GPU-Powered Modelling of Nonlinear Effects due to Head-On Beam-Beam Interactions in High-Energy Hadron Colliders.

The performance of high-energy circular hadron colliders, as the Large Hadron Collider, is limited by beam-beam interactions. The strongly nonlinear force between the two opposing beams causes diverging Hamiltonians and resonances, which can lead to a reduction of the lifetime of the beams. The nonlinearity makes the effect of the force difficult to study analytically, even at first order. Numerical models are therefore needed to evaluate the overall effect of different configurations of the machines. This report discusses results from an implementation of the weak-strong model, studying the effects of head-on beam-beam interactions. The assumptions has been shown to be valid for configurations where the growth and losses of the beam are small. The tracking has been done using an original code which applies graphic cards to reduce the computation time. The bunches in the beams have been modelled cylindrically symmetrical, based on a Gaussian distribution in three dimensions. This choice fits well with bunches in the Large Hadron Collider, and allows for certain consequences to be derived analytically. There has been good agreement between analytically expected effects and results from simulations. The effect that resonances have on the beam quality has been studied carefully. Resonances have been shown both analytically and numerically to be stronger at lower order and further from the design orbit of the beam. The strength of the beam-beam interaction will be higher after the upgrade from the Large Hadron Collider to the High-Luminosity Large Hadron Collider, and also in the next generation of machines, as the ultimate Future Circular Collider for hadron collisions. An equivalent increase has been shown to greatly limit the possible favourable working points. A scheme to try to cancel the resonances has been applied with noticeable effect

Head-On Beam-Beam Interactions in High-Energy Hadron Colliders - GPU-Powered Modelling of Nonlinear Effects

The performance of high-energy circular hadron colliders, as the Large Hadron Collider, is limited by beam-beam interactions. The strength of the beam-beam interactions will be higher after the upgrade to the High-Luminosity Large Hadron Collider, and also in the next generation of machines, as the Future Circular Hadron Collider. The strongly nonlinear force between the two opposing beams causes diverging Hamiltonians and drives resonances, which can lead to a reduction of the lifetime of the beams. The nonlinearity makes the effect of the force difficult to study analytically, even at first order. Numerical models are therefore needed to evaluate the overall effect of different configurations of the machines. For this thesis, a new code named CABIN (Cuda-Accelerated Beam-beam Interaction) has been developed to study the limitations caused by the impact of strong beam-beam interactions. In particular, 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. The bunches in the beams have been modelled based on a three-dimensional Gaussian distribution. The bunches in the Large Hadron Collider are well approximated by cylindrically symmetric Gaussian bunches, which allows for certain consequences to be derived analytically. The mapping of both round and flat beams have been implemented with the weak-strong model, considering one beam to stay fixed throughout the simulation, while the other beam is changing. The simulations have been run on graphic cards, well adapted for studying this highly parallelisable problem, to reduce the computation time. The beam-beam driven resonances have been shown both analytically and numerically to be stronger at lower order and further from the design orbit of the beam. Stronger beam-beam interactions cause a wider spread of the betatron frequencies/tunes within a single bunch, making it increasingly difficult to avoid resonances that cause detrimental effects on the beam quality. This has been seen in both simulations and experiments. In such scenarios, the common working point in the Large Hadron Collider, (Q_x , Q_y ) = (0.31, 0.32), is found to be suboptimal. Two alternative working points, (0.315, 0.325) and (0.475, 0.485), have been found to give better performance. Without long-range interactions, the beam quality is best preserved for zero crossing angle. Increasing the crossing angle activates odd resonances that can reduce the performance further, but it also reduces the tune spread within the bunch, making the bunch exposed to fewer strong resonances. Mixing between the longitudinal and transverse planes, caused by either a crossing angle, the hourglass effect or chromaticity, drives synchro-betatron resonances that also reduce the performance. However, a nonzero chromaticity is usually necessary to avoid coherent instabilities. A significant hourglass effect, σ_s /β_q^∗ = 2/3, has been found to reduce the detrimental effects caused by the chromaticity, and vice versa. A scheme designed to cancel beam-beam driven resonances, by applying a specific intermediate phase advance, has been found to have an extremely positive impact on the beam quality for zero crossing angle, but only a marginal impact for a nonzero crossing angle. A dedicated experiment with strong head-on beam-beam interactions has been performed in the Large Hadron Collider. Simulations run in CABIN for the same configurations show good quantitative agreement. A realistic maximum beam-beam tune shift from the LHC working point has been found to be ∆Q_Tot = 0.043 with zero crossing angle. With a Piwinski angle of φ_PIW = 1, this limit is reduced to ∆Q_Tot = 0.028, smaller than the largest beam-beam tune shift expected in the Future Circular Hadron Collider. These limits are slightly larger for the alternative working point, (0.315, 0.325), raised to ∆Q_Tot = 0.067 and 0.036 for zero and nonzero crossing angle respectively

Transverse Noise, Decoherence, and Landau Damping in High-Energy Hadron Colliders

High-energy hadron colliders are designed to generate particle collisions within specialized detectors. A higher number of collisions is achieved with high-quality beams of low transverse emittances, meaning a small transverse cross-section, and high intensity, meaning many particles per bunch. This thesis studies how noise negatively impacts the beam quality in high-energy hadron colliders, both in terms of beam instabilities and emittance growth. The impact is analyzed through the derivation of new theories, multi-particle tracking simulations, the numerical solving of partial differential equations, and dedicated experiments in CERN's Large Hadron Collider (LHC). The impact of noise on beam stability cannot be treated with the first-order, linear Vlasov equation, which is commonly used to study the thresholds of collective instabilities. Therefore, the Vlasov equation has in this thesis been expanded to second order in the perturbation of the beam distribution, finding a diffusion mechanism driven by the interplay between noise, decoherence, and wakefields. The diffusion leads to a local flattening of the distribution, which can cause a loss of Landau damping after a time delay referred to as the latency. An analytical formula for the latency and a specialized numerical diffusion solver were successfully benchmarked against the latency measurements in a dedicated experiment conducted in the LHC. Precaution in the machine operation has to be taken to account for this new mechanism. In particular, it is found that the machine must be operated with a margin to the linear stability threshold. For the case of the LHC, it has previously been found empirically that the octupole current during operation must be increased by about a factor 2, and this thesis provides the explanation as to why that is. Alternative operational settings are suggested to reduce the required octupole current in the LHC. In addition, the new theory allows for extrapolations to future machines, such as the High-Luminosity LHC, as well as the estimation of the impact of new devices, such as crab cavities. External noise and noise from the transverse beam feedback system cause an emittance growth rate due to decoherence of the noise kicks. Analytical theories for the suppression of the emittance growth rate with a bunch-by-bunch feedback have here been extended to a multi-bunch feedback. The numerical study of suppression during collision was conducted by means of a newly developed parallel multi-beam multi-bunch algorithm. For the typical case of low-frequency external noise and non-negligible feedback noise, a multi-bunch feedback has both analytically and numerically been found superior to a bunch-by-bunch feedback, as it can suppress the impact of the external noise equally well, while simultaneously reducing the noise generated by the feedback itself. Suggestions for a more optimal operation of the LHC are discussed, including a reduction of the upper cutoff frequency of the feedback system

Change of beam distribution due to decoherence in the presence of transverse feedback

The effect of Landau damping is often calculated based on a Gaussian beam distribution in all degrees of freedom. The stability of the beam is however strongly dependent on the details of the distribution. The present study focuses onthe change of bunch distributions caused by the decoherence of the excitation driven by an external source of noise, in the presence of both amplitude detuning and a transverse feedback. Both multiparticle tracking simulations and theoretical models show a similar change of the distribution. The possible loss of Landau damping driven by this change is discussed

High-performance scheduling of multi-beam multi-bunch simulations

Coherent multi-bunch interactions through beam-beam forces or wakefields can cause severe impacts on the beams in circular colliders, if not well understood and countered. COMBI is a parallel multiparticle tracking code developed to study such interactions. Its implementation greatly limits its efficiency when considering realistic configurations featuring effects with different computational requirements, such as the multi-bunch interaction through wakefields, beam-beam interactions, transverse feedback and lattice non- linearities. A new parallel scheduling method, pipelining the effects for each bunch, has greatly sped up the code. The new version of the code, ${\texttt{COMBIp}}$, is presented here

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 β^{*}=1.5σ_{s}. The ultimate parameters of the Future Circular Hadron Collider (FCC-hh) seem within reach, in absence of residual odd resonances

Loss of transverse Landau damping by noise and wakefield driven diffusion

Landau damping of coherent modes is strongly dependent on the exact shape of the particle bunches. One often assumes that the transverse distributions in high-energy hadron colliders can be approximated by Gaussian distributions, in acceptable agreement with measurements, but known to be only a first approximation. In this paper, it is investigated how a specific change of the transverse distributions can cause a loss of Landau damping. A mechanism is introduced where the coherent modes, which are excited by noise in the machine, act back on the individual particles through wakefields. The impact is modeled as a narrow diffusion in frequency space, and therefore also in action space due to amplitude dependent detuning, which leads to a local flattening of the distribution. This distribution evolution corresponds to the drilling of a borehole in the stability diagram, i.e. a local reduction of the imaginary part of the curve. Hence, initially stable regions are changed into unstable ones at the real frequencies of the coherent modes. To mitigate this instability mechanism, one must operate the machine with a stability margin of magnitude that depends on the noise amplitude and the coherent modes. In this model, the latency is defined as the time from the start of the noise excitation, on an initially Gaussian distributed bunch, to the bunch instability. The proposed model is found to agree with results in dedicated latency experiments performed in the LHC, where bunches eventually went unstable with more than twice the detuning strength required for the stabilization of a Gaussian distribution

Vlasov description of the beam response to noise in the presence of wakefields in high-energy synchrotrons: beam transfer function, diffusion, and loss of Landau damping

Noise can have severe impacts on particle beams in high-energy synchrotrons. In particular, it has recently been discovered that noise combined with wakefields can cause a diffusion that leads to a loss of Landau damping after a latency. Such instabilities have been observed in the Large Hadron Collider. This paper, therefore, studies the beam response to noise in the presence of wakefields, within the framework of the Vlasov equation. First, a wakefield beam eigenmode transfer function (MTF) is derived, quantifying the amplitude of a wakefield eigenmode when excited by noise. Then, the MTFs of all the wakefield eigenmodes are combined to derive the beam transfer function (BTF) including the impact of wakefields. It is found to agree excellently with multi-particle tracking simulations. Finally, the MTFs are also used to derive the single-particle diffusion driven by the wakefield eigenmodes. This new Vlasov-based theory for the diffusion driven by noise-excited wakefields is found to be superior to an existing theory by comparing to multi-particle tracking simulations. Through sophisticated simulations that self-consistently model the evolution of the distribution and the stability diagram, the diffusion is found to lead to a loss of Landau damping after a latency. The most important technique to extend the latency and thereby mitigate these instabilities is to operate the synchrotron with a stability margin in detuning strength relative to the amount of detuning required to barely stabilize the beam with its initial distribution

Long-term evolution of Landau damping in the presence of transverse noise, feedback, and detuning

The effect of Landau damping is often calculated assuming a Gaussian beam distribution in all transverse degrees of freedom, which agrees reasonably well with beam measurements. However, the stability of the beam is strongly dependent on the details of the distribution. The present study focuses on the slow evolution of the transverse bunch distribution, for a bunch excited by a coherent white noise source, damped by a transverse feedback system, and with detuning dependent on the transverse actions. The mechanism is modeled by the Fokker-Planck equation. It corresponds to a diffusion that is zero for particles with tune equal to the average tune of the bunch, and which is growing quadratically with the tune in the vicinity. The evolving distributions are then used to calculate the evolving stability diagrams, and thus the long-term evolution of the Landau damping. The relative effective octupole current is reduced faster for stronger noise, stronger linear detuning coefficients and weaker damper gain. The relevant parameters for this mechanism are scanned. With relevant parameter values for the LHC, this mechanism can cause a reduction of the effective octupole current by at most 10%/h

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 β∗=1.5σs. The ultimate parameters of the Future Circular Hadron Collider (FCC-hh) seem within reach, in absence of residual odd resonances