  Vortexed photons with finite orbital angular momentum have a distinct mode
profile with topological charge at the centre of the mode, while propagating in
a certain direction. Each mode with different topological charge of $m$ is
orthogonal, in the sense that the overlap integral vanishes among modes with
different values of $m$. Here, we theoretically consider a superposition state
among 3 different modes with left- and right-vorticies and a Gaussian mode
without a vortex. These 3 states are considered to be assigned to different
quantum states, thus, we have employed the $\mathfrak{su}(3)$ Lie algebra and
the associated SU(3) Lie group to classify the photonic states. We have
calculated expectation values of 8 generators of the $\mathfrak{su}(3)$ Lie
algebra, which should be observable, since the generators are Hermite. We
proposed to call these parameters as Gell-Mann parameters, named after the
theoretical physicist, Murray Gell-Mann, who established Quantum
Chromo-Dynamics (QCD) for quarks. The Gell-Mann parameters are represented on
the 8-dimensional hypersphere with its radius fixed due to the conservation law
of the Casimir operator. We have discussed a possibility to explore photonic
QCD in experiments and classified SU(3) states to embed the parameters in SO(6)
and SO(5).
