Some Issues with Tracking MADX vs Lifetrac Dmitry

Some Issues with Tracking: MADX vs. Lifetrac Dmitry Shatilov BINP, Novosibirsk Acknowledgements: H. Burkhardt, T. Charles 111 th FCC-ee Optics Design Meeting CERN, 31 January 2020

Preamble To perform beam-beam simulations for realistic lattice (with misalignments and corrections), the MADX output is converted to the Lifetrac input. But it was discovered that the lattice in Lifetrac does not quite match the original lattice in MADX. : • The vertical betatron tunes differ by 0. 02 0. 03 • The chromaticity (esp. vertical) is larger in Lifetrac • The vertical orbit deviation is much larger in Lifetrac Without misalignments, agreement is very good. Probably, the main source of discrepancy is the equilibrium orbit. Further investigations has shown that this is due to differences in drift tracking.

Coordinates and Drift in Lifetrac dx coord(2) = dx/ds = px/pz coord(4) = dy/ds = py/pz ds coord(6) = d. E/E 0 P/P 0 = SQRT (1 + coord(6) * (2 + coord(6)) / 0**2) / 0 = P/P 0 /(1 + coord(6)) Tracking: dx = ds * coord(2) dy = ds * coord(4) dz = ds * (1/ 0 - SQRT (1 + coord(2)**2 + coord(4)**2) / )

Coordinates and Drift in MADX dx coord(2) = px/P -? coord(4) = py/P -? ds coord(6) = d. E/P 0 -? dz is calculated by expansion up to 2 nd order (matrix + tensor) Maybe 2 nd order is not enough? Why not use exact solution? It is quite simple and faster than tensor. Hack in Lifetrac: dz calculated as in MADX, coord(2) = px/P Discrepancy greatly decreases (but not completely gone)

Discussion § In FCC-ee, where sextupoles are very strong and sensitivity to the orbit is much larger than in any previous machine, we need more accurate approach. Drift expansion to 2 nd order is probably not enough. § How it is made in SAD? § What will we do with this issue in MADX and/or Lifetrac? § Still there is no full understanding (some discrepancies remain).