Tune variations in the Large Hadron Collider

The horizontal and vertical betatron tunes of the Large Hadron Collider (LHC) mainly depend on the strength of the quadrupole magnets, but are also affected by the quadrupole component in the main dipoles. In case of systematic misalignments, the sextupole component from the main dipoles and sextupole corrector magnets also affect the tunes due to the feed down effect. During the first years of operation of the LHC, the tunes have been routinely measured and corrected through either a feedback or a feed forward system. In this paper, the evolution of the tunes during injection, ramp and flat top are reconstructed from the beam measurements and the settings of the tune feedback loop and of the feed forward corrections. This gives the obtained precision of the magnetic model of the machine with respect to quadrupole and sextupole components. Measurements at the injection plateau show an unexpected large decay whose origin is not understood. This data is discussed together with the time constants and the dependence on previous cycles. We present results of dedicated experiments that show that this effect does not originate from the decay of the main dipole component. During the ramp, the tunes drift by about 0.022. It is shown that this is related to the precision of tracking the quadrupole field in the machine and this effect is reduced to about 0.01 tune units during flat top.