The unusual glitch recoveries of the high magnetic field pulsar J1119-6127

Providing a link between magnetars and radio pulsars, high magnetic field neutron stars are ideal targets to investigate how bursting/magnetospheric activity and braking torque variations are connected to rotational glitches. The last spin-up glitch of the highly magnetised pulsar J1119$-$6127 back in 2007 was the first glitch in a rotationally powered radio pulsar to be accompanied by radiative changes. Moreover, it was followed by an uncommon glitch relaxation that resulted in a smaller spin-down rate relative to the prediction of the pre-glitch timing model. Here, we present 4 years of new radio timing observations and analyse the total of 16 years of timing data for this source. The new data uncover an ongoing evolution of the spin-down rate, thereby allowing us to exclude permanent changes in the external or internal torque as a stand-alone cause of the peculiar features of the glitch recovery. Furthermore, no additional variations of the radio pulse profile are detected, strengthening the association of the previously observed transient emission features with the glitching activity. A self-consistent measurement of the braking index yields a value $n\simeq2.7$, indicating a trajectory in the $P-\dot{P}$ plane inclined towards the magnetars. Such a potential evolutionary link might be strengthened by a, possibly permanent, reduction of $\sim15\%$ in $n$ at the epoch of the 2007 glitch.