Difference of measured proton and He3 EDMs: a reduced systematics test of T-reversal invariance

The excess of matter over anti-matter in the present-day universe suggests the existence of violations of P and T symmetries substantially greater than are required by the “standard model” of elementary particle physics. This provides the leading motivation for measuring EDMs, which must vanish if P and T symmetries are respected.

Ultimate proton EDM measurement has been thought to require simultaneously counter-circulating, frozen-spin 233 MeV protons in a “large” (depending on achievable electric field, and ranging ‘in size up to 800 m circumference) nominal all-electric storage ring. A step toward this goal is the “small” ($\sim$100 m circumference) PTR ring with superimposed electric and magnetic bending fields. Simultaneously counter-circulating frozen spin proton beams is one of the ultimate features that can be demonstrated in PTR.

The need for storage rings with predominantly electric bending is based on the requirement to “freeze the spins” of stored particle beams. This is the first step toward the measurement of electric dipole moments (EDMs) of the circulating particles, as well as for other symmetry violation investigations. Electrons (of either sign) along with the proton and triton baryons, are fundamental particles whose electric EDMs are potentially measurable in all-electric frozen spin storage rings. With superimposed magnetic bending other particles, especially deuterons and helions (He3 nuclei), can have frozen spins.

In all cases the EDM values are so small that suppression of systematic errors makes it essential for the storage ring to support simultaneous counter-circulating beams. With superimposed electric and magnetic bending, constructive bending superposition in one direction implies destructive superposition in the other direction. By itself, this does not preclude the possibility of simultaneously counter-circulating beams—but it does require major difference between the beams, either in particle type or momentum.

In contrast to a “single particle table-top trap”, the essential feature of a storage ring “trap” is that some $10^{10}$ particles can have their spins aligned in a polarized stored beam. As demonstrated by Hempelmann et al. at the COSY lab [12], this is a number of particles large enough for the beam polarization to be detected externally, and fed back to permit external control of the beam polarization. Though the table large enough for any such a “storage ring trap” would be quite large, the level of achievable spin control, though classical, not quantum mechanical, can be comparable to the control of one or a small number of polarized particles in a low energy trap. Precision spin tune control at COSY has been pioneered by Eversmann et al. [13].

In true “globally frozen spin” operation the bunch polarizations remain globally frozen, for example parallel to the circulating beam momentum vector. For all-electric bending this fixes all kinematic quantities uniquely. But with superimposed electric and magnetic bending, the frozen spin condition can be met over bands of E/B ratios, specific to particle type. Furthermore, some experiments can be performed with “pseudo-frozen spins” that appear frozen locally, but not globally.

The leading observable effect of a static particle EDM would be an “out-of-plane” spin precession, orthogonal to MDM-induced “in-plane” horizontal spin precession caused by whatever magnetic and/or electric fields cause the particle orbit to follow a sequence of horizontal circular arcs. With standard model EDM predictions being much smaller than current experimental sensitivities, detection of any particle’s non-zero EDM would signal discovery of New Physics.

A “nominal experimental proton EDM detectability target” has conventionally been defined to be $10^{-29}\,e$ cm. An EDM of this magnitude could help to account for the observed matter/antimatter asymmetry of our universe while, at the same time being plausibly (only one or two orders of magnitude) larger than existing standard model calculations.

Currently the proton EDM upper limit (as inferred indirectly by measuring the Hg atom EDM) is roughly $2.1\times 10^{-25}e\,$cm. [6] The original goal for PTR was to serve as prototype for the nominal, full-scale, all-electric proton EDM ring. It has been possible, however to make the PTR design flexible enough to support quite accurate EDM determinations and to achieve other goals, such as precision measurement of anomalous magnetic moments of light nuclear isotopes.

(Except for synchrotron radiation) first generation electron/positron colliders such as the Cornell Electron Storage Ring (CESR) have demonstrated CPT-symmetry, and robust backward/forward (bf)-symmetry. But the present paper considers only storage rings in which particle spins can be globally frozen. This rules out rings with pure magnetic bending—there are no candidates. One turns, therefore, to electric rings. Of special interest are the 232.8 MeV “nominal all-electric” all-electric proton EDM ring, which has been under discussion for more than a decade. Also of interest (especially at Wilson Lab) could be a (small) all-electric 14.50 MeV $e^{+}/e^{-}$- storage ring in which globally-frozen spin positrons and electrons can counter-circulate simultaneously.

For particles at rest “co-magnetometry” in low energy “table-top particle traps” has been essential. For example, Gabrielse [14] has (with excellent justification) described the measurement of the electron magnetic moment (with 13 decimal point accuracy) as “the standard model’s greatest triumph”, based on the combination of its measurement to such high accuracy and on its agreement with theory to almost the same accuracy.

Especially for the direct measurement of EDMs, storage ring technology with beam pairs that can counter-circulate simultaneously in a storage ring with superimposed electric and magnetic bending is required. In this context the term “mutual co-magnetometry” can be used to apply to “beam type pairings” for which both beams have frozen spins. Usually only one of the beams can be “globally” frozen or “magic”, however.

Surprisingly, counter-circulating beams can follow accurately identical orbits, even when the bending is performed by superimposed electric and magnetic fields. Here we show how this can enable the precise comparison of the spin tunes, and hence MDMs, of different elementary particle types—hence the name “spin tune comparator”. Both beams are assumed polarized and one of the beam type’s MDM may be much better known than the other; for example $e^{+}$ better than $p$, or $e^{-}$ better than $e^{+}$.

To the extent both MDMs are already known, the known ratio can be used to adjust a compensating element to match this ratio. Though EDM measurement is scarcely mentioned in this spin tune comparator section, this remains the goal and spin tune comparator operation will permit Koop-wheel rotation reversal with almost arbitrarily high, frequency domain, accuracy. Within each run a precision EDM measurement (including systematic error contribution) can then be obtained. Koop [15] argues (persuasively) that intentional roll of the Koop wheel can improve spin coherence duration. Extraction of the EDM value then relies on linear interpolation of data taken at negative and positive roll rates to cancel the torque applied to drive the “roll”.

All of the most important measurement methods for ultimate EDM precision can be developed in the PTR ring. Furthermore, one can be more relaxed about parameter tolerance specifications in a small “inexpensive” prototype than in an eight times larger eventual “ideal” ring. One of the tasks for PTR will be to investigate how tight the tolerances have to be, and how easily they can be met. In this spirit, the present section derives quite tight tolerances to be aimed for (but not necessarily achieved) for measuring EDMs in a small ring with accuracy rivaling what can be achieved in the full scale ring. This is work in progress. Unresolved technical issues are discussed in other sections.

Especially for the direct measurement of EDMs, storage ring technology with beam pairs that can counter-circulate simultaneously in a storage ring with superimposed electric and magnetic bending is required. In this context the term “mutual co-magnetometry” can be used to apply to “beam type pairings” for which both beams have frozen spins. Usually only one of the beams can be “globally” frozen or “magic”, however.

Compared to the electron, MDM accuracies of all other particles are inferior by three orders of magnitude or more. A task for a spin-tune-comparator can be to “transfer” some of the electron’s precision to the measurement of other magnetic dipole moments. For example, the positron EDM could perhaps be determined to almost the current accuracy of the electron’s, and the (presumed) equality (in magnitude) of electron and positron MDM’s could be improved by several orders of magnitude.

Another motivation for developing spin tune comparison capability is to study beam-based self-compensation and self-calibration. Accurately known counter-circulating MDM’s can be used to cancel unknown differences between electric and magnetic bending field shapes. “Introspectron” might be an apt task for such a ring—the purpose then being to investigate its own self-consistency. This is not without pure physics interest. With spins always a bit mysterious, confirmation (or not) to high accuracy, that CW and CCW spin tunes are identical could be informative. This possibility is pursued briefly in Appendix E, which suggests investigation of the Aharonov-Anandan phase. One supposes that any gravitational phase shift will be opposite for CW and CCW circulation.

The final section contains currently unresolved technical issues, the most important of which concerns the achievability of strong electric fields surrounded by weak magnetic fields. This is only an issue for the small prototype ring PTR. For an eight times larger ring, for example situated in the AGS or Wilson lab tunnel, electric field strengths much in excess of a million volts per meter are not required (at least for the $p,h$ EDM difference measurement).

Various authors [16][17][18] have pointed out that general relativity (GR) introduces effects that could be measurable at the precision level promised by proposed EDM rings. A paper by László and Zimborás [19] has calculated the GR influence on storage rings designed for MDM measurement, such as the PTR ring under discussion here.

The nominal EDM value can be compared to a general relativistic (GR) out-of-plane precession effect, mimicking an EDM of the order of $10^{-28}\,e$ cm, associated with the storage ring’s orientation in the earth’s magnetic field. Depending on storage ring details, this reliably calculable “background precession” will provide a “standard candle of convenient magnitude” calibration of any EDM measurement.

Other GR references include  [20][21][22][23]. These authors have calculated the GR effect of the earth’s gravitational field on the experiment referred to above as the nominal all-electric version of EDM measurement. They show that the GR effect mimics the EDM effect with a magnitude such that, if mistakenly ascribed to the proton EDM, would produce a spurious proton EDM value of $3\times 10^{-28}\,e$ cm. This is about thirty times greater than the precision anticipated for the ring and not inconsistent with the Orlov, Flanagan, and Semertzidis estimate [18].

Interpretation of gravitational effect as Aharonov-Anandan spin tune shift is discussed in Appendix E.

To aid the discussion of rings with superimposed electric and magnetic bending, (to be abbreviated as “EM-superposition”) fractional bending coefficients $\eta_{E}$ and $\eta_{m}$ can be defined by

$$\eta_{E}=\frac{q}{pc/e}\,\frac{E_{0}r_{0}}{\beta},\ \eta_{M}=\frac{q}{pc/e}\,cB_{0}r_{0},$$

(3.1)

neither of which is necessarily positive. These bending fractions satisfy

$$\eta_{E}+\eta_{M}=1\quad\hbox{and}\quad\frac{\eta_{E}}{\eta_{M}}=\frac{E_{0}/\beta}{cB_{0}}.$$

(3.2)

The “potencies” of magnetic and electric bending are in the ratio $cB_{0}/(E_{0}/\beta)$ because the electric centripetal force in electric field $E_{0}$ is stronger than the centripetal force produced by magnetic field $cB_{0}=E_{0}$ by the factor $1/\beta$, as regards bending charge $q$ onto an orbit with the given radius of curvature $r_{0}$. Also, when expressed in term of spin tunes, the “potencies” of magnetic and electrically induced MDM precessions are in the same ratio as the bending potencies. Defining a “normalized momentum”

$$\widetilde{p}=\frac{pc/e}{q},$$

(3.3)

the fractional bending coefficients can be expressed in terms of “Brho” and “Erho” factors;

$$\eta_{E}=\frac{1}{\widetilde{p}}\,E_{0}r_{0}/\beta=\frac{E_{0}r_{0}/\beta}{E_{0}r_{0}/\beta+cB_{0}r_{0}},\quad\eta_{M}=\frac{1}{\widetilde{p}}\,cB_{0}r_{0}=\frac{cB_{0}r_{0}}{E_{0}r_{0}/\beta+cB_{0}r_{0}}.$$

(3.4)

With superimposed electric and magnetic fields, the combination $E_{0}r_{0}/\beta+cB_{0}r_{0}$ plays the role that $cBr_{0}$ plays for purely magnetic lattices. This imports into EM-superposition the “geometrization of transverse optics”—which is to say “Newtonian optics with lens strengths quantified by focal lengths”. An essential inference from Eqs. (3.1) through (3.4) is that, if forward and backward focusing strengths are in the same ratio as E/M bending strengths then the forward and backward focal lengths are in the same ratio as the normalized $\tilde{p}$ values. This is strictly true for vertical focusing (irrespective of electric and magnetic bending fractions) but true only to a first approximation (i.e. vanishing in the $\eta_{M}=0$ limit) for horizontal focusing. With this limitation, forward and backward focal lengths will be the same, element by element and, for a lattice that is forward/backward symmetric the forward and backward $\beta_{y}$-functions will be identical and the $\beta_{x}$-functions approximately the same.

This issue is discussed in numerical detail in Section 6. It is shown that forward/backward symmetry can be satisfied closely but not quite exactly in the PTR ring. Though it is possible to superimpose electric and magnetic bending with high precision, it is much more difficult to superimpose electric and magnetic lumped quadrupole fields. For this reason, a significant fraction of the present paper is devoted to designing a PTR ring that is stable and robust, using toroidal field shaping electrodes and magnet coils along with “triplet, compromise quadrupoles” (described in detail in Appendix B) that approximately superimpose electric and magnetic lumped quadrupoles in the same ratio as the bending fields. To linearized order, the lattice properties will then be almost identical for forward and backward traveling beams.

In the idealized storage ring to be discussed, the electromagnetic fields are “cylindrical” electric ${\bf E}=E_{0}{\bf\hat{x}}r_{0}/r$ (with $E_{0}<0$) and, superimposed, uniform magnetic field ${\bf B}=B_{0}{\bf\hat{y}}$ (with $B_{0}>0$). The bend radius is $r_{0}>0$. Terminology is useful to specify the relative polarities of electric and magnetic bending: Cases in which both forces cause bending in the same sense will be called “constructive” or “frugal”; Cases in which the electric and magnetic forces subtract will be referred to as “destructive” or “extravagant”.

For a particle with spin circulating in a (horizontal) planar magnetic storage ring, its spin precesses around a vertical axis at a rate proportional to the particle’s anomalous magnetic dipole moment, $G$. For an “ideal Dirac particle” (meaning $G=0$) in a purely magnetic field the spin precesses at the same rate as the momentum—pointing always forward, for example. Conventionally the spin vector’s orientation is specified by the angle $\alpha$ between the spin vector ${\bf S}$ and the particle’s momentum vector ${\bf p}$ (which is tangential, by definition). For such a “not-anomalous” particle the spin-tune $Q_{M}$ (defined to be the number of $2\pi$ spin revolutions per particle revolution) therefore vanishes, in spite of the fact that, in the laboratory, the spin has actually precessed by close to $2\pi$ each turn.

In general, spin 1/2 particles are not ideal Dirac particles; the directions of their spin axes deviate at a rate proportional to their anomalous magnetic moments, $G$, and their spin tunes differ from zero even in a purely magnetic field. Note also, that a laboratory electric field produces a magnetic field in the particle rest frame, so a particle in an all-electric storage ring also has, in general, a non-vanishing spin tune $Q_{E}$. Along with $G$ and $Q$, all of these comments apply equally to the polarization vector of an entire bunch of polarized circulating particles.

By convention, in the BMT-formalism [24][25], the orientation of the spin vector ${\bf S^{\prime}}$ is defined and tracked in the rest frame of the circulating particle, while the electric and magnetic field vectors are expressed in the lab. The spin equation of motion with angular velocity $\omega$ is

$$\frac{d{\bf S^{\prime}}}{dt}=\omega\times{\bf S^{\prime}},$$

(3.5)

with orbit in the horizontal $(x,z)$ plane assumed, where

	$\displaystyle\omega$	$\displaystyle=-\frac{qe}{\gamma mc}\,\bigg{(}\Big{(}G\gamma\Big{)}cB_{0}+\Big{(}\big{(}G-\frac{1}{\gamma^{2}-1}\big{)}\gamma\beta^{2}\Big{)}\frac{E_{0}}{\beta}\bigg{)}\,{\bf\hat{y}}$
		$\displaystyle\equiv-\frac{qe}{\gamma mc}\,\bigg{(}(Q_{M})cB_{0}+(Q_{E})\,E_{0}/\beta\bigg{)}\,{\bf\hat{y}},$		(3.6)

With $\Delta\alpha$ being the in-plane spin precession advance relative to the beam direction each turn, the “spin tune” is defined to be $Q_{s}=\Delta\alpha/(2\pi)$. Spin tunes in purely electric and purely magnetic rings are given by

$$Q_{E}=G\gamma-\frac{G+1}{\gamma},\quad Q_{M}=G\gamma,$$

(3.7)

where $\gamma$ is the usual relativistic factor. Note that the sign of $Q_{M}$ is the same as the sign of $G$, which is positive for protons—proton spins precess more rapidly than their momenta in magnetic fields. Deuteron spins, with $G$ negative, lag their momenta in magnetic fields. With $G$ positive, $Q_{E}$ increases from -1 at zero velocity, eventually switching sign at the “magic” velocity where the spins in an all-electric ring are “frozen” (relative to the beam direction). When a particle spin has precessed through 2$\pi$ in the rest frame it has also completed one full revolution cycle from a laboratory point of view; so the spin-tune is a frame invariant quantity.

For brevity one can discuss just electrons (of either sign), protons($p$), deuterons($d$), tritons($t$), and helions($h$); or even just $p$, $d$, and $h$, based on the consideration that (with the exception of helion beam) most of the apparatus, and all of the technology, needed for their EDM measurement is presently available at COSY laboratory in Juelich, Germany. Polarized helion beams of appropriate energy will soon be available at BNL.

The circulation direction of a so-called “master beam” (of whatever charge $q_{1}$) is assumed to be CW or, equivalently, $p_{1}>0$. A secondary beam charge $q_{2}$ is allowed to have either sign, and either CW or CCW circulation direction.

Ideally both beam polarizations would be frozen “globally” (meaning spin tune $Q_{S}$ is zero and the angle $\alpha$ between polarization vector and momentum is constant everywhere around the ring). (Somewhat weaker) “doubly-frozen” can (and will) be taken to mean that a “primary beam” locked to $Q_{S}=0$, circulates concurrently with a “secondary” beam that is “pseudo-frozen”, meaning the spin tune is locked to an unambiguous, exact, rational fraction (other than zero). Only if the secondary beam rational fraction is exactly zero, would the terminology “doubly-magic” be legitimate.

Such particle pairings are expected to make direct MDM or EDM difference measurements of unprecedented precision possible. For any arbitrary pairing of particle types $(p,p),\ (p,d),\ (p,e+)$, etc., continua of such doubly-frozen pairings are guaranteed. This is taken to imply that both beam polarizations can be phase-locked, which is one of the requirements for mutual co-magnetometry.

A design particle has mass $m>0$ and charge $qe$, with electron charge $e>0$ and $q=\pm 1$ (or some other integer). These values produce circular motion with radius $r_{0}>0$, and velocity ${\bf v}=v{\bf\hat{z}}$, where the motion is CW (clockwise) for $v>0$ or CCW for $v<0$. With $0<\theta<2\pi$ being the cylindrical particle position coordinate around the ring, the angular velocity is $d\theta/dt=v/r_{0}$.

(In MKS units) $qeE_{0}$ and $qe\beta cB_{0}$ are commensurate forces, with the magnetic force relatively weakened by a factor $\beta=v/c$ because the magnetic Lorentz force is $qe{\bf v}\times{\bf B}$. By convention $e$ is the absolute value of the electron charge. When $e$ appears explicitly, usually as a denominator factor, its purpose in MKS formulas is to allow energy factors to be expressed in electron volts (eV) in formulas for which the MKS unit of energy is the joule. Newton’s formula for radius $r_{0}$ circular motion, expressed in terms of momentum and velocity (rather than just velocity, in order to be relativistically valid) can be expressed using the total force per unit charge in the form

$$\beta\frac{pc}{e}=\Big{(}E_{0}+c\beta B_{0}\Big{)}\,qr_{0},$$

(3.8)

Coming from the cross-product Lorentz magnetic force, the factor $q\beta cB_{0}$ is negative for backward-traveling orbits because the $\beta$ factor is negative.

A “master” or primary beam travels in the “forward”, CW direction. For the secondary beam, the $\beta$ factor can have either sign. For $q=1$ and $E_{0}=0$, formula (3.8) reduces to a standard accelerator physics “cB-rho=pc/e” formula. For $E_{0}\neq 0$ the formula incorporates the relative “bending effectiveness” of $E_{0}/\beta$ compared to $cB_{0}$. As well as fixing the bend radius $r_{0}$, this fixes the magnitudes of the electric and magnetic bend field values $E_{0}$ and $B_{0}$. To begin, we assume the parameters of a frozen spin “master”, charge $qe$, particle beam have already been established, including the signs of the electric and magnetic fields consistent with $\beta_{1}>0$ and $p_{1}>0$. In general, beams can be traveling either CW or CCW. For a CCW beam both $p$ and $\beta$ have reversed signs, with the effect that the electric force is unchanged, but the magnetic force is reversed. The $\beta$ velocity factor can be expressed as

$$\beta=\frac{pc/e}{\sqrt{(pc/e)^{2}+(mc^{2}/e)^{2}}}.$$

(3.9)

Introducing the abbreviations,

$$\mathcal{E}=qE_{0}r_{0},\quad\hbox{and}\quad\mathcal{B}=qcB_{0}r_{0},$$

(3.10)

it has been shown in the companion paper [1] that all circular orbits satisfy the equation

$$p_{m}^{4}-2cB_{0}(qr_{0})p_{m}^{3}+(\mathcal{B}^{2}-\mathcal{E}^{2})p_{m}^{2}-\mathcal{E}^{2}m^{2}=0,$$

(3.11)

where $p_{m}$ is the momentum of a particle of mass $m$.

Curiously, expressed as a quartic equation for $p_{m}$, the absence of a term linear in $p_{m}$ causes this equation to be “independent of $r_{0}$” (or rather of the product “$qr_{0}$”) in the sense that the product factor $(qr_{0})$ can be chosen arbitrarily without influencing any essential implications derivable from the equation. This and other properties can be confirmed by pure algebraic reasoning, based on the missing linear in $p_{m}$ factor, or by explicit partially-numerical factorization of the left hand side into the product of four factors linear in $p_{m}$.

Independence of the $qr_{0}$ has huge importance for planning any EDM ring prototype. In the present paper, Tables  2, 3, 5, and 6 give parameters for rings of bending radius, respectively, 11.0 m, 50.0 m, 95.5 m, and 85.0 m. In every case, in spite of the seeming complexity of these tables, $r_{0}$ can be changed arbitrarily without difficulty. The electric and magnetic fields need only to be scaled inversely to match the kinematic parameters to the bending radius. In this sense the 11.0 m PTR ring is a faithful prototype for the eight times larger “nominal full scale” rings.

These considerations have removed some, but not all of the sign ambiguities introduced by the quadratic substitutions used in the derivation of Eq. (3.11). The electric field can still be reversed, thereby doubling the set of solutions of the equation. Note that this change cannot be compensated by switching the sign of $q$, which also reverses the magnetic bending.

It is important also to realize, in cases where efficient polarimetry is available, that the sign of $B$ (or $E)$ can be reversed or reset to high accuracy without the need even to measure $B$ (or $E$). This feature, exploiting the constancy of electron MDMs, has enabled the masses of all resonances discovered in electron/positron colliders to be determined with such high accuracy.

The combined field spin tune $Q_{s}$ can be expressed in terms of the fractional bending coefficients;

$$Q_{S}=\eta_{E}Q_{E}+\eta_{M}Q_{M}.$$

(3.12)

Superimposed electric and magnetic bending permits beam spins to be frozen “frugally”; i.e. with a ring smaller than would be required for all-electric bending; for spin tune $Q_{S}$ to vanish requires

$$Q_{S}=\eta_{{}_{E}}Q_{E}+(1-\eta_{{}_{E}})Q_{M}=0.$$

(3.13)

Solving for $\eta_{{}_{E}}$ and $\eta_{{}_{M}}$,

$$\eta_{{}_{E}}=\frac{G\gamma^{2}}{G+1},\quad\eta_{{}_{M}}=\frac{1+G(1-\gamma^{2})}{G+1}=\frac{1-G\beta^{2}\gamma^{2}}{G+1}.$$

(3.14)

For example, with proton anomalous magnetic moment $G_{p}=1.7928474$, trying $\gamma=1.25$, we obtain $\eta_{{}_{E}}=1.000$ which agrees with the known proton 233 Mev kinetic energy value in an all-electric ring. For protons in the non-relativistic limit, $\gamma\approx 1$ and $\eta^{\rm NR}_{E}\approx 2/3$.

The electric/magnetic field ratio for the primary beam (with anomalous moment $G_{1}$) to be magic is

$$\frac{\eta_{E}}{\eta_{M}}=\frac{E_{0}/\beta}{cB_{0}}=\frac{G_{1}\gamma_{1}^{2}}{1-G_{1}\beta_{1}^{2}\gamma_{1}^{2}}.$$

(3.15)

For given $\beta_{1}$, along with this equation and the required bend radius $r_{0}$, this fixes the electric and magnetic fields to the unique values that globally freeze the primary beam spins. With $1\rightarrow 2$ subscript replacement, the same frozen beam formulas apply to the secondary beam; note, though, that the $\beta$ factor has opposite sign. To be “doubly-magic” both beams must satisfy this relation. At this time the only known dual solutions are $(p,p)$, $(p,h)$, and $(p,e^{+})$.

Responsible planning for an eventual “nominal all-electric” all-electric EDM storage ring with simultaneously counter-circulating $\mathcal{E}=232.8$ MeV frozen spin proton beams requires the construction of a prototype ring (PTR). To maintain flexibility the PTR lattice design initially is “scale-invariant” and site independent. But, for numerical examples: $\mathcal{C}$=102.5 m, roughly 1/8 scale relative to the nominal all-electric ring. Except for construction itself, PTR needs only present day technology and apparatus available in the COSY facility.

Though planned initially as a prototype, and constrained by cost minimization compromised performance, PTR will, itself, be capable of the performance promised in the title of the present paper. This is largely the result of improved understanding of the EDM measurement capability improvement produced by spin tune phase-locking, understanding the BSM power of measuring the difference of proton and helion EDMs, and recognizing that “quadrupole free” toroidal focusing will assure long spin coherence time, especially with betatron phase space “mixing” provided by running on the “difference resonance”.

There is a substantial literature describing the physical importance of measuring EDMs of elementary particles. The most recent comprehensive report is due to the CPEDM collaboration, based mainly on experience of the JEDI group using the COSY ring in Juelich, Germany, and described in a CERN Yellow Report [6] (referred to here as “CYR”) feasibility study for a storage ring to search for EDMs of charged particles.

Measurement of the proton EDM relies on the development of a storage ring with predominantly electric bending, capable of storing simultaneously counter-circulating frozen spin proton beams. A conclusion of the CYR study is that a low energy prototype EDM ring, here referred to as “PTR”, as economical as practical, is needed to investigate problematical issues.

The Phase I goals for PTR are, first, to store an intense (not necessarily polarized) proton beam and, second, to store a simultaneous counter-circulating beam. Phase II will superimpose magnetic bending in order to store and freeze an intense frozen-spin 45 MeV proton beam.

There is considerable overlap between Chapter 7 of the CYR report and the present paper. Figure 3 is a perspective mock-up of one sector of PTR copied from that report. Though the PTR superperiodicity is $N_{\rm super}=4$, it is sufficient for most purposes to display just one octant. The ring is also forward/backward symmetric. There have been substantial lattice design changes subsequent to the CYR manuscript commit date. The currently revised design is exhibited in Figure 2.

Quadrupoles are shown but, as mentioned previously, they are intended to be as weak as possible, consistent with their main roles, which are to enable operational procedures requiring large changes in ring optics, and to compensate for any unexpected systematic optical deficiencies associated with the fabrication of the toroidal electrodes.

Based on much the same optics that has enabled scanning transmission electron microscopy (TEM and STEM) of such spectacularly acute spatial resolution, toroidal focusing can be expected (for the same reason) to provide exceptionally long spin coherence times (SCT) for polarized beams. This can be expected to enable experimentation into the role played by particle spin in time-reversed situations.

The decision to rely entirely on toroidal focusing, carries with it a significant operational penalty resulting from the fact that, once fabricated, no smooth way of tuning the toroidal focusing $m$ value has been established. Theoretical discussion of these issues in detail is contained in companion paper [2]. Cancellation of spin polarization decoherence is greatly enhanced by toroidal focusing. As mentioned at the start of this paper there is considerable overlap of that paper with this paper. Here we list only essential references from that paper: [26], [27], [29], [30], [31], [32], [33], [34], [35]. In combination, along with other technical details, these papers explain how the requirement of conformal invariance is satisfied by the PTR design in the present paper.

The dominant bending and focusing is performed by the red and blue toroidal sectors shown in Figure 2. The $m$-values are displaced slightly, with $m_{\rm blue}\leqq m_{\rm red}$. The purpose for this is to preserve flexibility while the focusing design is being refined. At the time of final construction it will be preferable for red and blue sectors to be identical. Otherwise, every sector bend has identical superimposed electric and magnetic bending. As fabricated, all magnet bends will probably be powered in series (with possible small shunt compensation) and all electrode voltages only slightly adjustable (for closed orbit steering).

In spite of the small number of adjustable parameters, there is sufficient flexibility to tune PTR to some of the operational MODES mentioned in previous reports. Computationally the optical design has been performed using a MAPLE-BSM code, where “BSM” is an acronym for “Belt and Suspenders, Modifiable”. The lumped quadrupoles can provide the somewhat stronger (separated function) focusing needed for robust ring commissioning operations. However, a COLLIDING BEAM mode described in previous reports cannot be achieved. This is because backward/forward symmetric “low beta squeeze” (for increased luminosity) is impractical with superimposed electric and magnetic bending.

As well as violating forward/backward lattice function symmetry, lumped quadrupoles inevitably introduce chromaticity into ring optics. Customarily in storage rings this chromaticity is compensated by one, or typically, at least two, families of sextupoles. Such sextupoles prove to be acceptable for unpolarized beams, but their presence impairs the ring performance for polarized beams. Their presence is the leading cause of spin precession decoherence, which limits the spin coherence time (SCT). For ultimate EDM measurement precision, it will therefore be essential for the lumped quadrupoles to be as weak as possible.

As mentioned previously, the various lattice optical designs are scale invariant and site-independent. But, to make lattice parameters and plots more intuitive, the ring circumference is taken to be 102.5 m, matched exactly to the ring circumferences of the proposed bunch accumulator (BA) described below. As fixed by this circumference, in all subsequent lattice function plots, the horizontal axis coordinate is the longitudinal position coordinate $s$.

As regards the orientation of the beam polarization, it is essential to distinguish between “in-plane” and “out-of-plane” orientations, where “the plane” refers to the ring beam plane, which is presumed to be horizontal. In-plane precession, is routinely induced by ideal (rest frame) magnetic fields acting on beam particle magnetic dipole moments (MDMs). Ideally, out-of-plane precession can be induced only by P- and/or T-violating symmetry violating torques due to electric fields acting on particle EDMs. But, in practice, the inevitable existence of unintentional magnetic fields acting on particle MDMs can also induce out-of-plane precession. The average radial magnetic field $\langle B_{r}\rangle$ is expected to be the dominant source of systematic error in the determination of particle EDMs.

The leading strategy for reducing this source of systematic error is to produce simultaneously counter-circulating frozen spin beams on “identical” orbits. This condition is to be achieved by adjusting local vertical beam deflection corrections to cause the counter-circulting orbits to be identical. This cancels the dominant systematic error source $\langle B_{r}\rangle$ by canceling the average out-of-plane (vertical) orbit separation. We refer to this capability as “self-magnetometry”. The precision with which the orbits can be matched vertically depends on the precisions of the beam position monitors (BPMs) that measure the vertical beam positions, and on the ring lattice sensitivity to the magnetic field errors causing the orbits to be vertically imperfect.

Assuming both beam spins are frozen, at least the “primary” beam$-1$ will, by convention, be globally frozen; spin tune $Q_{s1}=0$. Ideally both beams would have $Q_{s}=0$ but, ordinarily, the “secondary” beam-2 will be only locally frozen; $Q_{s2}$ exactly equal to a rational fraction other than $0/1$. In this condition both beam polarizations can be phase-locked, allowing both beam spin tunes to be set and re-set with frequency domain precision, meaning that synchronism can be maintained for runs of arbitrary duration. Since the RF frequency can also be restored to arbitrarily high precision, conditions can be set and re-set repeatedly, without depending upon high precision measurement to the electric and magnetic bend fields. This allows, for example, the magnetic bending field to be reversed with high precision, as would be required to interchange CW and CCW beams. This capability can be referred to as stabilizing all fields by phase locking both revolution frequencies and both beam polarizations, using their own MDMs as “magnetometric gyroscopes”.

Figure 5 displays a conjectured placement of PTR in the polarized beam injection line of the (currently under review) BNL Center for Linac Isotope Production (CLIP), including the proposed light Ion linac (LIL). The anticipated polarized injected beam currents from LIL are a factor of roughly 6000 times larger than from the COSY cyclotron described previously. This is comparable with the 10,000 times increase in bunch current promised by the BA bunch accumulator shown in Figure 6.

Furthermore CLIP promises a gourmet menu of polarized nuclear isotopes, including most, if not all, shown on Koop’s menu, Figure 1. Most exciting of all the choices is the combination of protons, $p$, and helions, $h$. It is this pairing that can provide the most precise test of BSM physics that is likely to be achievable in the quite near future. Furthermore the range of available beam energies is luxurious enough to require no detailed discussion.

As mentioned previously, mutual co-magnetometry with PTR can be expected to permit convenient and precise MDM measurements for any nuclear isotope having lifetime in excess of a minute or so. Because of its concentration on the proton/helion EDM difference measurement, this capability is not discussed further in the present paper.

Transfer matrices from one symmetry point to the next of a storage ring with superperiodicity $N_{s}$ have the form

$${\bf M_{1}}=\begin{pmatrix}\cos\mu&\beta\sin\mu\\ -\frac{\sin\mu}{\beta}&\cos\mu\end{pmatrix}\equiv\begin{pmatrix}C&\beta S\\ -\frac{S}{\beta}&C\end{pmatrix},$$

(5.1)

where $\mu$ can stand for $\mu_{x}$ or $\mu_{y}$, $\beta$ can stand for $\beta_{x}$ or $\beta_{y}$ and similarly for $C\equiv\cos\mu$ and $S\equiv\sin\mu$. Squaring ${\bf M_{1}}$ and applying trigonometric identities produces

$${\bf M_{2}}\equiv{\bf M_{1}^{2}}=\begin{pmatrix}\cos 2\mu&\beta\sin 2\mu\\ -\frac{\sin 2\mu}{\beta}&\cos 2\mu\end{pmatrix}.$$

(5.2)

Squaring again and applying trigonometric identities produces

$${\bf M_{4}}\equiv{\bf M_{1}^{4}}=\begin{pmatrix}\cos 4\mu&\beta\sin 4\mu\\ -\frac{\sin 4\mu}{\beta}&\cos 4\mu\end{pmatrix}.$$

(5.3)

By applying these formulas separately to horizontal, $x$, and vertical, $y$ transverse coordinates produces

$${\bf M^{x}_{4}}\equiv{{\bf M^{x}_{1}}^{4}}=\begin{pmatrix}\cos 4\mu_{x}&\beta_{x}\sin 4\mu_{x}\\ \frac{\sin 4\mu_{x}}{\beta_{x}}&\cos 4\mu_{x}\end{pmatrix}\quad\hbox{and}\quad{\bf M^{y}_{4}}\equiv{{\bf M^{y}_{1}}^{4}}=\begin{pmatrix}\cos 4\mu_{y}&\beta_{y}\sin 4\mu_{y}\\ -\frac{\sin 4\mu_{y}}{\beta_{y}}&\cos 4\mu_{y}\end{pmatrix}.$$

(5.4)

What makes these formulas fascinating is that the parameters $\beta_{x}$ and $\beta_{y}$ can be chosen arbitrarily, and not necessarily equal. This is important for maintaining conformal mapping validity with a substantial fraction of the horizontal focusing being “geometric” which does not contribute to vertical focusing. Then, by constraining the betatron phase advances $\mu_{x}$ and $\mu_{y}$, for example as

$$\mu_{x}=\mu_{y}+\frac{\pi}{2},$$

(5.5)

one obtains simpler relations. In this case

$$4\mu_{x}=4\mu_{y}+2\pi\overset{\rm eff.}{\ =\ }4\mu_{y},$$

(5.6)

since increasing an angle by $2\pi$ has no detectable effect. In this paper lattice design meeting this constraint is referred to as “ideal, thick lens toroidal focusing”. Even simpler for tuning the PTR ring would be the choice $\mu_{x}\approx\pi/2$, in which case

$${\bf M^{x}_{4}}\approx\begin{pmatrix}1&\beta_{x}\sin 4\mu_{x}\\ -\frac{\sin 4\mu_{x}}{\beta_{x}}&1\end{pmatrix}\quad\hbox{and}\quad{\bf M^{y}_{4}}=\begin{pmatrix}1&\beta_{y}\sin 4\mu_{y}\\ -\frac{\sin 4\mu_{y}}{\beta_{y}}&1\end{pmatrix},$$

(5.7)

which are close to identity matrices which, in linear approximation, describe drifts of length $\beta_{x}\sin 4\mu_{x}$ and $\beta_{y}\sin 4\mu_{y}$ with (small) 2,1 elements representing weak net focusing). $\beta_{x}$ and $\beta_{y}$ are independently adjustable, and all matrix elements are simply expressible in terms of the two parameters, $C$ and $S$, defined in Eq. (5.1) and satisfying the condition $S^{2}+C^{2}=1$.

Currently the PTR lattice design shown in Figure 2 applies these formulas only up to ${\bf M^{x}_{2}}$ and ${\bf M^{y}_{2}}$. For operational flexibility, the ring optics is then controllable by the triplets (located at symmetry points) and masquerading as superimposed electric/magnetic quadrupoles. In this sense the ring behavior will resemble a separated function FODO lattice with free, arbitrarily strong, $F$ and $D$ focusing strength parameters. (All of this has to be consistent with $F$ and $D$ being turned off as nearly as possible during actual EDM measurement.)

The content of this paper has been limited to this point, mainly to resolving issues identified in the (pre-Covid) CERN Yellow Report. “Covid-enforced leisure” has resulted in the revolutionary promotion of PTR from “EDM prototype ring” to “ambitious research project” as explained so far and in the rest of the paper.

All-electric bending fields exist in the tall slender gaps between inner and outer, almost-plane-vertically, curved-horizontally, electrodes. The radial electric field dependence is

$$E=E_{r}\sim\frac{-1}{r^{1+m}},$$

(5.8)

Our convention has the electric field $E_{0}$ being negative since the nominal beam charge is positive and the force is centripetal. The corresponding electric potential parameterization specializes to inverse power law dependence on the bend radius $r$ with the electric potential defined to vanish on the design orbit; expressed as power series in $\xi=x/r$, it is given by

	$\displaystyle V(r)$	$\displaystyle=\frac{E_{0}r_{0}}{m}\,\bigg{(}(1-\xi)^{m}-1\bigg{)}$		(5.9)
		$\displaystyle=-E_{0}r_{0}\bigg{(}\xi+\frac{1-m}{2}\xi^{2}+\frac{(1-m)(2-m)}{6}\xi^{3}\dots\bigg{)}.$

This formula simplifies spectacularly for the Kepler $m$=1 case. As it happens, for relativistic particles, $m=1$ also marks the edge of a region to which the term “toroidal optics” applies. Beyond this edge orbits spiral in toward the center. The other edge is marked by the $m=0$ “cylindrical edge”; beyond this edge vertical orbits diverge to infinity. These edges are defined more accurately below by Eqs. (5.12).

For focusing index in the (approximate) range $0<m<1$ linearized horizontal evolution of initial displacement vector ${\bf V_{x0}}$ through a thick bending and focusing element element of length $w$ is given by

$${\bf V_{x1}}={\bf M_{4}}{\bf V_{x0}}=\begin{pmatrix}c_{x}&s_{x}&0&d_{x}N_{t}\\ -k_{x}^{2}s_{x}&c_{x}&0&s_{x}N_{t}/r_{0}\\ -s_{x}N_{t}/r_{0}&-d_{x}N_{t}&1&-N_{t}^{2}\\ 0&0&0&1\end{pmatrix}{\bf V_{x0}},\quad\hbox{where}\quad{\bf V_{x0}}=\begin{pmatrix}x_{0}\\ \theta_{x0}\\ s\\ \delta p/p\end{pmatrix}$$

(5.10)

$${\bf V_{y1}}={\bf M_{2}}{\bf V_{y0}}=\begin{pmatrix}c_{y}&s_{y}\\ -k_{y}^{2}s_{y}&c_{y}\end{pmatrix}{\bf V_{y0}},\quad\hbox{where}\quad{\bf V_{y0}}=\begin{pmatrix}y_{0}\\ \theta_{y0}\end{pmatrix}.$$

(5.11)

The parameters are given by

$$k_{x}=\frac{1}{r_{0}}\sqrt{1-m+\frac{\eta_{E}}{\gamma^{2}}},\quad k_{y}=\frac{1}{r_{0}}\sqrt{m},\quad N_{t}=1+\frac{\eta_{E}}{\gamma^{2}},$$

(5.12)

where $N_{t}$ accounts for time dilation. The transfer matrix components for elements of length $L$ are expressed in terms of expressions

$$c_{x}=\cos(k_{x}L),\ s_{x}=\sin(k_{x}L),\ d_{x}=\frac{1-c_{x}}{r_{0}k_{x}^{2}},\ t_{d}=\frac{k_{x}L-s_{x}}{r_{0}^{2}k_{x}^{2}},\\ c_{y}=\cos(k_{y}L),\ s_{y}=\sin(k_{y}L).$$

(5.13)

Due originally to Wollnik [36], but here expressed in more conventional modern notation, these formulas have been copied from Lebedev [37], except for having been further generalized to allow superimposed electric and magnetic bending. ³³3Subtle detail: ring tunes estimated from transfer matrices based on accepting large values of $L$ in advance in Eqs. (5.13) may be seriously incorrect due to incorrect aliasing with thick element focusing. Rearranging Eqs.4.6 from reference [1], and solving for $\tan\psi$ produces an expression for phase $\psi(s)$ in terms of matrix elements evaluated locally, for example at arbitrary origin; $$\psi(s)=\tan^{-1}\,\frac{{\bf M}_{1,2}(0,s)}{\beta_{0}{\bf M}_{1,1}(0,s)-\alpha_{0}{\bf M}_{1,2}(0,s)}.$$ (5.14) Like all inverse trigonometric formulas, this equation has multiple solutions. But, with the .sxf granularity being required to be fine enough, one can (in principle) sequentially obtain unique phases. With $\psi(s)$ starting from $\psi(0)=0$, as $s$ increases from $s_{i}$ to $s_{i+1}$ there is a unique solution of Eq. (5.14), $\psi(s_{i})\geq\psi(s_{i-1})$ such that the function $\psi(s)$ increases monotonically, as required. Because the interval from $s_{i=1}$ to $s_{i}$ is non-zero, $\psi$ can, superficially, advance discontinuously; the correct solution is the least discontinuous. The same calculation has to be done for both the $x$ and $y$ betatron sectors. Even though, theoretically, the phase advances monotonically, numerical errors can cause Eq. (5.14) to give local phase decrease. Requiring betatron phase $\psi(s)$ to increase monotonically (as shown most satisfyingly in Figure 7) can cause $\beta(s)$ shape variation to seem curious. This behavior was encountered especially while producing Figure 11. This issue is pursued further in Appendix E.

This accounts for the appearance of $\eta_{E}$ in Eqs. (5.12). (Recall that $\eta_{E}+\eta_{M}=1$, noting that one or the other of $\eta_{E}$ and $\eta_{M}$ can be negative, in which case the centripetal forces combine “destructively”. But the fractions must sum to 1 to produce the correct $E/B$ bend ratio.)

Operating on orbit displacement vector ${\bf V}$, ${\bf MV}(L)$ describes, to linear approximation, how ${\bf V}$ has evolved after propogation through bend element of length $L$. For pure magnetic bending $\eta_{E}=0$. For partial electric bending the $1+\eta_{E}/\gamma_{0}^{2}$ combination, which approaches 1+$\eta_{E}$ in the nonrelativistic limit, is a velocity dependent time dilation correction factor which modifies dilated time $t_{d}$ as well as the “geometric focusing” supplied by the curvature.

Notice that there is dependence, in Eqns. (5.12), on bending radius $R_{0}$ in both $k_{x}$ and $k_{y}$ and dependence on $\eta_{E}$ in $k_{x}$, though not in $k_{y}$. In the present paper the absence of dependence of $k_{y}$ on $\eta_{E}$ is referred to as “vertical only, geometricization”. This means that vertical focusing, and therefore $\beta_{y}$, is independent of beam momentum (provided the bending force matches the design ring radius). But horizontal focusing depends on beam momentum (or, rather, momentum/charge) through its dependence of $k_{x}$ on $\eta_{E}$.

In the case of doubly-frozen or doubly-magic counter-circulating beams, the vertical focusing is the same for both beams, independent of beam circulation direction. But the horizontal lattice optics depends on beam momentum and is different for counter-rotating beams. Fortunately, for motion in a purely horizontal plane, both spin tunes $Q_{x}$ and $Q_{y}$, as defined in Eqs. (3.7), are “constants of the motion”, independent of horizontal betatron oscillation amplitude to all orders. By symmetry, the spin tunes are independent of vertical betatron oscillation amplitude, but only to linear order. This makes it important, therefore, for $\beta_{y}(s)$ to be exactly independent of beam circulation direction, as it is. (This comment is pursued in Appendix C which discusses the so-called “hybrid option” for ring focusing.)

The matrix ${\bf M_{2}}$ has determinant 1, which is the only requirement for a 2x2 matrix to be symplectic. Block diagonalized into 2x2 matrices, the blocks on the main diagonal of ${\bf M_{4}}$ are individually symplectic, and the determinant of the blocks on the other diagonal vanish. Taken together, these conditions show that ${\bf M_{4}}$ is also “symplectic”.

The quotation marks here are intended to serve as a warning that, iterated indefinitely, ${\bf M_{4}}$ is actually divergent. The lower two rows of ${\bf M_{4}}$ include “chromatic” i.e. momentum-dependent effects—primarily dispersion and time-of-flight. Another idiosyncrasy of accelerator formalism is that (based on a fast-slow approximation) these chromatic effects are counted as “second” rather than “first” order effects. This is related to the fact that it is only the orbit shape that can be said to be stable for all time. (No meaning whatsoever is attached to orbit position as a function of time.)

The important feature of conformally-mapped toroidal optics is that “everything of importance” electrodes, equipotentials, on and off-momentum closed orbits remain perfect circles. The divergent behavior of iterating ${\bf M_{4}}$, represents the longitudinal beam spreading associated with the spread of particle velocities. So a bunched beam will gradually “de-bunch”, eventually becoming a uniform “coasting beam. As established by McMillen and Veksler, this debunching can be prevented by using an RF cavity to capture all (or most) of the particles into stable longitudinal “buckets”.

Parameters and lattice function values for the lattice design illustrated in Figure 2 are given in Table 1. Included in this table are parameters appropriate for Figure 6, which shows the positioning of PTR and “BA” which is a “bunch accumulator ring” needed to prepare polarized beam bunches for injection into PTR.

Figure 7 illustrates the result of lattice tuning in a storage ring with “no lumped quadrupoles”. The quotation marks here acknowledge that PTR will, in fact, have lumped quadrupoles. But their strengths will be adjustable such that (under unrealistically ideal conditions) their strengths will be tuned to exactly zero. The tuned-up situation is shown in the middle row of the figure; the upper/lower rows represent varying $m_{\rm nom}$ (where $m_{\rm nom}$ is another name for the $m$ parameter appearing in Eqs. (5.12)) below/above the tuned-up condition. For fixed ring geometry, quadrupole free, toroidal focusing required for long term stability in both horizontal and vertical planes depends on the value of this single parameter $m_{\rm nom}$.

From the point of view of operational practicality, this behavior is problematical in two ways; once constructed, neither the ring geometry nor the value of $m_{\rm nom}$ can be tuned conveniently. One can contemplate altering $m_{\rm nom}$ by using a smooth mechanical way of adjusting the vertical curvature of the electrodes producing the electric field—the horizontal curvature is fixed by the constant bending radius constraint—but no satisfactory design exists for adjusting the electrode shape. This is because an electrode supple enough to be easily adjustable, would almost certainly be insufficiently rigid to guarantee the long term mechanical and thermal shape constancy required for precision mutual co-magnetometry. The absence of lumped quadrupoles would mean that the entire ring design has to be “dead-reckoned”.

For this reason the quadrupoles shown in Figure 7 are critical to the design but need to be accurately tuneable to small values, consistent with achieving adequate ring stability. One might reasonably expect at least one iteration in which, having first established the quadrupole settings needed for stability, the electrodes could be mechanically deformed during a period of ring shutdown to enable the quadrupoles to be weaker.

This loss of operational flexibility has to be regarded as a serious impediment to achieving pure toroidal focusing. Nevertheless, the (eventual) benefits of toroidal focusing are so great, especially long SCT, that this “inconvenience” has to be accepted. Further investigation will be required to make sure the quadrupoles can be strong enough to overcome realistically large initial design, fabrication, and positioning errors.

For a $2\times 2$ transfer matrix ${\bf M}$ to be symplectic it must satisfy the algebraic relation

$${\bf M}^{-1}=-{\bf S}{\bf M}^{T}{\bf S},\quad\hbox{where}\quad{\bf S}=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix},$$

(9.1)

and the only imposed constraint for $2\times 2$ transfer matrices requires them to have unit determinant. For any higher (necessarily even) dimensions the same equation is satisfied with ${\bf S}$ being replicated along the diagonal. This introduces symplecticity conditions going beyond the unity determinant requirement.

Transfer matrices ${\bf M_{x}}$ and ${\bf M_{y}}$ for one PTR superperiod as well as powers up to ${\bf M_{x}^{8}}$ and ${\bf M_{y}^{8}}$, which describe two consecutive complete revolutions follow: $\\ {\bf Mx}=\left[\begin{array}[]{cccc}0.33015&9.04118&0.0&8.68780\\ -0.09855&0.33015&0.0&1.27816\\ -1.27816&-8.68780&1.0&204.00230\\ 0.0&0.0&0.0&1.0\end{array}\right]$

${\bf My}=\left[\begin{array}[]{cc}0.95513&12.95238\\ -0.02167&0.95513\end{array}\right]$

${\bf Mx^{2}}=\left[\begin{array}[]{cccc}-0.78201&5.96980&0.0&23.11206\\ -0.06507&-0.78201&0.0&0.84395\\ -0.84395&-23.11206&1.0&385.79588\\ 0.0&0.0&0.0&1.0\end{array}\right]$

${\bf My^{2}}=\left[\begin{array}[]{cc}0.63162&24.74233\\ -0.04139&0.63162\end{array}\right]$

${\bf Mx4}=\left[\begin{array}[]{cccc}0.22307&-9.33687&0.0&10.07647\\ 0.10177&0.22307&0.0&-1.31996\\ 1.31996&-10.07647&1.0&732.58076\\ 0.0&0.0&0.0&1.0\end{array}\right]$

${\bf My^{4}}=\left[\begin{array}[]{cc}-0.62514&31.25562\\ -0.05229&-0.62514\end{array}\right]$

${\bf Mx^{8}}=\left[\begin{array}[]{cccc}-0.90048&-4.16564&0.0&24.64854\\ 0.04541&-0.90048&0.0&-0.58890\\ 0.58890&-24.64854&1.0&1491.76252\\ 0.0&0.0&0.0&1.0\end{array}\right]$

${\bf My^{8}}=\left[\begin{array}[]{cc}-1.24343&-39.07858\\ 0.06537&-1.24343\end{array}\right]$

Even for the $4\times 4$ matrices (mentally-partitioned $2\times 2$) one can check mentally (at least at the 10% level) that all determinant values are close to 1. Displayed only to 6 place accuracy, each of these matrices satisfies Eq. (9.1) to machine precision. In fact, this too is an understatement; these matrices are, in fact, analytically exact. They have been obtained purely algebraically.

This is all that is required for the transverse propagation to be Hamiltonian, and hence loss-free. No paraxial approximation, nor small angle, nor small amplitude approximations have been made; no truncated power series have been employed, no linear, quadrupole, sextupole and higher orders introduced. Admittedly these statements are true primarily because the starting transfer matrix was assumed to be exact. This is an excellent (but not perfect) assumption for the toroidal focusing design, which has been adopted in modern day (Scanning) Transmission Electron Microscopy (STEM), as described most recently in a textbook by P. Erni [32], with clear elementary physical discussion, especially in Section 5.

In spite of this splendid transverse evolution description one cannot fail to notice that the horizontal 4x4 matrices are, in some sense, divergent. Concentrating on the $M_{34}$ elements, referring to Eq. (5.10), and visualizing $\delta$p/p=$10^{-5}$ as a possible fractional momentum offset, one finds that, after a single turn, the longitudinal arc length parameter $s$ will have advanced by 0.75 cm, relative to a central particle.

For a few turns in a ring of circumference 100 m this advance would be scarcely noticeable, but circulating at roughly a MHz rate for, say, two minutes, the beam spread will be of order $10^{6}\,$m. In other words the beam will have by then spread around the ring into a uniformly distributed coasting beam. To keep the beam bunched, which is required for phase-locking, the PTR beam has to be quickly captured into stable RF buckets. This should be straightforward.

One notes, from Eqs (3.7), that electric spin tune $Q_{E}$ and magnetic spin tune $Q_{s}$ depend kinematically only on the relativistic factor $\gamma$, which is phase locked and globally constant for stored beams. Though not stated previously, this simple behavior assumes the central orbit lies in a single (typically horizontal) plane. For constant $\gamma$ MDMs are therefore unaffected by horizontal bend errors to all orders. By up-down symmetry, MDMs are constant only to linear order in vertical displacement error. As a consequence, it is mainly vertical positioning that needs to be tightly specified; here to $\pm 0.1\,$mm, or less, as required.

An essential difference between neutron and proton EDM measurements comes from the fact that, though dominant for the neutron EDM measurement, the statistical proton EDM measurement error is negligible by comparison. A convenient way to exploit this is to note that the statistical error for a “nominal week long” ($2\times 10^{5}$ s) storage ring measurement will be about $\pm 10^{-29}\,e$ cm. This is an accuracy level which (conventional) thinking has suggested to be a plausible BSM proton EDM. For all practical purposes, the implication for storage ring EDM measurement is that we can ignore counting statistics and concentrate entirely on systematic errors.

• •

  CW/CCW beam reversal requires reversing the magnetic field.

• •

  Setting, reversing, and resetting magnetic field with frequency domain accuracy relies only on MDMs acting as “magneto-sensitive gyroscopes” for phase locking both betatron tunes.

• •

  This avoids the need for (unachievably precise) magnetic field measurement, while allowing systematic error reduction by averaging over CW/CCW reversal.

The doubly magic proton-He3 combination has been especially emphasized in the present paper. This is because, with equipment currently in hand, or at least described in careful design proposals, the detection of BSM physics, in the form of non-vanishing EDMs of fundamental particles is possible inexpensively in the near future, using proven operational techniques. Parameters for the (essentially unique) proton/He3 doubly-magic pairing are given in Table 3.

The EDM signature is out-of-plane MDM precession. The dominant systematic error is due to unintentional (and unknown) radial magnetic field error acting on particle MDMs, which mimic the effect of intentional electric field acting on particle EDMs. Such unknown magnetic sources are discussed in Chapter 3 of reference[38]. In particular, Section 3.3.4 of that book argues that the systematic storage ring proton EDM error can be estimated to be comparable to the systematic Ramsey neutron EDM error (coming from the same source). Copying from the first sentence of the abstract to the present paper, the current day neutron EDM systematic error is $\pm 0.2_{\rm sys}\times 10^{-26}\,e$ cm (five times smaller than the neutron EDM statistical error in the same experiment).

Another way of estimating the storage ring proton EDM systematic error is based on the precision with which the vertical separation of countercirculating beams can be minimized—zero average vertical beam separation would imply vanishing radial magnetic field average, $\langle B_{r}\rangle$ and, therefore, zero spurious out-of-plane precession. Uncertainties in this estimate come from the estimated beam position monitor (BPM) precision, and the extent to which the vertical BPM sensitivity can be increased by reducing the vertical focusing strengths. As regards order of magnitude, estimates by the author[38], calculations by Tahar and Carli [39] and by Lebedev [37] are consistent with the $\pm 0.2_{\rm sys}\times 10^{-26}\,e$ cm systematic error estimate given in the preceeding paragraph.

Treated as composite, the proton consists of quarks and gluons, but treated as a fundamental particle, the proton EDM should vanish, or at least be very small. Consisting of two protons and a neutron, the He3 nucleus is unambiguously composite. To the extent that T-symmetry is respected, as in the standard model, He3 would still be required to have vanishing EDM. Recognizing that it is quite challenging for the electric center and the mass center of two protons and one neutron to be identical, in an experiment looking for T-violation, one can conjecture that the magnitude of any He3 EDM is significantly greater than the proton EDM.

For the mutual co-magnetometry emphasized in the present paper, this suggests treating any out-of-plane proton precession as spurious (i.e. caused by spurious radial magnetic field acting on the proton EDM). Why not, therefore, utilize the proton for mutual co-magnetometry by phase locking the proton EDM into the horizontal plane, which amounts to cancelling $\langle B_{r}\rangle$. In this circumstance, any measureable out-of-plane precession of the He3 beam polarization can be interpreted as a measure of the He3 EDM. This is a picturesque way of visualizing our experiment designed to measure $\Delta$ = EDM[proton]-EDM[He3]. (Among the many other things that could go wrong) there remains the possibility that, by chance, proton and He3 EDMs are equal, but opposite in sign—this would be both surprising and unfortunate. In any case non-zero $\Delta$ would still imply BSM physics.

Based on the previous discussion, systematic errors anticipated in various configurations are listed in Table 4. In all cases one will reduce systematic error by averaging over CW and CCW circulation. The “consecutive” column systematic error assumes that simultaneous CW and CCW beam circulation has been established. Successive runs alternate between CW and CCW. “Concurrent” columns assume the simultaneous phase-locked circulation of CW and CCW beams, with at least one beam globally frozen in the “singly magic” case and both beams globally frozen in the “doubly magic” case. The final column pulls out all the stops, taking advantage of the cancellation of systematic error for $\Delta$ in the proton-He3 case.