Modeling of nonlinear effects due to head-on beam-beam interactions

The beam-beam interaction is one of the most severe limitations on the performance of circular colliders, as it is an unavoidable strong nonlinear effect. As one aspires for greater luminosity in future colliders, one will simultaneously achieve stronger beam-beam interactions. We study the limitations caused by strong incoherent head-on beam-beam interactions, using a new code (cabin) that calculates on a graphics processing unit (GPU), allowing for a detailed description of the long-term particle trajectories in 6D phase space. The evolution of the beam emittance and beam intensity has been monitored to study the impact quantitatively, while frequency map analysis has been performed to understand the impact qualitatively. Results from cabin have shown good quantitative agreement with dedicated experiments in the Large Hadron Collider (LHC). For large beam-beam tune shifts, alternatives to the LHC tunes have been found to improve the beam quality. Schemes devised to cancel beam-beam driven resonances, by use of specific intermediate phase advances between the interaction points, work very well with zero crossing angle, and the accuracy required is achievable. Due to lack of symmetry, these schemes have an almost negligible impact with a significant crossing angle. The hourglass effect has been found to reduce the detrimental effects caused by the chromaticity and vice versa. The optimal level of the hourglass effect has been achieved when ${\beta }^{*}=1.5{\sigma }_s$. The ultimate parameters of the Future Circular Hadron Collider (FCC-hh) seem within reach, in absence of residual odd resonances.

Figure 1

Illustration of the crossing of beam 1 and beam 2 at the IPs in circular colliders (not to scale).

Figure 2

Tune footprint caused by the beam-beam tune shift in both transverse planes simultaneously. The lines correspond to $x/{\sigma }_x\in \{0,1,2,3,4,5,6\}$ and $y/{\sigma }_y\in [0,6]$, or vice versa. The markers on the corners of the footprint correspond to the normalized amplitude in horizontal and vertical phase space. The red cross marks the working point.

Figure 3

Comparison of initial distributions of $N_{\mathrm{mp}}=1\times {10}^5$ macroparticles, with the actual ${\chi }_6^2$ distribution. The sum of the weight of all macroparticles is $N=1\times {10}^{11}$. The bottom right plot displays the same distribution as the plot above, but with a linear scale.

Figure 4

Example of evolution of beam emittance in both transverse planes, ${\epsilon }_x$ and ${\epsilon }_y$. The beam emittances are given based on normalized coordinates, being $\epsilon =2$ for a perfect initial Gaussian distribution. The straight lines represent the linear regressions giving the emittance growth rates.

Figure 5

Example of evolution of bunch intensity, relative to the initial intensity, $N_0$. The straight lines represent the linear regressions giving the loss rates.

Figure 6

Scan in $(Q_x,Q_y)$ space, for $\mathrm{\Delta }{Q}_{\mathrm{Tot}}({\varphi }_{\mathrm{PIW}})=0.03$, ${\beta }_q^{*}=40\mathrm{cm}$, ${\varphi }_{\mathrm{PIW}}=0.9$ and $Q^{\prime }=15$. The black cross ($\times$) marks the LHC working point, $(Q_x,Q_y)=(0.31,0.32)$. The green diamond () marks the suggested working point for this configuration, $(Q_x,Q_y)=(0.315,0.325)$.

Figure 7

Scan in $(Q_x,Q_y)$ space, for $\mathrm{\Delta }{Q}_{\mathrm{Tot}}({\varphi }_{\mathrm{PIW}})=0.03$, ${\beta }_q^{*}=40\mathrm{cm}$, ${\varphi }_{\mathrm{PIW}}=0.9$ and $Q^{\prime }=15$. At the working points marked by blue squares with white crosses, the entire beam was lost before the long-term loss rate could be calculated. The green diamond () marks a suggested working point in this tune area, $(Q_x,Q_y)=(0.475,0.485)$.

Figure 8

Beam quality reduction for different combinations of ${\varphi }_{\mathrm{PIW}}$ and ${\xi }_{\mathrm{Tot}}$ with different chromaticities, when $Q_x=0.31$, $Q_y=0.32$, and ${\beta }_q^{*}=0.4\mathrm{m}$.

Figure 9

FMAs showing effects of nonzero Piwinski angle and chromaticity for a working point of $(Q_x,Q_y)=(0.31,0.32)$, related to the beam quality study in Fig. 8. Possible resonance lines up to the 16th order have been plotted on top of the tune footprint in each figure. A few labels were added to guide the reader.

Figure 10

Beam quality reduction for different combinations of ${\varphi }_{\mathrm{PIW}}$ and ${\xi }_{\mathrm{Tot}}$ at different working points, when $Q^{\prime }=15$ and ${\beta }_q^{*}=0.4\mathrm{m}$.

Figure 11

FMAs illustrating activation of synchrobetatron resonances at the alternative working points, related to the beam quality study in Fig. 10. Possible resonance lines up to the 16th order have been plotted on top of the tune footprint. A few labels were added to guide the reader.

Figure 12

Beam quality reduction for multiple combinations of a transverse separation at the IPs, $\mathrm{\Delta }{x}_{\perp }$, and ${\xi }_{\mathrm{Tot}}$, when $Q_x=0.31$, $Q_y=0.32$, ${\beta }_q^{*}=40\mathrm{cm}$ and $Q^{\prime }=15$.

Figure 13

FMA illustrating the effect of a small transverse separation, $\mathrm{\Delta }{x}_{\perp }=0.25{\sigma }_{\mathrm{\Sigma }}$, for a working point of $(Q_x,Q_y)=(0.31,0.32)$, $Q^{\prime }=0$, ${\varphi }_{\mathrm{PIW}}=0$ and ${\xi }_{\mathrm{Tot}}=0.04$. Possible resonance lines up to the 16th order have been plotted on top of the tune footprints. A few resonance labels were added to guide the reader.

Figure 14

Beam quality improvement for intermediate phase advance close to $\mu /2$, for different values of ${\xi }_{\mathrm{Tot}}$, when $Q_x=0.31$, $Q_y=0.32$, and $Q^{\prime }=15$.

Figure 15

FMA illustrating canceling of resonances using an intermediate phase advance ${\mu }_1=\mu /2$, for a working point of $(Q_x,Q_y)=(0.31,0.32)$, $Q^{\prime }=0$ and ${\xi }_{\mathrm{Tot}}=0.03$. Possible resonance lines up to the 16th order have been added, in addition to a few labels meant to guide the reader.

Figure 16

Beam quality improvement for intermediate phase advance close to $\pi /2$, for different values of ${\xi }_{\mathrm{Tot}}$, when $Q_x=0.31$, $Q_y=0.32$, and $Q^{\prime }=15$.

Figure 17

FMAs illustrating canceling of resonances using an intermediate phase advance ${\mu }_1=0.512·\pi$, for a working point of $(Q_x,Q_y)=(0.31,0.32)$, $Q^{\prime }=0$ and ${\xi }_{\mathrm{Tot}}=0.03$. Possible resonance lines up to the 16th order have been added, in addition to a few labels meant to guide the reader.

Figure 18

Beam quality reduction for different combinations of ${\beta }_q^{*}$ and ${\xi }_{\mathrm{Tot}}$, when $Q_x=0.31$, $Q_y=0.32$, and ${\varphi }_{\mathrm{PIW}}=0$.

Figure 19

FMAs illustrating the effect of synchrobetatron resonances due to the hourglass effect, for a working point of $(Q_x,Q_y)=(0.31,0.32)$ and ${\xi }_{\mathrm{Tot}}=0.05$, related to the beam quality study in Fig. 18. Possible resonance lines up to the 16th order have been added, including a few labels meant to guide the reader.

Figure 20

Beam quality reduction for multiple combinations of the chromaticity, $Q^{\prime }$, and the beam-beam parameter, ${\xi }_{\mathrm{Tot}}$, when $Q_x=0.31$, $Q_y=0.32$. (a) ${\beta }_q^{*}=4\mathrm{m}$, ${\varphi }_{\mathrm{PIW}}=0$. (b) ${\beta }_q^{*}=\text{\hspace{0.17em}12\hspace{0.17em}}\mathrm{c}\mathrm{m}$, ${\varphi }_{\mathrm{PIW}}=0$.

Figure 21

Beam quality reduction for multiple combinations of the chromaticity, $Q^{\prime }$, and the beam-beam parameter, ${\xi }_{\mathrm{Tot}}$, when $Q_x=0.31$, $Q_y=0.32$. (a) ${\beta }_q^{*}=4\mathrm{m}$, ${\varphi }_{\mathrm{PIW}}=1$. (b) ${\beta }_q^{*}=\text{\hspace{0.17em}12\hspace{0.17em}}\mathrm{c}\mathrm{m}$, ${\varphi }_{\mathrm{PIW}}=1$.

Figure 22

Beam quality reduction for increasing beam-beam parameter, ${\xi }_{\mathrm{Tot}}$, until a threshold is found. Simulations are run for a large chromaticity, $Q^{\prime }=15$, and a significant hourglass effect, ${\beta }_q^{*}=12\mathrm{cm}$, changing the crossing angle, intermediate phase advance and working point. The outputs are presented as functions of $\mathrm{\Delta }{Q}_{\mathrm{Tot}}({\varphi }_{\mathrm{PIW}})$ calculated by Eq. (17).

Figure 23

Loss rates measured in the LHC, subtracted the estimated luminosity burn-off of $5\%{\mathrm{h}}^{-1}$, and loss rates ($L{R}_{\text{Beam}}$) calculated by cabin for the same configurations using the different implementations of the beam-beam interaction. The circles and stars correspond to the first and second fill respectively. Even resonance lines up to 14th order are plotted on top of the LHC measurements, with width proportional to the strength of the resonance coefficient [40].