Study of Coherent Tune Shift Caused by Electron Cloud in Positron Storage Rings

We discuss the transverse betatron tune shift of the coherent dipole motion of a beam interacting with an electron cloud. A positron beam which passes through a frozen charge distributionexperiences an electric field determined by the Coulomb law, and its tune shifts in the positive direction. The electrons in the actual cloud are not frozen, but move during the bunch passage. Thus, the electron distribution varies due to the interaction with the beam and it is “soft”. We study the dipole tune shift of the beam interacting with such a “soft” charge distribution. In positron storage rings, an electron cloud can be formed by photoemission and secondary emission, if the ring is operated with a long bunch train (more than 10 bunches) and short bunch spacing ( ns). The electron density first increases along the bunch train and then saturates due to the electron space charge field. The electron cloud induces a tune shift of the positron beam. At KEKB the measured tune shiftalongthe bunch trainincreases both in the horizontal and vertical plane [1]. In this report, we discuss the tune shift caused by the electron cloud using analytic and numerical methods. The tune shift is naively estimated by calculating the electric field for a frozen electron distribution. The electric field at for a uniform charge distribution with cylindrical symmetry is