On the low-beta optics for round and flat beams in colliders

Ideally the use of round beams with equal horizontal and vertical sizes, σv = σh should gain a factor of two in the luminosity of head-on colliders. This is due to the factor 1 + σv/σh in the luminosity limit as set by the beam-beam effect. This gain can, however, be offset by other difficulties related to round beams like: (1) lower tolerable beam-beam tune shifts although this remains a controversial subject, (2) shorter luminosity lifetime, and (3) difficulties of the required low beta in both planes. To quantify point (3) above, which is the object of the present note, the author determines the concurrent maximum beta values and related to this the contribution of the insertion to the chromaticity Q' = dQ/(dp/p). Both parameters depend on the quadrupole strength and on the minimum distance available between the first lens and the interaction point. For the purpose of comparison he defines 'standard conditions.' It then emerges that the maximum beta and the chromaticity are typically 3 to 5 times larger in round than in flat beams, all other parameters equal