two theories are
obtained, with which our follow-up analysis shows that they are equivalent in
such spacetime. Cylindrical cosmic strings are then considered as specific
examples of the metric. Three sets of solutions are obtained and the
corresponding string mass densities are evaluated. The results reveal how the
Gauss-Bonnet term in four dimensions contributes to the string geometry in the
new theory.
