  We study the dynamics of a quantum Brownian particle weakly coupled to a
thermal bath. Working in the Schwinger-Keldysh formalism, we develop an
effective action of the particle up to quartic terms. We demonstrate that this
quartic effective theory is dual to a stochastic dynamics governed by a
non-linear Langevin equation. The Schwinger-Keldysh effective theory, or the
equivalent non-linear Langevin dynamics, is insufficient to determine the out
of time order correlators (OTOCs) of the particle. To overcome this limitation,
we construct an extended effective action in a generalised Schwinger-Keldysh
framework. We determine the additional quartic couplings in this OTO effective
action and show their dependence on the bath's 4-point OTOCs. We analyse the
constraints imposed on the OTO effective theory by microscopic reversibility
and thermality of the bath. We show that these constraints lead to a
generalised fluctuation-dissipation relation between the non-Gaussianity in the
distribution of the thermal noise experienced by the particle and the thermal
jitter in its damping coefficient. The quartic effective theory developed in
this work provides extension of several results previously obtained for the
cubic OTO dynamics of a Brownian particle.
