  The magnetic moment ($a_\gamma$) and weak magnetic moment ($a_W$) of charged
leptons and quarks are sensitive to quantum effects of new physics heavy
resonances. In effective field theory $a_\gamma$ and $a_W$ are induced by two
independent operators, therefore, one has to measure both the $a_\gamma$ and
$a_W$ to shed lights on new physics. The $a_W$'s of the SM fermions are
measured at the LEP. In this work, we analyze the contributions from magnetic
and weak magnetic moment operators in the processes of $pp\to H \gamma$ and
$gg\to H \to \tau^+ \tau^- \gamma$ at the High-Luminosity Large Hadron
Collider. We demonstrate that the two processes could cover most of the
parameter space that cannot be probed at the LEP.
