BeamBeam–2013 : CERN 19.03.2013/ M.Vogt DESY-MFL : Analytic & Simulation Tools 1 ICFA Beam–Beam Workshop CERN 2013 Analytical and Numerical Tools for Beam–Beam Studies Mathias Vogt (DESY–MFL) • Intro • Weak–Strong Beam–Beam (WSBB) • A little bit on WSBB codes • Strong–Strong Beam–Beam (SSBB) • A little bit on SSBB codes . . . not necessarily in that strict order! FLASH Free−Electron Laser in Hamburg
BeamBeam–2013 : CERN 19.03.2013/ M.Vogt DESY-MFL : Analytic & Simulation Tools 2 Beam Beam Models (Basics) Immanent symmetry: “beam” ↔ “other beam” ⇒ “other beam” =: “beam ⋆ ” We don’t need the ⋆ to indicate IP–properties: “at-the-IP” is the default for beam–beam–stuff!! z ∈ R 2 n , n = 1 , 2 , 3 • Phase space: � • include long. phase space ( τ, δ ) ⇒ potential crossing angle { z i } i =1 ,..., 6 → → x, ( a := p x /p 0 ) , y, ( b := p y /p 0 ) , τ, δ . . . and more fun with beam–waists! • Indep. var. θ := 2 πs/C • Note of course : Hamiltonian ⋆ : H ⋆ = H 0 ⋆ + � N IP i =1 a 2 π⋆ ( θ − θ i ) H bb ⋆ • Hamiltonian: i H = H 0 + � N IP i =1 a 2 π ( θ − θ i ) H bb • H bb can be head–on or long–range i i • a 2 π ( θ ) = a 2 π ( θ + 2 π ) = (a.k.a. “parasitic” ) � δ 2 π ( θ ) : σ τ ≪ β x,y can be weak–strong (beam ⋆ fixed • H bb i loc. hump around 0 : otherwise from turn-to-turn) • H bb • a 2 π → δ 2 π ⇒ can be strong–strong (beam ⋆ i H bb → U bb (kick–potential) changes from turn-to-turn due to beam) i i • extended a 2 π : H bb = T free − space + U bb • Some collision schemes (RHIC, Teva- i i tron, LHC! ) need to consider more ← beam–waist → Hourglass–Effect than 1 bunch per beam!
BeamBeam–2013 : CERN 19.03.2013/ M.Vogt DESY-MFL : Analytic & Simulation Tools 3 Beam Beam Models (“Time”–Continuous) For the moment : only one short bunch per beam and head–on w/o crossing angle , only one IP. • Phase space densities : → so, why not skip the trajectories ?! z , θ ) & Ψ ⋆ ( � Ψ( � z , θ ) ∂ t Ψ = { H [Ψ ⋆ ] , Ψ } z Ψ) T J ( ∂ � z H [Ψ ⋆ ]) ≡ ( ∂ � • SSBB (the real thing!) : ∂ t Ψ ⋆ = { H [Ψ] , Ψ ⋆ } z Ψ ⋆ ) T J ( ∂ � ≡ ( ∂ � z H [Ψ]) dependence of H ( H ⋆ ) on Ψ ⋆ ( Ψ ) : → SSBB coupled Vlasov–Poisson eq’s H [Ψ ⋆ ] = H 0 + U ss [Ψ ⋆ ] → coupled system of 2 non–linear H ⋆ [Ψ] = H 0 ⋆ + U ss ⋆ [Ψ] 1-st order PIDEs � p , θ ) d n p • via ρ ( � q , θ ) := Ψ( � q , � → Can treat coherent (and incoherent) mo- & ρ ⋆ ( � � Ψ ⋆ ( � p , θ ) d n p q , θ ) := q , � tion and collective interactions � • U ss [Ψ ⋆ ]( � q ′ ) ρ ⋆ ( � q ′ ) d n q ′ , q ) ∝ G ( � q − � • WSBB : Ψ ⋆ given & fixed ∀ turns G : Green’s function → study only � z ( θ ) (and/or Ψ( � z , θ ) ) z ⋆ ( θ ) ⇒ Evolution of trajectories � z ( θ ) , � → U ws ( q ) ≡ U ss [Ψ ⋆ fixed ]( q ) needs up to date densities Ψ , Ψ ⋆ d z H ws ( � • dθ � z = J ∂ � z , θ ) ← Can. eq’s (both!) : ( J : symplectic structure) • ∂ t Ψ = { H ws , Ψ } ← Liouville eq. d z H [Ψ ⋆ ]( � dθ � z = J ∂ � z , θ ) → linear 1-st order PDE d z ⋆ = J ∂ � z H ⋆ [Ψ]( � z ⋆ , θ ) dθ � → Can NOT treat collective effects.
BeamBeam–2013 : CERN 19.03.2013/ M.Vogt DESY-MFL : Analytic & Simulation Tools 4 Beam Beam Models (“Time”–Discrete WSBB ) • WSBB : → linear(!) Perron–Frobenius Operator d • dθ � z = J ∂ � z H ( � z , θ ) M : Ψ �→ Ψ ◦ � M − 1 ← Hamiltonian Vectorfield • Discrete “time” maps : z ( θ f ) ≡ � ⇒ � z ( θ i ) �→ � M θ f ,θ i ( � z ( θ i )) restrict θ to discrete set { θ j } j =1 ,... ← Symplectic Flow � z j := � z ( θ j ) , Ψ j ( � z ) := Ψ( � z , θ j ) z 0 ) := ∂ � z 0 ∈ R 2 n M ( � M θ f ,θ i ( � z 0 ) ∈ Sp (2 n ) ∀ � � z ) := � M f,i ( � M θ f ,θ i ( � z ) M θ,θ = � � Id (identity) and forget about θ ∈ R . . . ⇒ Measure Preserving Flow : • OneTurnMap (OTM, monodromy map) µ Ψ ( A ) = µ Ψ ( � M ( A )) ∀A ∈ B 2 n � z ) := � T j ( � M θ j +2 π,θ j ( � z ) i.a.w.: Ψ = const . along trajectories • Since Sp (2 n ) is connected, all sym- ← this is why Liouville eq. holds! plectic C 1 maps are connected to � Id → Meth. o. Characteristics / P.F.–Meth. (identity) and thus can all be a flow. Ψ( � z , θ ) at point � z and “time” θ is given ⇒ extra freedom : use effective maps by Ψ( � M − 1 θ,θ 0 ( � z ) , θ 0 ) at an earlier “time” from θ i to θ f w/o caring what hap- θ 0 and the backward tracked point pens in–between! � z ) ≡ � M − 1 θ,θ 0 ( � M θ 0 ,θ ( � z )
BeamBeam–2013 : CERN 19.03.2013/ M.Vogt DESY-MFL : Analytic & Simulation Tools 5 Beam Beam Models (“Time”–Discrete SSBB) • from WSBB: Ψ f ( � z ) = ( M f,i Ψ i ) ( � z ) T [Ψ ⋆ ] − 1 = � K [Ψ ⋆ ] − 1 ◦ � ⇒ � L − 1 (inv. OTM) � � Ψ i ◦ � z ) = Ψ i ( � M − 1 = ( � M i,f ( � z )) f,i ⇒ T [Ψ ⋆ ] : Ψ �→ Ψ ◦ � T [Ψ ⋆ ] − 1 (P.F.) • SSBB : ⇒ Evolution from n -th turn to ( n +1) -st : • For every given decent ψ ( ∈ L 1 & normal- � K [Ψ ⋆n ] − 1 � �� � � L − 1 ( � Ψ n +1 ( � z ) =Ψ n z ) ized) J∂ � z H [ ψ ] is a perfectly Hamil- tonian V.F. and defines the perfectly � K [Ψ n ] − 1 � �� � � Ψ ⋆n +1 ( � z )=Ψ ⋆n L − 1 ( � z ) Symplectic Flow � M [ ψ ] • Extension to more IPs straight forward! ⇒ Thus (at least) the following model is • Example : HERA with “hadronic leptons” perfectly well defined: → needs only one bunch per beam • BB–Kick & Lattice (One IP) : 2 × 2 arcs: � L eW , � L eE , � L pW , � L pE • � T [Ψ ⋆ ] := � L ◦ � K [Ψ ⋆ ] 2 × 2 bb–kicks: � � � � q � � q � K e [Ψ p,N ] , � � K e [Ψ p,S ] , � K p [Ψ e,N ] , � K [Ψ ⋆ ] := K p [Ψ e,S ] �→ q U [ ρ ⋆ ]( � p � p − ∂ � � q ) � L represents the lattice w/o collective effects
BeamBeam–2013 : CERN 19.03.2013/ M.Vogt DESY-MFL : Analytic & Simulation Tools 6 “Time”–Discrete SSBB : HERA–Example • 2 × 2 arcs: � L eW , � L eE , � L pW , � L pE ( e ± , p ) × (West, East) • 2 × 2 bb–kicks: � K e [Ψ p,N ] , � K e [Ψ p,S ] , � K p [Ψ e,N ] , � K p [Ψ e,S ] ( e ± , p ) × (North, South) • Evolution of Ψ e and Ψ p over 2 n half turns : L eO − 1 L pW − 1 ◦ � n ] ◦ � ◦ � n ] ◦ � 1:N → S: Ψ e,S = Ψ e,N K e − 1 [Ψ p,N Ψ p,S = Ψ p,N K p − 1 [Ψ e,N n n n n L eW − 1 L pO − 1 ◦ � n ] ◦ � ◦ � n ] ◦ � 2:S → N: Ψ e,N Ψ p,N n +1 = Ψ e,S K e − 1 [Ψ p,S n +1 = Ψ p,S K p − 1 [Ψ e,S n n N ⇒ No fundamental difference between 2 IPs and 1 IP [Ψ ] N K [Ψ ] N K e p p e ⇒ Just more intricate dependence on M p E M W p the lattice parameters E W E M e M e W • There’s more complicated examples: RHIC, Tevatron, LHC!!! K [Ψ ] e [Ψ ] S S K • Also: approximate extended BB p e p waists with (kick → drift → ) k , k > 1 . S
BeamBeam–2013 : CERN 19.03.2013/ M.Vogt DESY-MFL : Analytic & Simulation Tools 7 The Rigid Bunch Model (RBM) . . . just for completeness: the Rigid Bunch Model (RBM) : • Quick and dirty: only centroid motion • However, well suited for first multi ( = N ) bunch & multi ( = M ) IP analysis : • One “macro particle” � z i per bunch i and WS–like interaction potential for crossing q i − � of i –th and j –th bunch at l -th IP U l ( � q j ) • Further simplification : linearization, no long. & uncoupled, kick → study ( x, a ) and ( y, b ) plane separately � � � � 1 0 0 ⇒ e.g. � z ⋆ ]( � z ⋆ ) K l [ � z ) = � z + and vice versa ( � z ↔ � + κ l q ⋆ − κ l 1 • Now glue together: bunches � Z := � z 1 ⊕ � z 2 ⊕ . . . ⊕ � z N , sections of lattice N and join with IPs K l (bunch-to-bunch coupling) M l := L 1 l ⊕ L 2 l ⊕ . . . ⊕ L l → linear stability analysis of 2 N × 2 N OTM T := K 1 M 1 . . . K M M M
BeamBeam–2013 : CERN 19.03.2013/ M.Vogt DESY-MFL : Analytic & Simulation Tools 8 The Absolutely Most Famous Results from Linear WSBB :-) • unperturbed linear OTM seen from IP ( α = 0 ): � � cos(2 πQ 0 ) β 0 sin(2 πQ 0 ) • T 0 := − sin(2 πQ 0 ) /β 0 cos(2 πQ 0 ) � � 1 0 • insert linear (focusing) WSBB kick K := before IP − κ 1 κ x,y = 2 N ⋆ r p ( σ ⋆x,y ( σ ⋆x + σ ⋆y )) − 1 • with κ from γ � � cos(2 πQ 0 ) − β 0 sin(2 πQ 0 ) κ β 0 sin(2 πQ 0 ) ⇒ T := T 0 K = − sin(2 πQ 0 ) /β 0 − cos(2 πQ 0 ) κ cos(2 πQ 0 ) 2 trace T = cos(2 πQ 0 ) − β 0 κ ⇒ cos(2 πQ ) = 1 2 cos(2 πQ 0 ) ⇒ Perturbed tune Q = Q 0 + β 0 κ 4 π + O ( κ 2 ) ξ := β 0 κ • Linear Beam–Beam Tuneshift Parameter 4 π
Recommend
More recommend
