Chromaticity compensation of a ghost collider
The GHOST collider final focus system is a pure quadrupole-drift beamline targeting ¿* = 2 mm at four serial interaction points, with peak ¿-functions reaching ~65 km in the final triplet. Beam tracking simulations reveal that a nominal bunch develops a pronounced C-shape in longitu- dinal phase space at IP1, with ¿¿ growing from 0.15 mm to ~2.3 mm — a factor of ~15× increase that directly re- duces luminosity. The mechanism is identified as chromat- ically amplified betatron path length: off-momentum par- ticles acquire enlarged betatron amplitudes in the high-¿ final triplet, generating excess path length via the geometric ¿¿ = - 1/2 ¿ (¿'2 + ¿'2) ¿¿ integral. Within the monoenergetic Balandin framework, ¿2¿2 with ¿ = - 1/2 ¿ ¿ ¿¿ dominates ¿2¿2 by over four million times; the full beam ¿¿ is a fur- ther factor of ~27 larger, driven by ¿¿ -induced chromatic amplitude growth. Phase-advance scans confirm that ¿¿ is insensitive to the apochromatic (¿ ˜ 0) condition. When combined with the geometric hourglass effect, this distor- tion poses a challenge to maximizing luminosity within the current lattice design.