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analytical estimates for particle dynamics in electric and magnetic rings with and without focusing have been given in a variety of papers and notes .
we have used these high precision estimates to provide benchmarks to test the accuracy of precision particle tracking programs .
the term `` focusing '' in this paper is used to denote `` weak vertical focusing '' unless otherwise indicated .
horizontal focusing is defined by the vertical focusing plus the geometry of the ring , always conforming with maxwell s equations .
these benchmarks include :    * pitch correction@xcite to particle precession frequency in a uniform b - field with and without focusing . * vertical oscillations and energy oscillations in a uniform b - field with no focusing , electric focusing , and magnetic focusing . * radial and vertical oscillations and energy oscillations in an all - electric ring with and without weak focusing . * synchrotron oscillations and momentum storage with a radio frequency cavity ( rf ) in a uniform b - field . * an edm signal and systematic error with an rf wien filter in a magnetic ring .    in the analytical estimates that follow
, we define @xmath0 as the lorentz factor of the design particle in the ring .
the vertical pitch angle @xmath1 of a particle is defined such that @xmath2 where @xmath3 in cylindrical coordinates .
the field focusing index is @xmath4 , with @xmath5 a number with range @xmath6 .
these analytical estimates provide a means of benchmarking particle tracking programs . a precision tracking method that successfully passes all benchmarks
can provide a baseline to benchmark faster programs .
we study the developed benchmarks in the context of runge - kutta / predictor - corrector integration .
the validity of the developed benchmarks extends to any precision tracking program .
precision experiments such as the muon ( @xmath7 ) and storage ring electric dipole moment ( edm ) experiments@xcite require measurements of sub - part per million ( ppm ) accuracy . in the case of a proton or deuteron storage ring edm experiment , a tracking program of extraordinary precision
is required to estimate the spin coherence time of the particle distribution and various lattice parameters , as well as to estimate the values of systematic errors associated with the experiment .
many commonly used beam and spin dynamics programs ignore , or erroneously account for , second and higher - order effects .
tracking in an electric storage ring poses the additional challenge of conforming with total - energy conservation while accounting for higher - order effects .    numerical integration with a sufficiently small step size allowed to run for
a sufficiently long time may reproduce the analytical result with high accuracy .
a tracking program to be used for estimates and investigations in precision experiments must be optimized to be as fast and accurate as possible .
this calls for a well - tested and robust procedure to benchmark the accuracy of tracking programs in situations relevant to the experiments .    we summarize and derive analytical solutions to the equations of motion of a particle in various electric and magnetic rings .
in several cases , comparison of the analytical estimate with precision tracking results identified discrepancies and indicated the need to improve the estimates . in this way
it was determined that the total correction due to vertical particle oscillations , the so - called pitch effect , can be significantly reduced@xcite .
these analytical estimates provide individual benchmarks for tracking programs .
a program well - benchmarked against these estimates would provide additional , more flexible means to benchmark faster programs in different conditions .
we benchmarked a program based on runge - kutta / predictor - corrector method@xcite against the developed analytical estimates .
runge - kutta / predictor - corrector integration should reproduce the analytical estimates to sub - ppm accuracy on a time scale on the order of hours , in order to be a feasible candidate for benchmarking faster programs .
together , the analytical estimates and a program based on runge - kutta / predictor - corrector integration provide a powerful tool for benchmarking precision tracking programs for muon @xmath8 , storage ring edm , and other precision physics experiments requiring high precision beam and spin dynamics simulation .
for a particle of mass @xmath9 and charge @xmath10 , there are two differential equations that govern particle and spin dynamics . for particle velocity @xmath11 and rest spin @xmath12 in external fields , the equations are@xcite : @xmath13,\ ] ] and the t - bmt equation , with an anomalous magnetic moment @xmath14 of the particle : @xmath15.\end{aligned}\ ] ]    the predictor - corrector integration was used with a step size of @xmath16ps to numerically solve the two differential equations with the corresponding initial conditions .
although the method is relatively slow , it is very simple and accurate .
this method uses the runge - kutta method to start the integration process and we refer to it as runge - kutta / predictor - corrector in this document .
a magnetic ring consists of a uniform magnetic field @xmath17 , taken to be in the vertical direction .
the correction @xmath18 to the precession frequency due to a vertical pitch is defined by @xmath19 , where @xmath20 is the ( @xmath7 ) frequency@xcite for a particle with anomalous magnetic moment @xmath14 .
the predicted correction is@xcite : @xmath21 with @xmath22 , where @xmath23 is the vertical ( pitch ) oscillation frequency .      when there is no focusing or when @xmath24 , the correction from equation [ mrnf1 ] becomes :    @xmath25 where for linear oscillations , @xmath26 , where @xmath27 is the maximum pitch angle of the particle trajectory .
for a particle with @xmath28 and a constant 1.0mrad vertical pitch as shown in figure [ pitch1 ] , the simulated correction to the ( @xmath7 ) precession frequency of @xmath29ppm is in very good agreement with the analytically predicted value of @xmath30ppm using equation [ mrnf2 ] .
[ .65]the particle path in cartesian coordinates in a uniform b - field with pitch angle @xmath31mrad , for a ring with a 5 m radius.,title="fig : " ]    checking over several values of @xmath1 confirms that the analytic expression and the pitch correction in the tracking simulation agree for small @xmath1 , as expected .
when there is magnetic focusing and when @xmath32 , the correction from equation  [ mrnf1 ] becomes :    @xmath33    the analytical estimate@xcite for the average particle radial deviation from the ideal orbit with radius @xmath34 , with weak magnetic focusing index @xmath4 , takes the form : @xmath35 for a vertical pitch frequency significantly greater than the ( @xmath7 ) precession frequency of the particle , where @xmath36 is the momentum compaction factor
.    equation [ mrmf1 ] predicts an average radial deviation @xmath37 of @xmath38 using @xmath39 1mrad and a field index @xmath40 , consistent with the tracking results shown in figure [ drmrmf ] to sub - part per billion ( ppb ) level .
the dependence of @xmath41 on the field index is shown to hold over a range of @xmath4 values in figure [ bfdr ] .
[ .29]the particle deviation from the ideal radial position over time , modulo 50@xmath42s .
the simulations used a maximum pitch angle of @xmath43mrad and magnetic focusing with field index @xmath40.,title="fig : " ]    [ 1.25]the average @xmath37 versus the field focusing index @xmath4 .
the solid line represents the predicted values while the points are the results of tracking .
the simulation used a maximum pitch angle of @xmath44mrad.,title="fig : " ]    in a continuous storage ring with weak focusing , field strength @xmath45 , and ring radius @xmath34 , the vertical and horizontal magnetic field components around the ideal trajectory can be expressed to second - order in the vertical position @xmath46 as : @xmath47@xmath48 where the nonlinearity arises from the application of maxwell s equations in cylindrical coordinates .
the horizontal and vertical tunes are given by @xmath49 and @xmath50 respectively .
we make use of the relations@xcite from equation [ mrmf1 ] : @xmath51 and @xmath52 , where @xmath53 is the maximum vertical excursion . from this
we see that : @xmath54    by considering the time - averaged relative b - field change , calling @xmath55 the modification due to the different b - field encountered by the particle , we find that the correction @xmath56 to the ( @xmath7 ) precession frequency in a magnetic storage ring with weak focusing , i.e. @xmath57 , is given by the expression : @xmath58 where we see that several terms of the inhomogeneous b - field correction and the correction in equation [ mrnf3 ] cancel , leaving a small correction .
the necessity of including the second - order inhomogeneous magnetic field contributions was overlooked by previous authors .
investigations with precision tracking identified the discrepancy and motivated the improvement of the analytical estimate .
this holds for a vertical pitch frequency much greater than the ( @xmath7 ) precession frequency of the particle , which for a weak focusing ring means @xmath59 .
equation [ mrmf2 ] implies that the pitch effect can , in principle , be made to vanish for @xmath60 , but the condition @xmath59 makes it rather difficult to achieve . to test the tracking program we introduced a particle with 10 times the muon mass , with same @xmath14 value as the muon , stored in a ring radius of 7.112 m .
the program indeed showed that the pitch correction vanishes with an uncertainty at the part per billion ( ppb ) level when @xmath61 was used .    for realistic muon parameters ,
the observed ( @xmath7 ) frequency is off from its correct value by + 0.109ppm , for a vertical maximum pitch angle @xmath62mrad , and @xmath63 , consistent with the offset shown in figure [ pitch2a ] to sub - ppb level .
[ .3]the pitch correction to the ( @xmath7 ) frequency with angle @xmath44mrad , @xmath64 , and @xmath63 , is 0.109ppm , consistent with equation [ mrmf2 ] to sub - ppb level .
, title="fig : " ]    a resonance of the pitch effect correction occurs when the vertical pitch frequency @xmath65 is equal to the ( @xmath7 ) precession of the stored particles @xmath20 .
the correction @xmath18 approaches equation [ mrnf2 ] for @xmath66 and equation [ mrnf3 ] for @xmath67 .
the full range of pitch corrections @xmath18 over a range of index values is shown in figure [ pitch2 ] . when all the fields are taken properly into account , as shown above ,
the tracking results reproduce the same curve to sub - ppb level for @xmath68 .
[ 0.68]parameter @xmath18 from equation [ mrnf1 ] for the pitch correction to the ( @xmath7 ) frequency over a range of @xmath4 values using @xmath69mrad and @xmath70 .
a resonance occurs when @xmath71 , at @xmath72 , as expected .
, title="fig : " ]    a comparison of the frequency shift predicted by equation [ mrmf2 ] and the results from tracking is given in table [ mrmft1 ] .
the analytical estimates of the pitch correction and the tracking results are in very good agreement , better than ppb level .
this level of precision is adequate for the muon @xmath7 experiments currently underway  @xcite , both aiming for better than 0.1ppm total systematic error .
.comparison of the frequency shift estimated using equation [ mrmf2 ] and the tracking results .
the tracking results assume a muon with @xmath64 , stored in a magnetic ring with magnetic focusing and a radius @xmath73 m .
the vertical angle used is @xmath74mrad .
the observed @xmath7 frequency is shifted higher by the small factors given below depending on the @xmath4-value used . [ cols="^,^,^ " , ]     [ prot ]    the wien filter provides another benchmark for testing the accuracy of the analytical estimates and the runge - kutta / predictor - corrector tracking method .
we again see very good agreement between the analytically predicted values and those calculated by tracking .
we have determined an array of analytical estimates for benchmarking tracking programs for precision storage ring experiments .
the benchmarks form a robust test for electric and magnetic rings , with and without focusing , as well as rf cavities and wien filters .
the inclusion of a high - order contribution to the weak magnetic focusing estimate resulted in the discovery of a method to reduce or eliminate the pitch effect , which was overlooked by other authors . together
these analytical estimates give a powerful tool to benchmark programs for studying particle motion and spin dynamics in a variety of storage ring configurations .
the runge - kutta / predictor - corrector integration appears to be an accurate , albeit slow , tool for precision tracking .
it has passed all the benchmarks that it was tested against , often to the part per billion level .
the tracking program was able to successfully simulate particle dynamics in electric and magnetic rings with and without weak focusing , in agreement with analytical estimation .
consequently , we conclude that the runge - kutta / predictor - corrector method can be used to benchmark faster tracking programs .
other tracking programs are also capable of providing high accuracy at least under certain conditions , e.g. ,  @xcite .
what we have presented here is an array of high precision analytical estimates that can be used to benchmark candidate simulation programs .
we would like to thank the department of energy and brookhaven national laboratory for their continued support of the high school and supplemental undergraduate research programs .
we especially thank the storage ring edm collaboration .
doe partially supported this project under bnl contract no .
de - sc0012704 .
ibs - korea partially supported this project under system code ibs - r017-d1 - 2014-a00 .
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a set of analytical benchmarks for tracking programs is required for precision storage ring experiments . to determine the accuracy of precision tracking programs in electric and magnetic rings , a variety of analytical estimates of particle and spin dynamics in the rings 
 were developed and compared to the numerical results of tracking simulations . 
 initial discrepancies in the comparisons indicated the need for improvement of several of the analytical estimates . as an example , we found that the fourth - order runge - kutta / predictor - corrector method was slow but accurate , and that it passed all the benchmarks it was tested against , often to the sub - part per billion level . 
 thus , high precision analytical estimates and tracking programs based on fourth - order runge - kutta / predictor - corrector integration can be used to benchmark faster tracking programs for accuracy .    
 analytical benchmarking , precision particle tracking , electric and magnetic storage rings , runge - kutta , predictor - corrector