  Azimuthal angle correlations of two jets in the process $pp\to HHjj$ are
studied. The loop induced $\mathcal{O}(\alpha_s^4 \alpha_{}^2)$ gluon fusion
(GF) sub-process and the $\mathcal{O}(\alpha_{}^4)$ weak boson fusion (WBF)
sub-process are considered. The GF sub-process exhibits strong correlations in
the azimuthal angles $\phi_{1,2}^{}$ of the two jets measured from the
production plane of the Higgs boson pair and the difference between these two
angles $\phi_1^{}-\phi_2^{}$, and a very small correlation in the sum of them
$\phi_1^{}+\phi_2^{}$. The impact of using a finite top mass $m_t^{}$ value on
the correlations is found crucial. The transverse momentum of the Higgs boson
can be used to enhance or suppress the correlations. The impact of a
non-standard value for the triple Higgs self-coupling on the correlations is
found much smaller than that on other observables, such as the invariant mass
of the two Higgs bosons. The peak shifts of the azimuthal angle distributions
reflect the magnitude of parity violation in the $gg\to HH$ amplitude and the
dependence of the distributions on parity violating phases is analytically
clarified. The WBF sub-process also produces correlated distributions and it is
found that they are not induced by the quantum effect of the intermediate weak
bosons but mainly by a kinematic effect. This kinematic effect is a
characteristic feature of the WBF sub-process and is not observed in the GF
sub-process. It is found that the correlations are different in the GF and in
the WBF sub-processes. As part of the process dependent information, they will
be helpful in the analyses of both the GF sub-process and the WBF sub-process
at the LHC.
