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Macroscopic Fluid Approach to the Coherent Beam-Beam Interaction

Building on the Radon transform of the Vlasov-Poisson equations, a macroscopic fluid model for the coherent beam-beam interaction has been developed. It is shown that the Vlasov equation, expressed in action-angle variables, can be reduced to a closed set of hydrodynamic (fluid) equations for the beam density and current velocity. The linearized one dimensional equations have been analysed, and explicit expressions for the coherent beam-beam tuneshifts are presented.


INTRODUCTION
In a colliding-beam storage ring device, the evolution of each beam is strongly affected by the electromagnetic force produced by the counter-propagating beam. A basic feature of this coherent interaction is linear mode coupling, also known as the coherent beam-beam resonance.
The problem of coherent beam-beam resonances in one dimension (the vertical direction) was first studied by Chao and Ruth [1] by solving the linearized Vlasov-Poisson equations. They considered the simplest case of a symmetric collider and obtained explicit expressions for the resonance stopbands. The purpose of the present paper is to extend their results to the case of an asymmetric circular collider.
Based on the Radon transform [2,3], a macroscopic fluid model of the coherent beam-beam interaction is developed. The linearized macroscopic fluid equations are then solved, and a generalized stability criterion for a coherent beambeam resonance of arbitrary order is derived.

THE RADON TRANSFORM
We begin with the one-dimensional Vlasov-Poisson equations describing the nonlinear evolution of the beams in the vertical (y) direction 3) Here, (k = 1, 2) labels the beam, f k (y, p; θ) is the distribution function, θ is the azimuthal angle, ν k is the betatron tune in vertical direction, R is the mean machine radius, r e is the classical electron radius, N 1,2 is the total number of particles in either beam, V k (y; θ) is the normalized beambeam potential, β * ky is the vertical beta-function at the interaction point, and L kx is the horizontal dimension of the beam ribbon [1]. The one-dimensional Poisson equation (2.3) can be readily solved to give Transforming to action-angle variables (J, ϕ), we rewrite Eqs. (2.1) and (2.2) in the form Next we perform the Radon transform defined as [2,3] and obtain the hydrodynamic equations where ̺ k is the Radon image of the distribution function f k . The integration variable ξ is regarded as a Lagrange variable, that keeps track of the detailed information about the action J. It us usually determined by the condition that the distribution function f k be equal to a specified distribution [3], from which J = v k (ϕ, ξ; θ). Taking into account Eq. (2.10), the beam density can be further eliminated from Eq. (2.11), which yields the result It is important to note that Eqs. (2.10) and (2.12) comprise a closed set, that is (as can be easily checked) equations for higher moments can be reduced to these two equations.
At this point we make the important conjecture that Eqs. (2.10) and (2.12) possess a stationary solution that is independent of the angle variable ϕ. Without loss of generality we choose v (0) (2.14)

SOLUTION OF THE LINEARIZED EQUATIONS
k , the linearized hydrodynamic equations can be written as Here ν k is the incoherently perturbed betatron tune, defined by where the angular bracket implies an average over the angle variable. Next we determine the derivatives of the firstorder beam-beam potential V (1) k entering the linearized hydrodynamic equations corresponding to Finally, we obtain the linearized equation for the beam density In order to solve Eq. (3.6), we note that the function ̺ (1) k may be represented as Assuming the function G(ξ) in Eq. (2.14) to be of the form for small vertical beam sizes σ k , we obtain If R k does not depend on the Lagrange variable ξ, making use of Eq. (3.9), we rewrite Eq. (3.6) as (3.11) Note that this approximation is valid if and only if the perturbed betatron tunes in Eq. (3.3) do not depend on ξ, which in general is not the case. This leads to an effect similar to Landau damping, well-known in plasma physics, which we shall neglect in what follows. Fourier transforming Eq. (3.10) yields where R k (n; θ) = 1 2π dϕ ′ e −inϕ sin ϕe imϕ ′ ×sgn(cos ϕ − cos ϕ ′ ). (3.14) In order to determine the infinite matrix M, we utilize the integral representation of the sign-function As a result, we obtain , for n + m = even, 0, for n + m = odd, (3.16) where use has been made of Here J n (z) is the Bessel function of the first kind of order n.

COHERENT BEAM-BEAM RESONANCES
Equation (3.12) can be formally solved to obtain the oneturn transfer map R k (n; 2π) = exp(−2πin ν k ) Consider now a coherent beam-beam resonance of the form where n 1 , n 2 and s are integers, and ∆ is the resonance detuning. Retaining only the ±n 1 and the ±n 2 elements in M nm , the transformation matrix of the coupled map equations (4.1) can be expressed as  The eigenvalues of the transfer matrix defined in Eq. (4. 3) are the roots of the secular equation Casting Eq. (4.6) in the form where c 1,2 = cos ψ 1 + cos ψ 2 2 ± ± 1 2 (cos ψ 1 − cos ψ 2 ) 2 − 4A sin ψ 1 sin ψ 2 , (4.8) we obtain the stability criterion |cos ψ 1 cos ψ 2 + A sin ψ 1 sin ψ 2 | < 1. (4.10) To conclude this section we note that in the case of a symmetric collider the stopbands calculated from Eq.

CONCLUDING REMARKS
Based on the Radon transform we have developed a macroscopic fluid model of the coherent beam-beam interaction. The linearized hydrodynamic equations are further solved and a stability criterion for coherent beam-beam resonances have been found in closed form.

ACKNOWLEDGMENTS
It is a pleasure to thank Prof. A. Chao and Dr. Y. Cai for helpful discussions concerning the subject of the present paper. This research was supported by the U.S. Department of Energy.