  Topological physics relies on the existence of Hamiltonian's eigenstate
singularities carrying a topological charge, such as quantum vortices, Dirac
points, Weyl points and -- in non-Hermitian systems -- exceptional points
(EPs), lines or surfaces. They appear only in pairs connected by a Fermi arc
and are related to a Hermitian singularity, such as a Dirac point. The
annihilation of 2D Dirac points carrying opposite charges has been
experimentally reported. It remained elusive for Weyl points and second order
EPs terminating different Fermi arcs. Here, we observe the annihilation of
second order EPs issued from different Dirac points forming distinct valleys.
We study a liquid crystal microcavity with voltage-controlled birefringence and
TE-TM photonic spin-orbit-coupling. Two neighboring modes can be described by a
two-band Hermitian Hamiltonian showing two topological phases with either two
same-sign or four opposite-sign Dirac points (valleys). Non-Hermiticity is
provided by polarization-dependent losses, which split Dirac points into pairs
of EPs, connected by Fermi arcs. We measure their topological charges and
control their displacement in reciprocal space by increasing the
non-Hermiticity degree. EPs of opposite charges from different valleys meet and
annihilate, connecting in a closed line the different Fermi arcs. This
non-Hermitian topological transition occurs only when the Hermitian part of the
Hamiltonian is topologically trivial (with four valleys), but is distinct from
the Hermitian transition. Our results offer new perspectives of versatile
manipulation of EPs, opening the new field of non-Hermitian valley-physics.
