Abstract
The soft-Gaussian approximation is often employed in computer simulations of strong-strong beam-beam interactions in storage rings. Its defect on the coherent oscillation frequency is pointed out and a possible remedy using Hermitian polynominals is discussed.
Highlights
The beam-beam interaction in storage rings is one of the subjects of beam physics to which analytic theories cannot provide sufficiently useful information for actual machines
Full details of interaction should in principle be obtained by the so-called particle-in-cell (PIC) simulation, it is very time consuming and suffers from numerical noises due to the finite number of macroparticles and to the finite mesh size
We have studied the nature of the soft-Gaussian approximation for beam-beam simulation with respect to the ratio l of the coherent oscillation frequency shift to the incoherent and found that (i) the soft-Gaussian approximation gives somewhat lower l, namely, lsoft-Gaussian ഠ1 1 lexact2, (ii) this might, in some cases, cause the Landau damping to occur more than it should, and (iii) this can be cured by using a distribution of the form Hermitian polynomials times Gaussian
Summary
The beam-beam interaction in storage rings is one of the subjects of beam physics to which analytic theories cannot provide sufficiently useful information for actual machines. A comparison with measurements is not easy because the theory predicts the value only when 2pj ø 1 (e.g., j , 0.01, otherwise, the dynamic beta effect can change l by, say, 10%), the bunch length is much shorter than the beta function, and the oscillation amplitude is much smaller than the beam size. Simulations by the soft-Gaussian approximation, which should be easier than PIC, seem to show smaller values systematically. The recent study of a detailed simulation for the Large Hadron Collider (LHC) by Zorzano and Zimmermann [10] quotes values between 1.09 and 1.15, depending on the aspect ratio. These values clearly disagree with the theory.
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