Numerical Integration of the Vlasov Equation of Two Colliding Beams

In a circular collider the motion of particles of one beam is strongly perturbed at the interaction points by the electro-magnetic field associated with the counter-rotating beam. For any two arbitrary initial particle distributions the time evolution of the two beams can be known by solving the coupled system of two Vlasov equations. This collective description is mandatory when the two beams have similar strengths, as in the case of LEP or LHC. The coherent modes excited by this beam-beam interaction can be a strong limitation for the operation of LHC. In this work, the coupled Vlasov equations of two colliding flat beams are solved numerically using a finite difference scheme. The results suggest that, for the collision of beams with equal tunes, the tune shift between the $\sigma$- and $\pi$- coherent dipole mode depends on the unperturbed tune $q$ because of the deformation that the so-called dynamic beta effect induces on the beam distribution. Only when the unperturbed tune $q\rightarrow 0.25$ this tune shift is equal to $Y\times \xi$, with $Y$ the Yokoya factor as predicted from the linearized Vlasov theory. Colliding beams with unequal tunes brings the tunes of the dipole modes back into the continuum, but it also generates a flip-flop asymmetry in the transverse beam size. It will be shown how coherent resonances can excite the amplitude of the coherent modes and induce variations in the beam transverse size (size growth, period-n oscillations) as well as significant deformations of the beam shape

Measurements of Coherent Tune Shifts and Head-tail Growth Rates

Measurements of the coherent tune shifts with intensity and of head-tail growth rates have been performed with single proton bunches in the SPS at 26 GeV. From these measurements, the real and imaginary part of the transverse impedance can be estimated. A reproducibility at the 20% level was achieved for the value of the effective vertical impedance inferred from the coherent tune-shift measurements

Spatio-temporal patterns driven by autocatalytic internal reaction noise

The influence that intrinsic local density fluctuations can have on solutions of mean-field reaction-diffusion models is investigated numerically by means of the spatial patterns arising from two species that react and diffuse in the presence of strong internal reaction noise. The dynamics of the Gray-Scott (GS) model with constant external source is first cast in terms of a continuum field theory representing the corresponding master equation. We then derive a Langevin description of the field theory and use these stochastic differential equations in our simulations. The nature of the multiplicative noise is specified exactly without recourse to assumptions and turns out to be of the same order as the reaction itself, and thus cannot be treated as a small perturbation. Many of the complex patterns obtained in the absence of noise for the GS model are completely obliterated by these strong internal fluctuations, but we find novel spatial patterns induced by this reaction noise in regions of parameter space that otherwise correspond to homogeneous solutions when fluctuations are not included.Comment: 12 pages, 18 figure

Intrabeam scattering and the coasting beam in the HERA proton ring

Undesired coasting beam of protons has been detected in the HERA proton ring in high energy storage operation. This mainly disturbs the operation of the HERA-B experiment, and can have some impact on H1 and ZEUS where it generates background spikes (depending also on the collimator settings). In this work we present a collection of data and facts, to be taken as starting point for further theoretical and experimental studies. We propose Intra Beam Scattering as a possible physical mechanism for creating the coasting beam and discuss the implications of this longitudinal dynamics model on observables such as the bunch length, energy spread, dc current and reaction rate at the HERA-B wires. The results seem to be in qualitative agreement with the measurements

Simulations of Coherent Beam-Beam Modes at the LHC

The transverse coherent oscillations of the two colliding LHC proton beams are studied via multi-particle tracking, using the beam-beam force of a Gaussian distribution with variable barycenters and rms sizes. In addition to head-on collisions, our simulation optionally includes the effect of long-range collisions and an external impedance. Simulation results are the coherent and incoherent oscillation frequencies, the emittance growth of either beam, and evidence for the existence or absence of Landau damping. We find that with equal beam sizes and equal tunes Landau damping acts on the coherent modes for current ratios smaller than 60%. For equal current ratio, Landau damping of the coherent dipole oscillations is lost. However it can be restored by separating the tunes, provided the external impedance is sufficiently small

Coherent Beam-Beam Oscillations at the LHC

The transverse coherent motion of the two colliding LHC beams is studied by multi-particle tracking, where the beam-beam force is calculated assuming a Gaussian beam distribution with variable barycentres and rms beam sizes. The simulation yields the coherent and incoherent oscillation frequencies, the emittance growth of either beam, and evidence for the existence or lack of Landau damping. The transverse beam sizes change with the fractional part of the tune as expected from the dynamic beta effect. For head-on collisions, we find that the pi-mode frequency lies outside of the continuum frequency spread if the ratio of the beam-beam parameters exceeds 0.6, in accordance with predictions [1]. For smaller ratios, the pi-mode is Landau damped. When long range interactions are also included, undamped coherent modes do still exist outside the continuum, both with and without alternating crossing planes at two interaction points. However, the simulation shows that separating the tunes of the two beams can restore the Landau damping

Coherent Dipole Modes for Multiple Interaction Regions

In the Large Hadron Collider (LHC) two proton beams of similar strength will collide at several interaction points. For a single interaction point it is known that the head-on collision of two equally strong beams with the same betatron tune, excites two coherent dipole modes whose frequencies are different from the frequencies of oscillation of individual particles in the beam. Because of this frequency difference Landau damping does not act on the dipole modes and the beams can be unstable. In this paper we extend these studies to several interaction points and explore the possibility of cancellation of the dipole coherent modes by carefully adjusting the phase difference between the beams from one collision to the next. We also study the collision of the two beams with LHC optics V 6.1. Special attention should be paid to coherent resonances that are excited due to local phase advance correlations. It will be shown also that a tune split of 0.03 between the two beams suppresses these coherent dipole modes

Emittance Growth of the LHC Beam due to the Effect of Head-on Beam-Beam Interaction and Ground Motion

The influence of ground motion on the LHC beam is estimated by applying the existing theories of particle diffusion due to a weak-strong beam-beam collision with random offset at the interaction point. Noise at odd harmonics of the betatron frequency contributes significantly to particle diffusion. Extrapolating the characteristics of the random offset from the ground motion spectrum at the LHC site shows a fast fall-off with frequency and the amplitude is very small even at the first harmonic. We find that the head-on beam-beam force in the weak-strong approximation and ground motion by themselves do not induce significant diffusion over the lifetime of the beam

Simulations of coherent beam-beam modes at the Large Hadron Collider

The transverse coherent motion of two colliding proton beams at the Large Hadron Collider is studied by multiparticle tracking. We use the beam-beam force for a Gaussian beam distribution with variable barycenters and rms beam sizes, and optionally include the effect of long-range collisions and external impedance. The simulation yields the coherent and incoherent oscillation frequencies, the emittance growth of either beam, and evidence for the existence or lack of Landau damping. For head-on collisions of beams with equal sizes, we find that the pi -mode frequency lies outside of the continuum frequency spread, if the ratio of the beam-beam parameters exceeds 0.6, in accordance with predictions. For smaller ratios of the beam- beam parameters, or if, for equal beam-beam parameters, the beam sizes are widely different, the pi mode is Landau damped. When long- range collisions are also included, undamped coherent modes do still exist outside the continuum, both with and without alternating crossing planes at two interaction points. However, separating the tunes of the two beams restores the Landau damping, provided the external impedance is sufficiently small. (19 refs)

Complex noise in diffusion-limited reactions of replicating and competing species

We derive exact Langevin-type equations governing quasispecies dynamics. The inherent multiplicative noise has both real and imaginary parts. The numerical simulation of the underlying complex stochastic partial differential equations is carried out employing the Cholesky decomposition for the noise covariance matrix. This noise produces unavoidable spatio-temporal density fluctuations about the mean field value. In two dimensions, the fluctuations are suppressed only when the diffusion time scale is much smaller than the amplification time scale for the master species.Comment: 10 pages, 2 composite figure