Analytic Derivation of Vertical Chromaticity in the Fermilab Muon $g{-}2$ Storage Ring

Abstract

We derive the vertical chromaticity $ξ_y$ of the Fermilab Muon g-2 storage ring in closed analytic form. Expanding the Hamiltonian as a Taylor polynomial in the dynamical variables and integrating the equations of motion order by order, we obtain the vertical second-order aberrations of the homogeneous magnetic dipole ($\mathtt{DI}$) and the combined-function dipole-and-electrostatic-quadrupole element ($\mathtt{DIQ}$) used in the muon $g{-}2$ ring. Composing the per-element maps over the periodic dispersion orbit yields a closed-form expression for the vertical chromaticity $\xichromy$ of the continuous-ring $\mathtt{DIQ360}$ model, in direct functional analogy with the horizontal result of our earlier work on the same ring (Ref.~\refcite{ChromCPO11}). Comparison against COSY INFINITY differential-algebra computation shows agreement at the $10^{-11}$ level across all three ring models ($\mathtt{DIQ360}$ closed form and the modular $\mathtt{DIEQ\_ON}$, $\mathtt{DIEQ}$ via per-element composition) for muon $g{-}2$ electrostatic-quadrupole (ESQ) voltages $\Vesq \in [10, 26]\,\mathrm{kV}$.