  A model of (3+1)-dimensional leptogenesis, proposed previously by the
authors, requires a CPT Violating (CPTV) background of the Kalb-Ramond (KR)
axion field. The KR axion is a pseudoscalar, which is dual to the field
strength of the spin-one field present in the massless gravitational multiplet
in the theory of closed bosonic strings (compactified to four dimensions).
Microscopic models for the emergence of such backgrounds are provided both by
low-energy string-inspired gravitational theories with torsion (including
(primordial) gravitational and axial gauge anomalies) and by Einstein-Cartan
gravity, a closely related simpler model. In this work we examine the
pseudoscalar quanta of the KR axion in this background using the methods of
effective field theory. In our model for leptogenesis there is a single sterile
{right-handed} neutrino (RHN) with mass $m_N$. At energies lower than $m_N$, an
axion potential is derived by integrating out {at one loop} the sterile
neutrino in the spirit of effective field theory. The stability of this axion
potential is important for the viability of our model. The vacuum of this
potential is generally metastable. The stability of the vacuum is determined by
the ratio of the torsion-induced-axion coupling (which depends on the string
mass scale) to $m_N$, which should be larger or equal to one, for the validity
of our effective field theory. The vacuum is metastable only for axion
couplings much larger than the sterile neutrino mass (large string mass scales,
e.g. comparable to the four-dimensional Planck mass), with a lifetime much
larger than the age of the observable Universe. By contrast, if axion couplings
are comparable to the RHN mass the false vacuum is highly unstable, with a
lifetime much smaller than the age of the observable Universe; in this case the
CPTV leptogenesis scenario is not viable.
