Coherent beam-beam effects X. Buffat Content Coherent vs. - PDF document

Intensity limitations in Particle Beams Coherent beam-beam effects X. Buffat Content  Coherent vs. incoherent  Self-consistent solutions  Coherent modes of oscillation  Decoherence  Impedance driven instabilities  Summary

Weak-strong treatment  The electromagnetic interaction felt by a particle traveling through a counter rotating beam is very non-linear → resonances, losses, emittance growth  The other beam is not perturbed by the passage of the particle → weak-strong 2 − x Δ x ' ( x )= − 2 r 0 N 1 2 ) ≈ 4 π ξ x 2 σ x ( 1 − e approximation γ r Self-consistent solutions Strong beam  Optics Beam-beam  Beam parameters forces Weak beam  Disturbed optics  Disturbed beam parameter

Self-consistent solutions Strong beam  Disturbed optics Beam-beam  Disturbed beam forces parameter Strong beam  Disturbed optics Beam-beam forces  Disturbed beam parameter Self-consistent solutions δ x = δ x' β cot ( π Q )  Weak-strong : d δ x = Δ x coh ' ( d ) β cot ( π Q ) δ x'

Coherent beam-beam force  The average force felt by the particles in the beam is called the coherent force (1) ∞ Δ x' coh ( Δ x )= ∫ Δ x' ( Δ x − X ) ρ ( X ) dX −∞ 2 −Δ x = − 2 r 0 N 2 ) ≈ 4 π ξ 1 4 σ Δ x ( 1 − e Δ x γ r 2 2 − x Δ x' ( x )= − 2 r 0 N 1 2 ) ≈ 4 π ξ x 2 σ x ( 1 − e γ r Self-consistent solutions δ x = δ x' β cot ( π Q )  Weak-strong : d δ x = Δ x coh ' ( d ) β cot ( π Q ) δ x' { δ x' δ x B 1 = Δ x coh ' ( d + δ x B 1 + δ x B 2 ) β B 1 cot ( π Q B 1 )  Strong-strong : δ x B 2 = Δ x coh ' ( d + δ x B 1 + δ x B 2 ) β B 2 cot ( π Q B 2 )  Similar treatment applies to the optical functions (e.g. dynamic β effect (2) )  These effects were already covered in T. Pieloni's lectures, but : → Simple formulas become non-linear system of equations  Iterative methods are used to evaluate these effects (3)  Prohibits several single beam measurement techniques  The solution of the non-linear equations is not always unique

Observations Orbit effect  Displacement of the luminous region  Different bunches experience different beam-beam long-range interactions → they have different orbits  Also observed in LEP with bunch trains Observations Dynamic β : Flip-flop VEPP-2000 (4) :  Low ξ : The two beams have identical transverse sizes  High ξ : Two equivalent equilibrium configurations :  Electron beam is blown up  Positron beam is blown up

Coherent modes of oscillation Rigid bunch model ( x 1 ' ) t + 1 cos ( 2 π Q ) ) ( x 1 ' ) t = ( x 1 cos ( 2 π Q ) sin ( 2 π Q ) x 1 − sin ( 2 π Q ) Coherent modes of oscillation Rigid bunch model ( x B 2 ' ) cos ( 2 π Q ) ) ( x B 2 ' ) = ( x B 1 x B 1 cos ( 2 π Q ) sin ( 2 π Q ) 0 0 x B 1 ' − sin ( 2 π Q ) cos ( 2 π Q ) 0 0 x B 1 ' x B 2 0 0 cos ( 2 π Q ) sin ( 2 π Q ) x B 2 0 0 − sin ( 2 π Q ) t + 1 t

Coherent modes of oscillation Rigid bunch model 2 −Δ x Δ x' B 1 = − 2 r 0 N 2 ) ≈ k ( x B 1 − x B 2 ) 1 (Small amplitude 4 σ Δ x ( 1 − e γ r approximation) ( x B 2 ' ) cos ( 2 π Q ) ) ( x B 2 ' ) = ( x B 1 x B 1 cos ( 2 π Q ) sin ( 2 π Q ) 0 0 x B 1 ' − sin ( 2 π Q ) cos ( 2 π Q ) x B 1 ' 0 0 0 0 cos ( 2 π Q ) sin ( 2 π Q ) x B 2 x B 2 0 0 − sin ( 2 π Q ) t + 1 t Coherent modes of oscillation Rigid bunch model 2 −Δ x Δ x' B 1 = − 2 r 0 N 2 ) ≈ k ( x B 1 − x B 2 ) 1 (Small amplitude 4 σ Δ x ( 1 − e γ r approximation) ( x B 2 ' ) ⋅ M lattice ( x B 2 ' ) = ( 1 ) x B 1 x B 1 1 0 0 0 x B 1 ' x B 1 ' + k 1 − k 0 0 0 1 0 x B 2 x B 2 − k 0 + k t + 1 t

Coherent modes of oscillation Rigid bunch model In-phase oscillations Out of phase oscillations → σ mode → π mode  x 1 = -x 2 at every  x 1 = x 2 at every interaction interaction → Q π ~ Q – ξ (*) → Q σ = Q (*) ξ << 1 and for tunes away from resonances Collective resonance Q σ = n/2 ( x i ' ) t + 1 = M lattice ⋅ M BB ( x i ' ) t Resonance x i x i conditions : Q π = n/2  The rigid dipole mode can be unstable under resonant conditions  Higher order resonances can also drive the beam-beam coherent modes unstable (2)

Coherent modes of oscillation Vlasov perturbation theory (5) Rigid bunch model : Vlasov perturbation theory : Each beam phase space distribution Each beam centroid position and momentum x 1 ,x' 1 and x 2 ,x' 2 ( 1 ) , F ( 2 ) F Liouville's thorem : Equation of motion :   { ( 1 ) ∂ F ( 1 ) , H ( F ( 2 ) )]= 0 ( x i ' ) t + 1 = M lattice ⋅ M BB ( x i ' ) t ∂ t +[ F x i x i ( 2 ) ∂ F ( 2 ) , H ( F ( 1 ) )]= 0 ∂ t +[ F Non Linear beam-beam map : Hamiltonian   (lattice + beam-beam) 2 −Δ x Δ x' coh = − 2 r 0 N 2 ) 1 4 σ Δ x ( 1 − e ( 1 ) , Ψ ( 1 ) , F ( 2 ) ,t ) H ( J γ r First order perturbation Linearized kick :   ( i ) = F 0 + F 1 ( i ) Δ x ' coh = 4 πξ F Δ x 2 Formulate the linearized system  Write one turn matrix and find  as a linear operator → find eigenvalues / eigenvectors eigenvalues / eigenfunctions Coherent mode spectrum Q σ = Q Q π = Q - ξ Rigid bunch :  The Yokoya factor Y is usually between 1.0 and 1.3 depending on the type of interaction (Flat, round, asymmetric, long-range, …) (5)

Coherent mode spectrum Self-consistent Q σ = Q Q π = Q - Y ξ Model :  The Yokoya factor Y is usually between 1.0 and 1.3 depending on the type of interaction (Flat, round, asymmetric, long-range, …) (5) Incoherent spectrum  The non-linearity of beam-beam interactions result in a strong amplitude detuning  The single particles generate a continuum of modes, the incoherent spectrum

Incoherent spectrum  The non-linearity of beam-beam interactions result in a strong amplitude detuning  The single particles generate a continuum of modes, the incoherent spectrum Incoherent spectrum  The non-linearity of beam-beam interactions result in a strong amplitude detuning  The single particles generate a continuum of modes, the incoherent spectrum  Both the σ and π mode are outside the incoherent spectrum

Incoherent spectrum  The non-linearity of beam-beam interactions result in a strong amplitude detuning  The single particles generate a continuum of modes, the incoherent spectrum  Both the σ and π mode are outside the incoherent spectrum → Absence of Landau damping ! Observations (6) LEP TRISTAN PETRA LHC Perfect agreement RHIC  with fully self- consistent models  SPPS ?  Tevatron ?

Multiparticle tracking (see K. Li's lectures)  Model the beam distribution with a discrete set of macro- particles  Track the particles, solving for each beam's fields at each interaction  Non-linear beam-beam map  Gaussian fit : soft-Gaussian approximation 2 − x i Δ x' i = − 2 r 0 N ( x i ' ) t + 1 = M lattice ⋅ M BB ( x i ' ) t 1 x i x i 2 ) 2 σ ( 1 − e γ r x i Numerical Poisson solver  Beam-beam coherent mode spectrum Self-consistent field solver Soft-Gaussian approximation  The soft-Gaussian approximation underestimate the Yokoya factor → Need to fully resolve the particles distribution

Decoherence : weak-strong  Multiparticle tracking simulation, with a single beam and a fixed beam-beam interaction → weak-strong regime :  Start the simulation with a beam offset with respect to the closed orbit and let it decohere Decoherence : weak-strong

Decoherence : weak-strong Decoherence : weak-strong

Decoherence : weak-strong Decoherence : weak-strong

Decoherence : weak-strong Decoherence : weak-strong

Decoherence : weak-strong Decoherence : weak-strong

Decoherence : weak-strong Decoherence : weak-strong The amplitude detuning due to beam-beam interaction leads to  decoherence identically to other lattice non-linearities 1 d ϵ 2 dt = Δ → ϵ 0 2 Decoherence time ~1/ ξ 

Decoherence : strong-strong  Similar setup but :  Two independent beams  Non-linear beam-beam map based on the charge distribution  Start the simulation with both beams offset in the same direction with respect to the closed orbit  Let the mode decohere ? Decoherence : σ -mode

Decoherence : σ -mode Decoherence : σ -mode

Decoherence : σ -mode Decoherence : σ -mode

Decoherence : σ -mode Decoherence : σ -mode

Decoherence : σ -mode Decoherence : σ -mode

Decoherence of the σ mode  The single particle motion is the linear composition of the centroid position and the position with respect to the centroid position → The single particle motion does not change the coherent force  The incoherent and coherent motion are decoupled → Absence of decoherence Decoherence : strong-strong  Identical setup :  Two independent beams  Non-linear beam-beam map based on the charge distribution  Start the simulation with both beams offset in opposite directions with respect to the closed orbit  Let the mode decohere ?

Decoherence of the π mode Decoherence of the π mode

Decoherence of the π mode Decoherence of the π mode

Decoherence of the π mode Decoherence of the π mode

Decoherence of the π mode Decoherence of the π mode

Decoherence of the π mode Decoherence of the π mode