Systematic error studies for the charged particle EDM measurement proposal
SSystematic error studies for the charged particle EDM measurement proposal
M. Haj Tahar and C. Carli
CERN, Geneva, Switzerland
Proposals aimed at measuring the Electric Dipole Moment (EDM) for charged particles requirea good understanding of the systematic errors that can contribute to a vertical spin buildup mim-icking the EDM signal to be detected. In what follows, a method of averaging emanating from theBogoliubov-Krylov-Mitropolski method is employed to solve the T-BMT equation and calculate theBerry phases arising for the storage ring frozen spin concept. The formalism employed proved tobe particularly useful to determine the evolution of the spin at the observation point, i.e. at thelocation of the polarimeter. Several selected cases of lattice imperfections were simulated and bench-marked with the analytical estimates. This allowed the proof of the convergence of the numericalsimulations and helped gain better understanding of the systematic errors.
I. INTRODUCTION
The quest to challenge the standard model of particle physics is on-going with a very diverse set of experimentalinvestigations aimed at finding new physics. The direct approach relies on particle colliders through possibleproduction of new particles. Nevertheless, due to the, so far, negative results of searches for new particles with theLarge Hadron Collider, a new program has been established at CERN, the so called Physics Beyond Colliders (PBC)program [1].Among the potential projects that are being considered by this program is the quest for precise measurements ofthe permanent Electric Dipole Moment (EDM) of fundamental particles or subatomic systems, widely consideredas a sensitive probe for physics beyond the standard model [2] and among the essential scientific activities thatwas recommended by the 2020 European strategy group for particle physics [3]. The quest to measure such anasymmetric charge distribution within the particle volume has gained attractiveness and enthusiasm over the lastfew decades since a non null EDM would be a sign of CP (Charge Parity) violation. The latter is one of the threeconditions that could explain why a universe containing initially equal amounts of matter and antimatter shall evolveinto a matter-dominated universe, as formulated by Andrei Sakharov in 1967 [4].To this end, the search for such a small-scale quantity has been pursued by several research groups and significantcontributions made over the years [5, 6]. In particular, neutral systems such as neutrons, neutral molecules or atomshave been privileged in many cases due to the ease of constructing a trapping system where the electromagneticfields have minimum impact on the translational motion [7–10]. Another approach is indirect measurements withcharged particles exploiting the strong electric fields in some molecules. For instance, the most sensitive upperlimit to an EDM of any elementary particle or nucleus comes from indirect measurement relying on a cryogenicmolecular beam of the heavy polar molecule thorium monoxide (ThO) and yielded an upper limit of the electronEDM, | d e | < . × − e.cm at 90% confidence level [11]. However, since a single indirect EDM measurementcannot decide on the source of CP-violation even if detected, several measurements with a variety of systems arewidely considered necessary in order to elucidate the nature of the EDM and its underlying mechanisms [5, 12].To circumvent such a difficulty of attaining high precision direct measurements for charged particles, the method of“magic energy” concept has been successfully applied to measure the anomalous Magnetic Dipole Moment (MDM) ofmuons [13] and represents an attractive solution to search and measure the EDM of muons as well as other chargedparticles [12, 14–16]. The concept relies on a storage ring where polarized particles are injected and recirculated attheir magic momentum [15] so that the orientation of the particle spin with respect to its momentum direction ispreserved with the well-known MDM torque. Since the EDM of a particle is aligned with its spin vector, measuringa spin build-up by coupling with radial electric fields will be a direct observation of a non null EDM signal. Forprotons, an attractive solution exists to build a low energy all-electric ring [18, 19] since the magic kinetic energy tofreeze the spin is E kin = 232 . − e.cm [20, 21]. To give amore intuitive perception, this is equivalent to measuring a separation between the centre of mass of the proton andits centre of charge with an accuracy of 10 − cm [8].However, to reach the desired sensitivity level, it is crucial to understand and mitigate the systematic errors due tomachine imperfections that can yield a fake signal mimicking the EDM one. Typical machine imperfections of anall-electric proton EDM ring are residual magnetic fields penetrating the shield and the limited positioning accuracyand mechanical tolerances of electric bends and focusing quadrupoles. The objective of this paper is to contribute a r X i v : . [ phy s i c s . acc - ph ] A ug to a better understanding regarding that matter: starting from the spin precession equation, we will establish theformalism and all necessary quantities to compute the spin evolution in a storage ring. Then, using a perturbationmethod, an approximate solution to this equation is derived and benchmarked with BMAD tracking simulations. Theapplication example is focused on the case of the all-electric proton EDM ring [22]. However, the formalism developedapplies to any storage ring relying on the frozen spin technique among which the hybrid ring lattice where magneticfields are used for focusing and electric fields for deflection [23] or other concepts for which the spin is frozen by meansof combined electrostatic and magnetic deflectors [14, 15].In particular, it will be shown that, even at the magic energy, machine imperfections lead to various effects generatinga vertical spin component build-up and thus a fake signal. In particular, the geometric phases, often also referred to asthe Berry phases, constitute one leading contribution to such an effect. The latter will be calculated and benchmarkedwith the tracking simulations.This paper is divided as follows: first, we start by recalling the spin precession equation in storage rings and thechoice of convenient coordinate system to simplify the analysis. Then, a perturbation approach will be invoked tosolve the equation in the vicinity of the magic energy. This will allow to establish and distinguish the different classesof leading systematic errors. Finally, the analytical expressions will be benchmarked with tracking simulations of anEDM ring with selected imperfections. II. THOMAS-BARGMANN-MICHEL-TELEGDI EQUATION
The variation with time of the classical spin vector S (such that | S | = 1) can be described by a vector equation,the so called Thomas-Bargmann-Michel-Telegdi (T-BMT) equation [24–26]: d S dt = ( Ω MDM + Ω EDM ) × S (1)where Ω MDM = − qmc (cid:20)(cid:18) G + 1 γ (cid:19) c B − Gγcγ + 1 ( β.B ) β − (cid:18) G + 1 γ + 1 (cid:19) β × E (cid:21) (2)is the precession vector due to the particle’s magnetic moment and Ω EDM = − qmc η (cid:20) E − γγ + 1 ( β . E ) β + c β × B (cid:21) (3)is the precession vector due the particle’s finite electric dipole moment. S is defined in the rest frame of the particlewhile B , E , t denote the magnetic fields, electric fields and time defined in the laboratory frame of reference, G isthe particle’s anomalous gyro-magnetic factor often quoted as G = ( g − /
2. In addition, q , m , c have their standardmeanings while γ and β denote the Lorentz factor as well as the velocity of the particle normalized in units of c . Thedimensionless factor describing the size of the EDM is given by η . III. CONVENIENT COORDINATE SYSTEM
In accelerator physics, the particle coordinates are generally expanded around a reference frame sketched in fig. 1,following the reference particle orbit. We denote the three unit vectors attached to such a frame ( u x , u y , u z ) and s the curvilinear abscissa along the reference orbit, not necessarily equal to the distance traversed by the particle. Ina storage ring where the reference orbit is closed, the coordinate system privileged to describe the spin is the same,i.e. the one in which the xy plane attached to the reference particle is rotating at a convenient reference angularfrequency. Such a frame is heavily employed for magnetic resonance problems as well [17]. The angular velocity vectordescribing the rotation of this coordinate system (due to the acceleration experienced by the particle as it moves underthe action of electromagnetic forces) is denoted by ω , sometimes also referred to as the Darboux vector.Thus, if ∂/∂t represents the differentiation with respect to such a rotating coordinate system, then, by a well-knowntransformation [17] ∂ S ∂t = d S dt − ω × S = Ω rot × S (4) ReferenceOrbitActualOrbitCenter ofCurvature s ρ s = 0Particle ReferenceParticle u y u x u z FIG. 1: The local reference coordinate system used for analytical derivations and for comparative tracking studiesusing BMAD [27]. The reference orbit lies in the (theoretical) median plane of the accelerator: u z is the unit vectorpointing along the momentum direction of the reference particle, u x points radially outwards and u y is the verticalunit vector defined as: u y = u z × u x .where Ω rot = Ω MDM + Ω EDM − ω (5)and ω = − ds/dtρ u y = − β z cρ + x u y (6) ρ being the bending radius of the reference orbit. Now, writing the relativistic form of Newton’s second law in aperfect machine without any imperfections, the bending radius of the closed orbit can be expressed as a function ofthe applied bending fields: 1 ρ = − qmγβ c E x + qmγβc B y (7)Note that the subscripts i denote the projected components of the field, normalized velocity as well as the spin vectorin such a frame.In order to simplify our analysis of the systematic errors, a vanishing EDM contribution is assumed, i.e. η = 0.Expanding the projected components of the spin precession vector Ω rot = (Ω x , Ω y , Ω z ) and keeping terms up to thesecond order only, yields: Ω x = − qmc (cid:18) G + 1 γ + 1 (cid:19) β z ( E y − y (cid:48) E z ) − qm (cid:18) G + 1 γ (cid:19) B x + qm G (cid:18) − γ (cid:19) x (cid:48) B z Ω y = qmc (cid:18) G + 1 γ + 1 (cid:19) β z ( E x − x (cid:48) E z ) − qm (cid:18) G + 1 γ (cid:19) B y + qm G (cid:18) − γ (cid:19) y (cid:48) B z + β z cρ + x Ω z = qmc (cid:18) G + 1 γ + 1 (cid:19) β z ( x (cid:48) E y − y (cid:48) E x ) − qm Gγ B z + qm G (cid:18) − γ (cid:19) ( x (cid:48) B x + y (cid:48) B y ) (8)Finally, by making use of Eq. (7), and assuming a particle in a perfect machine following the reference orbit( x = x (cid:48) = y = y (cid:48) = 0), the expression of the vertical component can be further simplified:Ω y = qmc (cid:18) G − γ − (cid:19) βE x − qm GB y (9)whereas the other two components vanish Ω x = Ω z = 0.From relation (9), one can see that, for each energy, there exists ( E x , B y ) combinations that shall preserve theorientation of the particle spin with respect to its momentum direction. This is called “frozen spin” condition and isachieved by setting Ω y to zero [14, 15]. In particular, for particles possessing a positive G-factor, this can be obtainedfor an all-electric ring and for one specific momentum that we generally refer to as the magic momentum p m : p m = mc √ G (10)For protons, this corresponds to p m = 700 .
74 MeV/c i.e. to a particle kinetic energy of 232 . IV. METHOD OF AVERAGES
In our approach, we are interested in determining the impact of perturbations on the beam polarization evolution:the proximity to the magic energy leads to the assumption that the derivative ∂ S /∂t is small, an assumption that isintrinsic to the choice of such an energy for which the spin precession components shall vanish and that we will referto as the nearly frozen spin condition. In Matrix notation, the T-BMT equation writes as follows: ∂ S ∂t = Ω ( t ) S ( t ) = − Ω z Ω y Ω z − Ω x − Ω y Ω x S x S y S z (11)Thus, when the above condition is fulfilled, the Bogoliubov-Krylov-Mitropolski (BKM) method of averages can beinvoked whereby the evolution of S is decomposed as the sum of two terms obeying two timescales: a slowly varyingterm ξ , due to the smallness of Ω i , and small rapidly oscillating terms due to the presence of t in Ω i , i.e. describingthe spin precession changes within the elements. The basic idea of this approach was first developed by Krylov andBogoliubov (1934) [31]. Later on, in 1958, Bogoliubov and Mitropolski established the general scheme and a morerigorous treatment for this method [32]. Finally, in 1969, Perko almost completed the theory with error estimates forthe periodic and quasi-periodic cases [33].In the formalism that we employ throughout this paper, ξ accounts for the polarization buildup due to the averagesof the spin precession components while φ represents the oscillatory behavior of the beam polarization. Thus, it isassumed that the spin angular frequencies possess an average value (with respect to the explicit variable t ) that isdenoted by the angular brackets as follows: (cid:104) Ω i (cid:105) = lim T →∞ T (cid:90) T Ω i ( t ) dt ; i = x, y, z (12)In addition, the integrating operators˜and (cid:101) ˜are defined as follows,˜Ω i ( t ) = (cid:90) [Ω i ( t ) − (cid:104) Ω i (cid:105) ] dt (cid:101) ˜Ω i ( t ) = (cid:90) (cid:104) ˜Ω i ( t ) − (cid:68) ˜Ω i (cid:69)(cid:105) dt A. 1st order approximation
The first order approximate solution of the T-BMT equation, obtained applying the BKM method [32], is given by: S ( t ) = (cid:104) + ˜ Ω ( t ) (cid:105) ξ ( t ) (13)where the integrating operator is acting on all the elements of the matrix and ξ ( t ) is the solution of the averagedT-BMT equation, i.e. ∂ ξ ∂t = (cid:104) Ω (cid:105) ξ ( t ) = − (cid:104) Ω z (cid:105) (cid:104) Ω y (cid:105)(cid:104) Ω z (cid:105) − (cid:104) Ω x (cid:105)− (cid:104) Ω y (cid:105) (cid:104) Ω x (cid:105) ξ x, ξ y, ξ z, (14)the subscript ξ x,i denoting the i th order of the approximation.A solution of the above equation is readily obtained using the Euler-Rodriguez formula: ξ ( t ) = e (cid:104) Ω (cid:105) t ξ (0) = (cid:34) + (cid:104) Ω (cid:105) sin( (cid:104) Ω (cid:105) t ) (cid:104) Ω (cid:105) + (cid:104) Ω (cid:105) − cos( (cid:104) Ω (cid:105) t ) (cid:104) Ω (cid:105) (cid:35) ξ (0) (15) (cid:104) Ω (cid:105) = (cid:113) (cid:104) Ω x (cid:105) + (cid:104) Ω y (cid:105) + (cid:104) Ω z (cid:105) (16)where one assumes an initial value of the spin vector given by S (0) = ξ (0) = ( ξ x , ξ y , ξ z ).In the limit where (cid:104) Ω (cid:105) t (cid:28)
1, consistent with a nearly frozen spin condition, Eq. (13) re-writes by keeping terms upto the first order in Ω i : S ( t ) = ξ ( t ) + φ ( t )= [ + (cid:104) Ω (cid:105) t ] S (0) + ˜ Ω S (0) (17)where φ represent the first order rapidly oscillating terms that vanish after each period completion.Now, expanding the first order linear solution relevant for a turn-by-turn analysis of the spin build-up yields: ξ x, ( t ) = ξ x + [ (cid:104) Ω y (cid:105) ξ z − (cid:104) Ω z (cid:105) ξ y ] tξ y, ( t ) = ξ y + [ (cid:104) Ω z (cid:105) ξ x − (cid:104) Ω x (cid:105) ξ z ] t (18) ξ z, ( t ) = ξ z + [ (cid:104) Ω x (cid:105) ξ y − (cid:104) Ω y (cid:105) ξ x ] t B. 2nd order approximation
To obtain the second order approximation, the method of successive approximations is applied by re-injecting thefirst order approximation (17) into the exact T-BMT equation and re-integrating it again. This writes as follows: ∂ S ∂t = Ω ( t ) S ( t )= (cid:104) Ω + Ω (cid:104) Ω (cid:105) t + Ω ˜ Ω (cid:105) S (0) (19)Following the integration steps in Appendix A, the second order approximation is established: S ( t ) = ξ ( t ) + φ ( t ) (20)where ξ ( t ) = (cid:34) + (cid:110) (cid:104) Ω (cid:105) + (cid:104) Ω (cid:105) (cid:68) ˜ Ω (cid:69) − (cid:68) ˜ Ω (cid:69) (cid:104) Ω (cid:105) + (cid:68) ( Ω − (cid:104) Ω (cid:105) ) ˜ Ω (cid:69)(cid:111) t + (cid:104) Ω (cid:105) t (cid:35) S (0) (21)and φ ( t ) = (cid:104) ˜ Ω + (cid:103) Ω ˜ Ω + (cid:16) t ˜ Ω − (cid:101) ˜ Ω (cid:17) (cid:104) Ω (cid:105) (cid:105) S (0) (22)In particular, if (cid:104) Ω (cid:105) = 0, then the only remaining contribution to the vertical (or radial) spin build-up is due to thegeometric (or Berry) phases [34, 35] such as: ξ ( t ) = (cid:104) + (cid:68) Ω ˜ Ω (cid:69) t (cid:105) S (0) (23)To verify the validity of the previous analytical solutions, several cases were simulated by solving the T-BMT equationusing explicit Runge Kutta tracker in MATHEMATICA. The expanded Matrix form is shown in appendix B.Finally, it should be noted that the rapidly oscillating terms φ i for a specific order have no impact on the measuredpolarization if we restrict the approximation to that order. By construction, these terms vanish after each turncompletion, i.e. at the location of the polarimeter corresponding to a longitudinal position s = 0. However, they arecrucial to refine the approximation to higher orders as shown previously. In particular, one can observe that the 2ndorder approximation revealed some additional terms in comparison with the first order approximation. Those termswill be discussed in section V that focuses on the case of an initial longitudinal beam polarization. C. Case of longitudinally polarized beam
In the frozen spin scenario, the idea is to inject a beam which is initially polarized longitudinally i.e. S (0) = (0 , , ξ y, ( t ) = − (cid:104) Ω x (cid:105) t + (cid:104) Ω z (cid:105) (cid:68) ˜Ω y (cid:69) t − (cid:104) Ω y (cid:105) (cid:68) ˜Ω z (cid:69) t + (cid:68) (Ω z − (cid:104) Ω z (cid:105) ) ˜Ω y (cid:69) t + (cid:104) Ω y (cid:105) (cid:104) Ω z (cid:105) t (24)This will be our main focus for the remaining part of this paper. In addition, unless otherwise specified, the oscillatingcontribution to the spin evolution, i.e. φ ( t ), is disregarded.At this point, it is worthwhile to specify the level of accuracy with which the spin evolution shall be determined inorder to reduce the systematic errors to the level of the desired EDM signal. As mentioned earlier, for an aimedsensitivity of 10 − e.cm, corresponding to η = 1 . · − , the vertical spin build-up will be: ∂S y ∂t = − (cid:104) Ω x (cid:105) = qmc η (cid:104) E x (cid:105) (25)Thus, assuming an average field of (cid:104) E x (cid:105) = − .
27 MV/m, corresponding to a C = 500 m circumference ring, thisyields a build-up of 1 . D. Error analysis
The above second order approximation to the T-BMT equation is based on the assumption that the average spinprecession component is small on the timescales of the EDM experiment. If the spin coherence time is T coh = 1000 sas is generally assumed to reach the aimed statistical sensitivity (of 10 − e.cm) to measure the EDM within 4 yearsof operation time [36], then a necessary but non sufficient condition can be formulated as follows: (cid:104) Ω (cid:105) T coh = (cid:113) (cid:104) Ω x (cid:105) + (cid:104) Ω y (cid:105) + (cid:104) Ω z (cid:105) T coh (cid:28) ⇒ (cid:104) Ω x (cid:105) , (cid:104) Ω y (cid:105) , (cid:104) Ω z (cid:105) (cid:28) T coh ≈ − s − (26)This signifies that, the larger the EDM build-up time, the smaller are the required averages of the spin precessioncomponents to guarantee a linear regime of the polarization signal. In particular, the condition (26) justifies the needfor the second order approximation in order to account for the systematic errors that can yield a signal at the levelsof the EDM one.From the above scheme we can infer that the general frozen solution to the T-BMT equation in the interval [0 , T coh ]can be classified into three main regimes depending on the value of the average spin precession: • If (cid:104) Ω i (cid:105) (cid:38) /T coh for all i , then the spin evolution is governed by the averages of its precession components.Therefore, in many cases Eq. (15) gives sufficiently accurate results. • If 0 < (cid:104) Ω (cid:105) (cid:28) /T coh then the non-linear increase with time can be neglected on the timescales of the EDMexperiment. And using the 2 nd order approximation based on the BKM method of averages, i.e. Eq (21), it canbe seen that: ξ y ( t ) = ξ y, ( t ) + O ( (cid:15) ) t (27)where (cid:15) can be established by pushing the approximation to the third order. The latter is invoked in somepeculiar cases such as the one hereafter. • In the limit where (cid:104) Ω (cid:105) = 0, i.e. (cid:104) Ω i (cid:105) = 0 for all i , the geometric phases are the only contribution to the spinbuild-up. The latter is governed by the non-commutativity of the rotation around different axes. Using themethod of successive approximations to establish the third order approximation, it can be easily shown that: ξ y ( t ) = ξ y, ( t ) + O (cid:18)(cid:28) Ω z (cid:94) Ω z ˜Ω x (cid:29) t − (cid:68) Ω z ˜Ω x (cid:69) (cid:68) ˜Ω z (cid:69) t + (cid:68) Ω x ( ˜Ω y ) (cid:69) t (cid:19) (28)Such a result is particularly instructive to illustrate how the higher order terms of the Berry phases can arisein a lattice even when the particle is continuously at the magic energy, i.e. Ω y ( t ) = 0 and its average spinprecession components are all vanishing i.e. (cid:104) Ω (cid:105) = 0. The following diagram shows how a vertical spin build-upcan be generated in such a case: S z = 1 − Ω x −−−→ S y = φ y, = − ˜Ω x − Ω z −−−→ S x = ξ x, + φ x, = (cid:68) Ω z ˜Ω x (cid:69) t + (cid:94) Ω z ˜Ω x Ω z −−→ S y = ξ y, + φ y, In general, when realistic misalignment errors are taken into account, the above condition (26) is not satisfied as isdiscussed in section VI. Nevertheless, the 2nd order approximation can serve as an important benchmarking test ofthe tracking simulations on short timescales t such that (cid:104) Ω (cid:105) (cid:28) /t holds and is crucial to understand the differentsources of imperfections to mitigate.It follows from the T-BMT equation that the magnitude of the spin shall be constant. Nevertheless, it is importantto note that the Hermiticity of the approximate frozen solution is not conserved for the 2nd order approximation. Forinstance, if one computes the Euclidean norm of the frozen solution at times t = kT , i.e. after each turn completion,one obtains for the special case where (cid:104) Ω (cid:105) = 0: (cid:107) ξ ( t = kT ) (cid:107) = ( ξ x, + ξ y, + ξ z, ) / = (cid:18) (cid:20)(cid:68) Ω z ˜Ω x (cid:69) + (cid:68) Ω z ˜Ω y (cid:69) (cid:21) t (cid:19) / ≈ (cid:68) Ω z ˜Ω x (cid:69) + (cid:68) Ω z ˜Ω y (cid:69) t (29)Such an effect is negligible for the timescales of the EDM experiment. However, the Hermiticity can be improved bykeeping the higher order terms in the expansion of the sinusoidal functions of the 1 st order solution. This will not bepursued here. V. ON THE DIFFERENT CLASSES OF SYSTEMATIC ERRORS
From the second order approximation given by Eq. (24), and under the assumption that the condition (26) holds,one can infer five different classes of leading systematic errors:1. The first term, − (cid:104) Ω x (cid:105) t , is due to a non-vanishing average radial spin precession that rotates the initial longitu-dinal polarization into the vertical plane. This accounts for the EDM effect to be measured due to the averageradial electric field in the ring. Another contribution is an average radial magnetic field, which is probably themost severe systematic effect limiting the smallest EDM to be identified.2. The second term, (cid:104) Ω z (cid:105) (cid:68) ˜Ω y (cid:69) t , is due to a non-vanishing average longitudinal spin precession that rotates theoscillating horizontal polarization into the vertical plane.3. The third contribution, − (cid:104) Ω y (cid:105) (cid:68) ˜Ω z (cid:69) t , is due to the slowly linearly varying term of the radial polarizationcomponent which leads to “periodic” vertical spin oscillations with increasing amplitude described by ˜Ω z . Thelatter is sensitive to the location of the perturbations in the ring.4. The fourth contribution, (cid:68) (Ω z − (cid:104) Ω z (cid:105) ) ˜Ω y (cid:69) t , accounts for the geometric phases whereby an oscillating horizontalpolarization is transferred into the vertical plane by means of another oscillating longitudinal spin precession.This is due to the non-commutativity of spin rotations around different axes.5. The last term, (cid:104) Ω y (cid:105) (cid:104) Ω z (cid:105) t , accounts for the rotations around the average of the angular frequency with lon-gitudinal and vertical components: (cid:104) Ω y (cid:105) generates radial spin which is rotated into the vertical by means of (cid:104) Ω z (cid:105) .In the presence of field imperfections and misalignment errors, and in the absence of any feedback system, the directionof the spin starts to depart from the horizontal plane. The resulting polarization signal is thus a mixture of all theabove. Probably the most challenging contribution to cure is the static radial magnetic field since the latter mimicsthe EDM signal even combining measurements for both clockwise and counter-clockwise beams.Although the leading terms of the geometric phases are derived, the procedure established above can be reiterated todetermine the higher order terms.In the next section, several cases of field imperfections and misalignment errors are discussed. Our focus is on theall-electric proton EDM ring. VI. BENCHMARKING WITH NUMERICAL SIMULATIONS
In order to establish the validity of the analytical solution and how effective it can be in explaining the leading sourcesof systematic errors, we apply it to a model accelerator which is based on the all-electric proton ring lattice proposedby V. Lebedev [22] and underlying several recent publications [21]. The proposed ring consists of 4 superperiods, eachincluding 5 FODO cells with 3 cylindrical deflectors per half cell. The ring has a circumference of C = 500 m chosento obtain reasonable maximum electric fields of 8 MV/m for operation at the proton “magic energy”. The main ringparameters are summarized in table I and the lattice functions determined with the tracking code BMAD [27] areplotted in fig 2. The chosen optics are characterized by a weak vertical focusing, resulting in large vertical betatronTotal beam energy 1.171 GeVRing circumference C
500 mFocusing structure FODO N cells , number of cells 20Deflector shape cylindricalNumber of deflectors per cell 6Bending radius ρ ±
120 kVHorizontal tune Q x Q y η -0.192TABLE I: Table of the ring parameters of the proton EDM experiment. Note that, for protons, G=1.7928474.oscillations with a maximum of β ymax = 216 m. The underlying reason is to enhance the vertical separation due toaverage radial magnetic fields of CW and CCW circulating beams. The measurement of this orbit difference withspecial high sensitivity pick-ups to estimate and correct the average radial magnetic field is an important ingredientfor the concept. In addition, as pointed out in [22], operation below transition helps reduce the Intra-beam scatteringgrowth rates which is crucial in order to allow for a large spin coherence time of the order of 1000 s.
40 60 80 100 120 140 160 180 200 220 0 100 200 300 400 500 β x , β y , D x [ m ] Path Length [m]2* β x [m] β y [m]4*D x [m] FIG. 2: Twiss parameters and dispersion for the entire circumference of the all electric proton EDM ring.The aim of this section is to benchmark the BMAD spin tracking simulations against the previously establishedanalytical formula. The analysis is restricted to a particle whose motion is following the closed orbit, i.e. not executingany betatron or synchrotron oscillations. This is a simpler case than particles executing both oscillations. Yet, itcomprises most phenomena generating systematic effects that can limit the possible sensitivity of the experiment.Thus, for each simulated case, the analysis departs by searching for the closed orbit in order to determine the fieldsexperienced by the particle on such a trajectory. From this, the spin precession components as well as their averagesare calculated in an independent python routine to obtain the nearly frozen spin solution given by Eq. (21) andprobe the leading classes of systematic errors. Finally, the BMAD spin tracking simulations based on the built-infourth order Runge Kutta integration algorithm are compared with the analytical estimates based on the one turncomputation of the averages. The comparison is focused on the turn-by-turn data since this is the signal to be detectedby the polarimeter. For all cases considered, the initial beam polarization is longitudinal.
A. Selected cases of lattice imperfections
1. Average radial magnetic field
The particle equation of motion allows to establish the relationship between the electromagnetic fields and thephase space momenta. For the vertical plane, this writes as follows:1 q ( p y ( t ) − p y (0)) = 1 q (cid:90) t dp y dt = (cid:90) t ( E y + β z cB x ) dt (30)The latter is set to zero on the closed orbit so that the effective average fields acting on the spin of the particle arefurther constrained.As a first benchmarking test, one considers the impact of residual radial magnetic field imperfections on the verticalspin. Making use of the relation between the applied fields on the closed orbit established herein, (cid:104) E y (cid:105) = − β z c (cid:104) B x (cid:105) ,the rate of the vertical spin build-up is derived using Eq. (8): ∂S y ∂t ≈ − (cid:104) Ω x (cid:105) = qm (cid:20)(cid:18) G + 1 γ (cid:19) (cid:104) B x (cid:105) + (cid:18) G + 1 γ + 1 (cid:19) β z (cid:104) E y (cid:105) c (cid:21) = qm (cid:20)(cid:18) G + 1 γ (cid:19) − (cid:18) G + 1 γ + 1 (cid:19) β z (cid:21) (cid:104) B x (cid:105) = qm G (cid:104) B x (cid:105) = (1 . · Hz/T) (cid:104) B x (cid:105) (31)0 -8 -7 -6 -5 -4 -3 -2 -1 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 ∂ S y / ∂ t [r a d / s ] [T] AnalyticalTracking simulations FIG. 3: Vertical spin buildup as a function of the average residual radial magnetic field on the closed orbit andcomparison with the analytical estimate given by Eq. (31).where, for the last transformation, the relation G = 1 / ( γ − (cid:104) B x (cid:105) (cid:28)
10 pT. To further achieve the aimed sensitivity level (equivalentto 1 .
2. Quadrupole misalignments
If the particle is injected with a momentum offset δ , then, in presence of vertical motion, vertical spin precession willoccur. For instance, assuming a net vertical misalignment of one quadrupole and no contribution due to magnetic fieldimperfections, one shall calculate the vertical spin buildup. For this, the total energy conservation is a crucial aspectof the simulation [37, 38] since it leads to strong variation of the momentum offset ∆ p/p m within the electrostaticelements (see appendix C). As illustrated in fig. 4, where an initial momentum offset δ = 10 − is assumed, the leadingterm of the vertical spin buildup is the quadratic increase term. Such a quadratic increase in the vertical plane isdue to a linear radial spin buildup which in itself is due to the deviation from the magic energy as is established inappendix C: recalling Eqs. (C10) and (C11) and noting that the horizontal closed orbit x co is, to the first order,proportional to the amplitude of the horizontal misalignment error, the radial spin build-up can be evaluated: ∂S x ∂t ≈ (cid:104) Ω y,disp (cid:105) + (cid:104) Ω y,mis (cid:105) ≈ ( − . · Hz) δ + ( − . · Hz/m)∆ x mis (32)such as, in this example, S x ( t ) ≈ − .
10 Hz t.By making use of Eq. (8) where the vertical slope y (cid:48) is obtained by means of a standard closed orbit search, one alsoevaluates (cid:104) Ω z (cid:105) = − .
18 Hz which is due to the vertically misaligned quadrupole, ∆ y = 100 µ m, generating a verticalslope inside the electrostatic deflectors as shown in Ref [39]. Thus, the condition (26) is not fulfilled and the verticalspin build-up is S y ( t ) ≈ (cid:104) Ω y (cid:105) (cid:104) Ω z (cid:105) t ∝ (cid:104) E x ∆ p/p (cid:105) (cid:104) y (cid:48) E x (cid:105) ∝ δ · ∆ y (33)1 -5 -4 -4 -4 -4
0 0.002 0.004 0.006 0.008 0.01 S p i n v e r ti ca l [r a d ] Time [s] ∆ y = 10 µ m ∆ y = 50 µ m ∆ y = 100 µ m FIG. 4: Comparison of the tracking simulations with the analytical estimate (solid lines) for the case of onequadrupole misaligned vertically by ∆ y (∆ p/p = 10 − ).which is confirmed through tracking simulation results shown in fig. 4. Nevertheless, the above behavior changes at theproximity to the magic energy i.e. when δ →
0, and gives rise to a linear build-up instead. To show this, let’s considerthe same lattice where the beam is injected at the magic energy and where two quadrupoles are misaligned as follows:in the first quarter of the ring, a defocusing quadrupole is misaligned vertically and horizontally by (+∆ x, +∆ y ). Inthe third quarter, i.e. 180 degrees out of phase, a second defocusing quadrupole is misaligned by ( − ∆ x, − ∆ y ). Thus,the average misalignment vanishes in this configuration. Such misalignments generate closed orbit perturbations inboth the horizontal and vertical direction: The horizontal orbit perturbations produce a change of the kinetic energywhich is dominant within the electrostatic bends [37]. Consequently radial spin oscillations arise such as S x ≈ ˜Ω y .The latter is transferred into the vertical plane by means of a longitudinal spin precession. For instance, assuming∆ x = ∆ y = 10 µ m, one obtains by making use of Eq. (8): S y ( t ) ≈ − (cid:104) Ω x (cid:105) t + (cid:68) Ω z ˜Ω y (cid:69) t − (cid:104) Ω y (cid:105) (cid:68) ˜Ω z (cid:69) t + (cid:104) Ω y (cid:105) (cid:104) Ω z (cid:105) t ≈ ∗ t − . ∗ − t + 2 . ∗ − t + 1 . ∗ − t (34)Thus, the vertical spin buildup is mainly due to the geometric phases that can be approximated by: ∂S y ∂t ≈ (cid:68) Ω z ˜Ω y (cid:69) ∝ E x ∆ pp y (cid:48) ∝ ∆ x ∗ ∆ y (35)Such an effect is proportional to the product of the displacements of both quadrupoles: the horizontal displacement ofthe quadrupoles yields larger radial spin oscillations due to the variation of the kinetic energy in the electrostatic bendswhile the vertical displacement of the quadrupoles yields a vertical slope inside the electrostatic bends, therefore alongitudinal spin precession which rotates the radial spin into the vertical plane. Such an effect yields a non-vanishingaverage value, therefore the frozen spin is proportional to both displacements as verified by tracking simulations infig. 5 (and similarly if one replaces ∆ x by ∆ y ).
3. Geometric phases due to magnetic field perturbations
In this case, one assumes alternating longitudinal and vertical magnetic field imperfections which are 90 degrees outof phase as illustrated in fig. 6 and such that the integrated localized field imperfections are ± y represents the integral of Ω y − (cid:104) Ω y (cid:105) therefore accounts for the presence of vertical magnetic fields yieldingoscillating radial spin components. The latter are rotated into the vertical plane by means of longitudinal magnetic2 -3.0*10 -10 -2.5*10 -10 -2.0*10 -10 -1.5*10 -10 -1.0*10 -10 -5.0*10 -11
0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 S p i n v e r ti ca l [r a d ] Time [s]( ∆ x, ∆ y)=(10,10) µ m( ∆ x, ∆ y)=(20,10) µ m( ∆ x, ∆ y)=(30,10) µ m FIG. 5: Vertical spin buildup due to a special case of quadrupole misalignment in both planes causing geometricphase effects and comparison with the analytical estimate. -80 -60 -40 -20 0 20 40 60 80 100 120-180-160-140-120-100-80-60-40-20 0 20-1.6e-17-1.4e-17-1.2e-17-1e-17-8e-18-6e-18-4e-18-2e-18 0 2e-18Y [m] Azx y -By+By -Bz+Bz Ideal closed orbitPerturbed closed orbitProjected perturbed orbitX [m] Z [m]
FIG. 6: Spin and orbit evolution for a lattice with alternating magnetic field imperfections: a vertical magnetic fieldyields a horizontal spin component which is rotated into the vertical plane by means of a longitudinal fieldcomponent. The closed orbit of the perturbed motion is shown in blue and the particle motion is clockwise startingfrom Point A. The orbit displacement from the ideal one is amplified for the sake of clarity.3
Path length Ω y , z Ω y Ω zy ~Ω FIG. 7: Illustrations of the longitudinal and vertical components of the spin precession vector due to alternatinglongitudinal and vertical magnetic field imperfections. The vertical tilde component ˜Ω y represents the integral of thevertical component and accounts for the rapidly oscillating terms of the radial spin component. The average of theproduct of ˜Ω y and Ω z yields a non-vanishing vertical spin component. S p i n v e r ti ca l [r a d ] Time [s]
FIG. 8: Vertical spin buildup from tracking simulations and comparison with the analytical estimate based onEq. (36).fields therefore a non null Ω z . The product of these two components yields the linear vertical spin build-up due tothe geometric phases. By making use of Eq. (8), one obtains: ∂S y ∂t ≈ (cid:68) Ω z ˜Ω y (cid:69) ≈ cβ z C (cid:16) qm (cid:17) (cid:18) G + 1 γ (cid:19) Gγ ( B y L )( B z L )= (cid:2) . · Hz/(T.m) (cid:3) ( B y L )( B z L ) (36)which is proportional to the amplitude of the field perturbations. Comparison with the tracking simulation results isfinally shown in fig. 8 where one obtained good agreement.4
4. Parametric scan of energy and misalignment errors
One objective of the above developed formalism is to allow fast and reliable parametric studies of the impact of thefield imperfections on the systematic errors for the EDM measurement. As shown earlier, the approach which, for themoment being, relies on computation of the averages on the closed orbit, yielded results in good agreement with theBMAD Runge Kutta tracking simulations. As an instructive exercise, we vary simultaneously the beam energy in thevicinity of the magic one as well as the vertical misalignment of one quadrupole and compute the radial and verticallinear spin build-up simultaneously. The radial spin build-up is particularly useful as a tool to probe the deviation ofthe particle from the magic energy and can help the feedback system to find the optimum condition to freeze the spin[40]: Such a feedback system will measure the radial polarization with a polarimeter and rotate the spin vector backto the longitudinal direction by acting for example on the RF frequency and/or adding a small vertical magnetic field(or both to adjust the radial spin of both the CW and the CCW rotating beams).From what preceded, the linear build-up rates of the spin with respect to the momentum vector at the location ofthe polarimeter are given by Eq. (21): ∂S y ∂t = − (cid:104) Ω x (cid:105) + (cid:104) Ω z (cid:105) (cid:68) ˜Ω y (cid:69) − (cid:104) Ω y (cid:105) (cid:68) ˜Ω z (cid:69) + (cid:68) (Ω z − (cid:104) Ω z (cid:105) ) ˜Ω y (cid:69) ∂S x ∂t = (cid:104) Ω y (cid:105) + (cid:104) Ω z (cid:105) (cid:68) ˜Ω x (cid:69) − (cid:104) Ω x (cid:105) (cid:68) ˜Ω z (cid:69) + (cid:68) (Ω z − (cid:104) Ω z (cid:105) ) ˜Ω x (cid:69) The latter are computed by making use of Eq. (8) and the contour lines for both quantities are simultaneouslydisplayed in fig. 9. As expected, the radial spin build-up is more important than the vertical one and is mainlydependent on the deviation from the magic energy as given by (cid:104) Ω y,disp (cid:105) ≈ ( − . · Hz) δ (see Eq. (C11)). Inaddition, the effect of the quadrupole misalignment on the radial spin starts to play a role for larger misalignmenterrors and is due to a mixing between the first order and the second order effects. In particular, even for a beaminitially injected at the magic energy, a radial spin component will be generated if misalignment is present since thelatter alters the magic energy within the electrostatic elements.For the vertical spin, the linear build-up is mainly due to the second order effects since no magnetic field imperfectionsare considered for this study; hence (cid:104) Ω x (cid:105) can be neglected here.The boundary of the aimed EDM sensitivity is shown in gray and is of particular interest since it provides anestimate of the level of control required for the beam energy as well as the misalignment error (the counter-rotatingbeams approach is omitted in this discussion): for instance, for a given vertical misalignment error of ∆ y = 100 µ m orless, a control of the linear radial spin build-up to the level below 8 · − rad/s, shall guarantee that the vertical linearbuild-up falls below 1 . − e.cm. VII. CONCLUSION AND COMMENT ON THE NECESSITY OF A FEEDBACK SYSTEM
In this paper, general expressions were derived to evaluate the systematic effects on “magic energy” EDM rings, i.e.the phenomena other than EDM but caused by machine imperfections leading to a vertical spin build-up. This allowsto better understand mechanisms limiting the achievable sensitivity and, hopefully, to define mitigation measures.Several formula were established and benchmarked with selected cases of lattice imperfections. In particular, itappears that the second order approximation based on successive approximations starting from the first order BKMmethod of averages, is very useful to calculate and probe the sources of vertical spin build-up for a nearly frozen spinlattice. Nevertheless, it is clear that under realistic errors, a feedback system is necessary in order to achieve thelinear regime where the averages of the spin precession components are small such as condition (26) holds.The latter is not sufficient as was established later on through tracking simulations. In particular, residual radialmagnetic fields shall be controlled down to 10 aT level to achieve the desired sensitivity of 10 − e.cm. In addition,eliminating the radial spin build-up by means of a feedback system is not a sufficient condition in order to achievethe frozen spin lattice for its vertical component. The reason lies in the fact that a frozen radial spin, when achievedin an imperfect machine, does not guarantee that the beam is at the magic energy. Hence, strict control of machineimperfections which might require a beam-based alignment approach intending to make the beam orbit as planar as5 y [ m ] - . e - - . e - - . e - - . e - - . e - . e - . e - . e - . e - . e - - . e - - . e - - . e - . e - . e - . e - FIG. 9: Contour plot of the radial and vertical linear spin build-up (in units of [rad/s]) as a function of the initialmomentum offset δ and the vertical misalignment of one quadrupole in the ring. The number along with the red andblue lines are the radial and vertical spin build-up, respectively. The gray area defines the boundary of the aimedEDM sensitivity of ± . Acknowledgements
We acknowledge useful discussions with members of the CPEDM and JEDI collaboration, specifically Mike La-mont, Sig Martin, Selcuk Hacı¨omero˘glu, Andreas Lehrach, Yannis Semertzidis, Ed Stephenson, Hans Stroeher andRichard Talman. Special thanks to David Sagan and Yann Dutheil for helping with BMAD and Gianluigi Arduinifor proofreading the manuscript.6
Appendix A: Identities
Let’s assume that Ω i ( t ) is a well defined function that possesses an average value. Ω i ( t ) can be expressed in thefollowing way: Ω i ( t ) = (Ω i ( t ) − (cid:104) Ω i (cid:105) ) + (cid:104) Ω i (cid:105) = ddt ˜Ω i + (cid:104) Ω i (cid:105) (A1)and (cid:90) t dτ Ω i ( τ ) = (cid:104) Ω i (cid:105) t + ˜Ω i ( t ) (A2)Thus, by means of an integration per parts, the following expressions can be simplified: (cid:90) t dτ Ω i ( τ ) τ = (cid:90) t dτ (cid:104) Ω i (cid:105) τ + (cid:90) t dτ ddτ ˜Ω i ( τ ) τ = (cid:104) Ω i (cid:105) t + (cid:104) τ ˜Ω i (cid:105) t − (cid:90) t dτ ˜Ω i = (cid:104) Ω i (cid:105) t + t ˜Ω i ( t ) − (cid:68) ˜Ω i (cid:69) t − (cid:101) ˜Ω i ( t ) (A3)Similarly, one can establish the following identity: (cid:90) t dτ Ω i ( τ ) ˜Ω i ( τ ) = [ ˜Ω i ( t )] (cid:104) Ω i (cid:105) (cid:68) ˜Ω i (cid:69) t + (cid:104) Ω i (cid:105) (cid:101) ˜Ω i ( t ) (A4)Finally, the same operations acting on all the elements of the Matrix Ω yield: Ω = (cid:104) Ω (cid:105) + ddt ˜ Ω (cid:90) t dτ Ω = (cid:104) Ω (cid:105) t + ˜ Ω (cid:90) t dτ Ω ˜Ω = (cid:68) Ω ˜Ω (cid:69) t + (cid:103) Ω ˜Ω (cid:90) t dτ Ω τ = (cid:104) Ω (cid:105) t + t ˜ Ω − (cid:68) ˜ Ω (cid:69) t − (cid:101) ˜ Ω (A5) Appendix B: Second order approximation
Based on Eq. (21), the second order polarization can be written in the Matrix form as follows: ξ ( t ) = (cid:2) + M t + M t (cid:3) ξ (0) (B1)where M and M are the transport matrices for the linear and quadratic polarization build-up respectively,7 M = (cid:104) Ω (cid:105) + (cid:68) Ω ˜ Ω (cid:69) − (cid:68) ˜ Ω (cid:69) (cid:104) Ω (cid:105) = − (cid:104) Ω z (cid:105) + (cid:68) Ω y ˜Ω x (cid:69) − (cid:104) Ω x (cid:105) (cid:68) ˜Ω y (cid:69) (cid:104) Ω y (cid:105) + (cid:68) Ω z ˜Ω x (cid:69) − (cid:104) Ω x (cid:105) (cid:68) ˜Ω z (cid:69) (cid:104) Ω z (cid:105) + (cid:68) Ω x ˜Ω y (cid:69) − (cid:104) Ω y (cid:105) (cid:68) ˜Ω x (cid:69) − (cid:104) Ω x (cid:105) + (cid:68) Ω z ˜Ω y (cid:69) − (cid:104) Ω y (cid:105) (cid:68) ˜Ω z (cid:69) − (cid:104) Ω y (cid:105) + (cid:68) Ω x ˜Ω z (cid:69) − (cid:104) Ω z (cid:105) (cid:68) ˜Ω x (cid:69) (cid:104) Ω x (cid:105) + (cid:68) Ω y ˜Ω z (cid:69) − (cid:104) Ω z (cid:105) (cid:68) ˜Ω y (cid:69) (B2)and M = (cid:104) Ω (cid:105) − (cid:104) Ω y (cid:105) + (cid:104) Ω z (cid:105) (cid:104) Ω x (cid:105) (cid:104) Ω y (cid:105) (cid:104) Ω x (cid:105) (cid:104) Ω z (cid:105) (cid:104) Ω y (cid:105) (cid:104) Ω x (cid:105) − (cid:104) Ω z (cid:105) + (cid:104) Ω x (cid:105) (cid:104) Ω y (cid:105) (cid:104) Ω z (cid:105) (cid:104) Ω z (cid:105) (cid:104) Ω x (cid:105) (cid:104) Ω z (cid:105) (cid:104) Ω y (cid:105) − (cid:104) Ω x (cid:105) + (cid:104) Ω y (cid:105) (B3) Appendix C: Spin precession component simplification
In what follows, we express the vertical spin precession component as a function of the horizontal misalignmenterrors as well as the momentum offset at injection.To begin with, let us write E x ≈ E bx + ( ∂E x /∂x ) x where E bx represents the radial electric field of the ideal lattice, i.e.constant within the electrostatic deflectors and vanishing everywhere else. In addition, making use of the followingrelation between the radial electric field of the ideal lattice and the radius of curvature of the corresponding idealtrajectory: qE bx = − γ m β m ρ mc (C1)the expression of Ω y simplifies toΩ y = qmc (cid:18) G + 1 γ + 1 (cid:19) β z ( E x − x (cid:48) E z ) − qm (cid:18) G + 1 γ (cid:19) B y + qm G (cid:18) − γ (cid:19) y (cid:48) B z + β z cρ + x = qmc (cid:18) G + 1 γ + 1 − γ m β m (cid:19) β z E bx + β z c (cid:18) ρ + x − ρ (cid:19) + qmc (cid:18) G + 1 γ + 1 (cid:19) β z (cid:18) ∂E x ∂x x − x (cid:48) E z (cid:19) − qm (cid:18) G + 1 γ (cid:19) B y + qm G (cid:18) − γ (cid:19) y (cid:48) B z (C2)Furthermore, it can be shown that: K = G + 1 γ + 1 − γ m β m = − γ m + 1 + 1 γ + 1 ; G = 1 β m γ m = − γ m + 1 + 1( γ m + 1) 1[1 + ( γ − γ m ) / ( γ m + 1)]= − γ − γ m ( γ m + 1) + ( γ − γ m ) ( γ m + 1) − ( γ − γ m ) ( γ m + 1) + ... (C3)8Now, recalling that β = p c/ E and E = p c + m c where E is the total energy of the particle, the expression of theLorentz factors as a function of the particle momentum offset from the magic one can be established [27]: β = 1 + ∆ p/p m (cid:104) (1 + ∆ p/p m ) + G (cid:105) / ; γ = (cid:20) G (1 + ∆ p/p m ) (cid:21) / (C4)so that in the paraxial approximation, β z = β x/ρ [(1 + x/ρ ) + x (cid:48) + y (cid:48) ] / ≈ β (C5)Injecting Eq. (C4) into the expression of K and keeping terms up to the second order in ∆ p/p m finally yields: γ m − γ ≈ − G ( G + 1)] / (cid:34) ∆ pp m + 12 (cid:18) ∆ pp m (cid:19) (cid:35) K ≈ − Gγ m ( γ m + 1) (cid:18) ∆ pp m (cid:19) + γ m − − G ( γ m + 1)2 γ m ( γ m + 1) G ( G + 1) (cid:18) ∆ pp m (cid:19) (C6)Recalling that x = x co − ∆ x mis + x β + x D (C7)where the reference trajectory (in the absence of any misalignment errors) corresponds to x co = 0, ∆ x mis representsthe horizontal misalignment errors in the ring, x β the horizontal displacement due to the betatron oscillations (whichwe neglect for the present study since the spin build-up is limited to the closed orbit) and x D is the horizontaldisplacement due to the dispersive effects which is given by x D = Dδ , D being the periodic dispersion function and δ the momentum offset at injection. This is generally referred to as the “non-local dispersion” [27] since it is definedwith respect to the changes in energy at the beginning of the machine. The last step in our analysis is thus to expressthe variation of the momentum offset inside the ring as a function of the momentum offset at injection. Recalling theconservation of the total energy [37]:∆ pp m = δ + qE bx β m cp m x − qE bx / (2 ρ ) + qG/ β m cp m x + qG/ β m cp m y ≈ (cid:20) − Dρ (cid:21) δ − ρ ( x co − ∆ x mis ) (C8)Finally, retaining the relevant terms (and omitting some of the algebra), it can be shown that in the absence of verticalmagnetic fields or longitudinal fields: Ω y = Ω y,disp + Ω y,mis (C9)where Ω y,mis ≈ (cid:20) − βcG (1 + G )( γ m + 1) ρ − βcρ + qmc (cid:18) G + 1 γ m + 1 (cid:19) β ∂E x ∂x (cid:21) ( x co − ∆ x mis ) (C10)Ω y,disp ≈ (cid:20) βcG (1 + G )( γ m + 1) ρ (cid:18) − Dρ (cid:19) − βcρ D + qmc (cid:18) G + 1 γ m + 1 (cid:19) β ∂E x ∂x D (cid:21) δ (C11)and ∂E x ∂x = − E bx ρ = γ m β m mc q ρ if bend g q if quadrupole (C12)9 [1] J. Jaeckel, M. Lamont and C. Vall´e, The quest for new physics with the Physics Beyond Colliders programme, Nat. Phys.16, 393401 (2020).[2] T. Fukuyama, Searching for new physics beyond the Standard Model in electric dipole moment, Int. J. Mod. Phys. 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