Optimizing integrated luminosity of future hadron colliders

The integrated luminosity, a key figure of merit for any particle-physics collider, is closely linked to the peak luminosity and to the beam lifetime. The instantaneous peak luminosity of a collider is constrained by a number of boundary conditions, such as the available beam current, the maximum beam-beam tune shift with acceptable beam stability and reasonable luminosity lifetime (i.e., the empirical “beam-beam limit”), or the event pileup in the physics detectors. The beam lifetime at high-luminosity hadron colliders is largely determined by particle burn off in the collisions. In future highest-energy circular colliders synchrotron radiation provides a natural damping mechanism, which can be exploited for maximizing the integrated luminosity. In this article, we derive analytical expressions describing the optimized integrated luminosity, the corresponding optimum store length, and the time evolution of relevant beam parameters, without or with radiation damping, while respecting a fixed maximum value for the total beam-beam tune shift or for the event pileup in the detector. Our results are illustrated by examples for the proton-proton luminosity of the existing Large Hadron Collider (LHC) at its design parameters, of the High-Luminosity Large Hadron Collider (HL-LHC), and of the Future Circular Collider (FCC-hh).

Figure 1

HL-LHC luminosity evolution as a function of time, for a single fill, without (red), and with leveling at a pileup of 140 events per crossing (blue), compared with the LHC design (black). An inelastic pileup cross section of 85 mbarn is assumed for the mapping between number of pileup events and luminosity, while a total cross section of 100 mbarn was assumed for evaluating the proton burn off rate during the store.

Figure 2

HL-LHC luminosity evolution as a function of time, during several length-optimized fills over 30 h, without (dashed red), and with leveling at a pileup of 140 events per crossing (dashed blue). A turnaround time of 5 h has been assumed. The corresponding time-averaged luminosity values are indicated by the solid red and blue lines.

Figure 3

Bare (red) and integrated HL-LHC luminosity (blue) as a function of time, during two length-optimized fills over 30 hours, with leveling at a pileup of 140 events per crossing (dashed blue).

Figure 4

Instantaneous luminosity, integrated luminosity, bunch intensity, emittance, total beam-beam tune shift, and event pileup as a function of time during 24 h, for FCC-hh phases 1 and 2 from Table 2.

Figure 5

Left: Luminosity evolution for FCC-hh phase 2 from Table 2 for three different values of the transverse emittance damping time (nominal value: 1 hour). Right: Integrated luminosity per year for FCC-hh phases 1 and 2 from Table 2 as a function of turnaround time.

Figure 6

Luminosity, integrated luminosity, bunch intensity, emittance, total beam-beam tune shift, and event pileup as a function of time during 24 h, for the parameters of FCC-hh phase 2 (see Table 2) with and without a limit ${\mu }_{\max }=700$, for the maximum event pileup.