  We demonstrate how a class of non-supersymmetric $SO(10)$ GUT with asymmetric
left-right theory $SU(2)_L \times U(1)_R \times U(1)_{B-L} \times SU(3)_C$ and
Pati-Salam theory $SU(2)_L \times SU(2)_R \times SU(4)_C$ as intermediate
symmetry breaking steps leads to successful gauge coupling unification
satisfying proton decay constraints. The motivation behind this work is two
fold: firstly to study the renormalization group evolution equations for gauge
couplings by keeping right-handed neutral gauge boson $Z_R$ around LHC energy
range leading interesting dilepton searches at collider while fixing charge
partner of the gauge boson $W_R$ at very high scale; secondly to explain
neutrino masses and associated lepton number violating process like
neutrinoless double beta decay in three possible cases depending on how
$SU(2)_L \times U(1)_R \times U(1)_{B-L} \times SU(3)_C$ breaks down to SM. We
include one extra fermion singlet per generation in order to implement gauged
extended seesaw where light neutrino mass is governed by natural type-II seesaw
mechanism whereas type-I seesaw contribution is exactly canceled out. Since
light neutrino mass formula is independent of Dirac neutrino mass matrix, the
value of Dirac neutrino mass is taken to be up-type quark mass matrix which is
a characteristics of Pati-Salam symmetry relating quarks with leptons. We
present analytic relation for effective Majorana mass parameter and
corresponding half-life arising from new physics contributions due to purely
left-handed currents through exchange of heavy right-handed neutrinos and
sterile neutrinos. We numerically estimate effective Majorana mass parameter
and half-life vs. lightest neutrino mass and derive lower bound on lightest
neutrino mass by saturating with experimental bounds like GERDA Phase-II,
KamLANDZen and EXO.
