Beam Optics Studies in the Large Hadron Collider: Observations on an Anomalous Octupolar Resonance Line in the LHC -- and -- Accuracy & Feasibility of the $\beta^*$ Measurement for LHC and HL-LHC Using K-Modulation

While linear LHC dynamics are mostly understood and under control, non-linear beam dynamics will play an increasingly important role in the challenging regimes of future LHC operation. In 2012, turn-by-turn measurements of large betatron excitations of LHC Beam 2 at injection energy were carried out. These measurements revealed an unexpectedly large spectral line in the horizontal motion with frequency $-Q_x-2Q_y$. Detailed analyses and simulations are presented to understand the nature of this spectral line. -- ABSTRACT II -- The future regimes of operation of the LHC will require improved control of $\beta^*$ measurements to succesfully level the luminosities in the interaction points. The method of K-modulation has been widely used in other machines such as, LEP, HERA, Tevatron and ALBA to measure lattice beta functions. In the LHC, K-modulation of the last quadrupoles of the IP is the method to measure $\beta^*$ in the IP. This paper highlights the challenge of high precision tune measurements (up to $10^{-5}$ for the correct determination of $\beta^*$ in the LHC and the HL-LHC. Furthermore it presents a new analytical method for the calculation of $\beta^*$ using K-modulation, and compares it to the current numerical method.