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\begin{document}
\title{Analytical studies of beam-beam effects for the HL-LHC}
\author{D. Kaltchev }
\maketitle
%%%%%%%%%%%%%%%%%%
Around the experimental regions of the Large Hadron Collider (LHC)
beams travel in a common vacuum chamber and therefore experience the
fields of the opposing beams, so-called long range interactions. The
number of these parasitic encounters depends on the lengths of the
common regions before the beams are sufficiently separated by dipole
magnets, and on the bunch spacing. For a fixed encounter, the effect of
the long-range interaction depends on the beam separation normalized to
transverse beam size.
The TRIUMF-CERN collaboration on High-Luminosity Upgrade of the LHC
(HL-LHC) is focused on investigations of these beam-beam interactions
and their influence on beam quality which in turn affects the maximum
achievable luminosity. Although beam-beam effects are dominant to
decrease the available dynamic aperture of the HL-LHC beams at
collision, for correct calculation of dynamic aperture they must be
considered in combination with the other nonlinearities present.
Our main activity remains particle tracking in a latest version HL-LHC
lattice which includes beam-beam effects and field errors. The aim is to
compute the dynamic aperture (DA), its dependence on parameters and
tune-scans: DA values for a dense set of points in horizontal/vertical
tune space allowing to see decremental effects of betatronic resonances.
Another important objective is advancing the theory of motion in
presence of multiple long-range beam-beam interactions. This should
allow to understand better the combination of factors influencing DA.
Most promising are analytical calculations of the combined action of all
long-range collisions (as an effective Hamiltonian, or approximate
invariant of motion) and the interpretation of the distortion of this
invariant (emittance smear) as early indicator of chaotic motion, The
one-dimensional version of this theory, developed and reported in the
past, was recently confirmed to explain the resonance locations:
\cite{ipacone} and \cite{talklbl}.
The above, in terms of Hamiltonian mappings, is in fact equivalent to
finding the lowest order ``normal form''. It therefore requires a
two-dimensional beam-beam Hamiltonian to be expressed in the so called
resonance (action-angle) basis. As this was done, the resultant
Fourier-expansion coefficients appeared to be nearly identical to some
little known mathematical objects -- two-dimensional Bessel functions.
Low-order such functions were shown to describe the tune shift with
amplitude, beam-beam footprint, in exact agreement with HL-LHC tracking,
\cite{ipactwo}. The higher-order ones possess, same as their standard 1D
counterparts recursive properties that were used in \cite{ipactwo} to
compute them fast numerically. Finally, the symmetries and recursions
obeyed by the beam-beam Hamiltonian coefficients may be of direct
interest since the combined effect of all long-range encounters around
the ring depends strongly on the left-right (anti) symmetries of the
HL-LHC insertions.
\begin{thebibliography}{99}
\bibitem{ipacone}
D.~Kaltchev, D.~Pellegrini, N.~Karastathis, Y.~Papaphilippou, E.~McIntosh,
{\it Extended-domain
tune-scans for the HL-LHC Dynamic aperture in presence of Beam-Beam effects},
Proc. of IPAC 2018, Vancouver BC, Canada
\bibitem{talklbl} D. Kaltchev,
{\it Tune-scans and working point optimization for colliding beams in HL-LHC},
Presentation at the Beam-Beam Effects Workshop, Berkeley CA, Feb 5 -- 7, 2018
\bibitem{ipactwo} D.~Kaltchev, {\it Fourier Coefficients of Long-Range Beam-Beam
Hamiltonian via Two-Dimensional Bessel functions}, Proc. of IPAC
2018, Vancouver BC, Canada
\end{thebibliography}
\end{document}