content_image/435071.jpg
# Potential for a new muon $_{g}$-2 experiment 

Alexander J. Silenko ∗ 

Research Institute of Nuclear Problems, Belarusian State University, Minsk 220030, Belarus 

(Dated: November 7, 2018) 

A new high-precision experiment to measure the muon g -2 factor is proposed. The developed experiment can be performed on an ordinary storage ring with a noncontinuous and nonuniform field. When the total length of straight sections of the ring is appropriate, the spin rotation frequency becomes almost independent of the particle momentum. In this case, a high-precision measurement of an average magnetic field can be carried out with polarized proton beams. A muon beam energy can be arbitrary. Possibilities to avoid a betatron resonance are analyzed and corrections to the g -2 frequency are considered. 

Measurement of the anomalous magnetic moment of the muon is very important because it can in princi- ple bring a discovery of new physics. Experimental data dominated by the BNL E821 experiment, a exp µ ± = 116592080(63) × 10 − 11 (0 . 54 ppm), are not consistent with the theoretical result, a the µ ± = 116591790(65) × 10 − $^{11}$, where a = ( g − 2) / 2. The discrepancy is 3.2 σ : a exp µ ± − a the µ ± = +290(90) × 10 − 11 [1]. In this situation, the exis- tence of the inconsistency should be confirmed by new ex- periments. The past BNL E821 experiment [2] was based on the use of electrostatic focusing at the “magic” beam momentum p$_{m}$ = $_{mc/}$√$_{a}$ ( γ$_{m}$ = √ 1 + 1 /a ≈ 29 . 3). An upgraded (but not started up) experiment, E969 [3], with goals of σ$_{syst}$ = 0 . 14 ppm and σ$_{stat}$ = 0 . 20 ppm is based on the same principle. 

Since the muon g -2 experiment is very important, a search for new methods of its performing is necessary. One of new methods has been proposed by Farley [4]. Its main distinctions from the usual g -2 experiments are i ) noncontinuous magnetic field which is uniform into cir- cular sectors, ii ) edge focusing, and iii ) measurement of an average magnetic field with polarized proton beams instead of protons at rest. A chosen energy of muons can be different from the “magic” energy. Its increasing prolongs the lab lifetime of muons. As a result, a mea- surement of muon g -2 at the level of 0.03 ppm appears feasible [4]. 

In the present work, we develop the ideas by Farley. We adopt his propositions to measure the average mag- netic field with polarized proton beams and to use a ring with a noncontinuous field for keeping the independence of the spin rotation frequency from the particle momen- tum. We also investigate the most interesting case when the beam energy can be arbitrary. However, we propose to perform the high-precision muon g -2 experiment on 

PACS numbers: 13.40.Em, 14.60.Ef, 29.27.Hj Keywords: anomalous magnetic moment; muon; polarized beams 

## I. INTRODUCTION 

an ordinary storage ring with a nonuniform field created by superconducting magnets. We prove that the inde- pendence of the spin rotation frequency from the parti- cle momentum can be reached not only in a continuous uniform magnetic field [2, 3] and a noncontinuous and lo- cally uniform one [4] but also in a usual storage ring with a noncontinuous and nonuniform magnetic field. In the last case, the total length of straight sections of the ring should be appropriate. We also analyze possibilities to avoid the betatron resonance ν$_{x}$ = 1 ( ν$_{x}$ is the horizontal tune) and consider corrections to the g -2 frequency. 

The system of units ℏ = c = 1 is used. 

## II. $_{g}$-2 RING WITH A NONCONTINUOUS MAGNETIC FIELD AND MAGNETIC FOCUSING 

Let us consider spin dynamics in a usual storage ring with a noncontinuous magnetic field and magnetic focus- ing. The general equation for the angular velocity of spin precession in the cylindrical coordinates is given by (see Ref. [5]) 

<img src='content_image/435071.jpg'>

Eq. (1) is useful for analytical calculations of spin dy- namics with allowance for field misalignments and beam oscillations. This equation does not contain small terms which can be neglected. η = 4 dm/e is an analogue of the g factor for the electric dipole moment, d . The sign ‖ denotes a horizontal projection for any vector. There- inafter, the electric dipole moment will be disregarded. The vertical magnetic field, B$_{z}$ , is the main field in the muon g -2 experiment. The spin precession caused by this 

