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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 on the 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. INTRODUCTION In synchrotrons, the beam is kept stable partially by Landau damping due to the tune spread within each bunch. The stability diagram in plane j ∈ {x, y} is calculated from [1] 1 ΔQcoh,j = − ∞ ∫ 0 dJx ∞ ∫ 0 dJy Jj dΨ(Jx,Jy) dJj Q − Qj(Jx, Jy) , (1) where ΔQcoh,j, Jj and Qj are the coherent tune shift, action and tune, respectively, in plane j, Q ∈ (−∞, ∞), and Ψ is the distribution. The stability can be changed significantly by a small change of the distribution [2–5]. In a recent experiment in the LHC, Landau damping was lost due to a noise driven diffusion [6]. Here we will introduce an analytical theory that explains how the distribution changes after an initial offset, due to the combined effects of a tune spread and a transverse feedback. The goal is to find how the distribution changes, and how the stability diagram evolves as a result. THEORY The calculation consists of 4 steps: (i) Derive an expression for the change of the action for each particle after a kick, taking into account the balance between the tune spread and the transverse feedback; (ii) Consider the change of action as a Wiener process with a drift, and derive the FokkerPlanck equation for the particle density distribution of the bunch [7]; (iii) Solve the Fokker-Planck equation to get the time evolution of the distribution; (iv) Calculate numerically the stability diagram with PySSD [8], as the distribution evolves. This approach has the advantage that it is modular, each step can be modified if necessary. Furthermore, the 4-step calculation may be applied to various sources of tune spread. Here we shall discuss the case when the tune spread is caused by Landau octupoles. ∗ sondre.vik.furuseth@cern.ch Transverse Feedback and Decoherence We apply normalized, canonical coordinates [9] x = 1 √βε0 X = √2J cos(φ) , p = − 1 √βε0 (αX + β dX ds ) = −√2J sin(φ) , (2) where X is the offset from the design orbit, s is the position in the beamline, α and β are Twiss parameters, ε0 is the initial beam emittance and φ is the canonical conjugate of J. If a bunch is kicked by Δp = k, the action changes to Jk = J0 + k√2J0 sin(φ0) + 2k 2 , (3) where J0 and φ0 are the action and phase of the particle prior to the kick. There exists an expression for the subsequent emittance growth, when there is both a transverse tune spread and a transverse feedback, and we will take a similar approach [10]. We refer to the centroid of the bunch as z = ⟨x⟩ + i⟨p⟩, where the angle brackets signify the average over the distribution. The tune of the centroid is Qc, and its transverse offset will each turn be reduced by a factor g, called the gain. Assuming a perfect, immediate feedback, the initial centroid offset z0 = ik will after n turns be zn = z0⋅ec⋅(1− g 2) n n→∞ −−−−→ z0⋅ec⋅e − 2 n , (4) with a damping time of τ = 2/g turns. It is assumed that the reduction of the centroid amplitude due to the tune spread is negligible compared to that of the transverse feedback. The position of an individual particle, with a constant tune of Qc + ΔQ, is referred to as y = x + ip. After many turns, when the centroid tends to the origin in the limit ng ≫ 1, the position will become yn = e−i2π(Qc+ΔQ)n ⎛⎜ ⎝ r0+z0 ⋅ (1− 2) (1 − e i2πΔQ) 1 − (1− 2) e i2πΔQ ⎞⎟ ⎠ , (5) where r0 = x0 + ip0 is the position prior to the kick, and y0 = r0 + z0 is the position just after the kick z0. The change of the action in the limit ΔQ ≪ 1, ng ≫ 1 is thus ΔJ =k 2 2 (1− 2) 2 4π2ΔQ2 (2) 2 + (1− 2) 4π2ΔQ2 + k√2J0 (1− g 2) × cos(φ0) ( g 2) 2πΔQ + sin(φ0) (1− g 4) 4π 2ΔQ2 (2) 2 + (1− 2) 4π2ΔQ2 =2k 2L2 + k√2J0 [M cos(φ0) + N sin(φ0)] (6) =2k 2L2 + k√2J0√M2 + N2 cos(φ0 − atan(N )) , Th is is a pr ep ri nt — th e fin al ve rs io n is pu bl ish ed w ith IO P 10th Int. Partile Accelerator Conf. IPAC2019, Melbourne, Australia JACoW Publishing ISBN: 978-3-95450-208-0 doi:10.18429/JACoW-IPAC2019-MOPGW078 MC5: Beam Dynamics and EM Fields D05 Coherent and Incoherent Instabilities Theory, Simulations, Code Developments MOPGW078 281 Co nt en tf ro m th is w or k m ay be us ed un de rt he te rm so ft he CC BY 3. 0 lic en ce (© 20 19 ). A ny di str ib ut io n of th is w or k m us tm ai nt ai n at tri bu tio n to th e au th or (s ), tit le of th e w or k, pu bl ish er ,a nd D O I

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