  We use a hydrodynamic approach to investigate dynamic spin susceptibility of
the antiferromagnetic spin-$1/2$ Heisenberg chain with a uniform
Dzyaloshinskii-Moriya (DM) interaction in the presence of an external magnetic
field. We find that transverse (with respect to the magnetic field) spin
susceptibility harbors two (respectively, three) spin excitation modes when the
magnetic field is parallel (respectively, orthogonal) to the DM axis. In all
cases, the marginally irrelevant backscattering interaction between the spinons
creates a finite energy splitting between optical branches of excitations at $k
= 0$. Additionally, for the orthogonal geometry, the two lower spin branches
exhibit avoided crossing at finite momentum which is determined by the total
magnetic field (the sum of the external and internal molecular fields) acting
on spinons. Our approximate analytical calculations compare well with numerical
results obtained using matrix-product-state (MPS) techniques. Physical
consequences of our findings for the electron spin resonance experiments are
discussed in detail.
