Synchrobetatron resonance of crab crossing scheme with large crossing angle and finite bunch length

Crab crossing scheme is an essential collision scheme to achieve high luminosity for the future colliders with large crossing angles. However, when bunch length of one or both colliding beams is comparable with the wavelength of the crab cavity voltage, the nonlinear dependence of the crabbing kick may present a challenge to the beam dynamics of the colliding beams and impact the beam quality and the luminosity lifetime. In this paper, the results of nonlinear dynamics in the crab crossing scheme are presented, using both analytical and numerical approaches. The result indicates that higher-order synchrobetatron resonances may be excited in the crab crossing scheme with large crossing angle, which causes the beam quality deterioration and luminosity degradation. The studies also reveal possible countermeasures to suppress the synchrobeta resonance, hence mitigate the degradation of beam quality and luminosity.

Figure 1

The horizontal tilting effect at IP for different parameters: $k_c{\sigma }_z$ and ${\theta }_{\mathrm{P}}$. $k_c$ is the wave number of the crab cavity, ${\sigma }_z$ the bunch length, and ${\theta }_{\mathrm{P}}$ is the Piwinski angle.

Figure 2

Beam-beam kick for particles with different horizontal coordinates. The blue curves are calculated by tracking while the discrete points are calculated by a truncated power series. ${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$ are beam sizes. $M$ and $N$ are truncated orders in Eq. (23). All vertical axes are normalized by the same value.

Figure 3

The luminosity evolution as function of time by SS simulation for crab-crossing and head-on cases, the luminosity data is normalized by its average value of the first 1000 turns in the simulation while the degradation rates are normalized by the head-on SS number.

Figure 4

The beam size evolution with crab crossing scheme from SS simulation.

Figure 5

The luminosity evolution as function of time by HWS simulation for crab-crossing and head-on cases, the luminosity data is normalized by its average value of the first 1000 turns in the simulation while the degradation rates are normalized by the head-on SS number. The linear fitting uses the last 40000 turns weak-strong simulation data only.

Figure 6

The beam size evolution with crab crossing scheme from HWS simulation.

Figure 7

Luminosity degradation with different beam-beam parameter of the proton beam (top figure) and of electron beam (bottom figure). The proton/electron beam-beam parameter ${\xi }_{\mathrm{p}/\mathrm{e}}$ is scaled by varying the opposite beam intensity.

Figure 8

Beam size evolution of different beam-beam parameter of the proton beam (top figure) and of electron beam (bottom figure). The proton/electron beam-beam parameter ${\xi }_{\mathrm{p}/\mathrm{e}}$ is scaled by varying the opposite beam intensity.

Figure 9

Luminosity evolution with different crab cavity frequency (top figure) and bunch length (bottom figure) of the proton beam.

Figure 10

The scaling relation of the luminosity degradation and emittance growth as functions of ${\sigma }_z^3{f}_c^2$. Here ${\sigma }_{z0}$ and $f_{c0}$ are the nominal parameters in Table 1.

Figure 11

Luminosity evolution for different longitudinal tune of proton beam.

Figure 12

The tune variation of 1000 particles in SS simulation.

Figure 13

Tune deviation from the average versus the window shift (in turns) for particles with different diffusion index: small (blue) and large (yellow). The solid lines give horizontal tune while the dashed lines show the vertical.

Figure 14

Phase space trajectories for particles with different diffusion index: small (blue) and large (yellow). Upper and lower pictures are for horizontal and vertical plane, respectively.

Figure 15

The frequency map for nominal parameters in Table 1, the top figure is for crab crossing scheme and the bottom figure is for head-on scheme. Particles with large diffusion index are labeled by red color while the small ones are in blue.

Figure 16

Particle distribution in longitudinal phase space. Upper and lower pictures are for crab crossing and head-on collision scheme.

Figure 17

Emittance evolution as function of turns, $r$ is defined by Eq. (35).

Figure 18

Frequency maps tracked by SS simulation. The upper is for a smaller beam-beam parameter while the lower is for a smaller longitudinal tune.

Figure 19

The average driving terms versus longitudinal coordinates for crab crossing and head on collision scheme. The design parameters are used during calculation.

Figure 20

The average driving terms versus longitudinal coordinates for crab crossing and head on collision scheme. The beam parameters are extracted from strong-strong simulation.

Figure 21

SS simulation results for different tune choice of the proton beam. The longitudinal tune is kept as ${\nu }_z=0.01$.

Figure 22

SS simulation results in the presence of second-order harmonic crab cavity. $\alpha$ is the relative strength of the second-order harmonic crab cavity.

Figure 23

Weak-strong simulation results for two IPs. The transverse fractional tunes of the proton ring are (0.310,0.305) in all three cases. Here the $\mathrm{\Delta }{\psi }_x$ is the horizontal betatron phase advance between the IPs.