Re-evaluation of Spin-Orbit Dynamics of Polarized e+ e- Beams in High Energy Circular Accelerators and Storage Rings: an approach based on a Bloch equation

We give an overview of our current/future analytical and numerical work on the spin polarization in high-energy electron storage rings. Our goal is to study the possibility of polarization for the CEPC and FCC-ee. Our work is based on the so-called Bloch equation for the polarization density introduced by Derbenev and Kondratenko in 1975. We also give an outline of the standard approach, the latter being based on the Derbenev-Kondratenko formulas.


Contents 1 Introduction 4 1 Introduction
This paper is an update on a talk by K. Heinemann at the IAS Mini-Workshop on Beam Polarization in Future Colliders on January 17, 2019, in Hong Kong. 1 Our ultimate goal is to examine the possibility of high polarization for CEPC and FCC-ee. We will first briefly review the "standard" approach which is based on the Derbenev-Kondratenko formulas. 2 These formulas rely, in part, on plausible assumptions grounded in deep physical intuition. So the following question arises: do the Derbenev-Kondratenko formulas tell full story? In fact there is an alternative approach based on a Bloch-type equation for the polarization density 3 which we call 3 the Bloch equation (BE) and which we believe can deliver more information than the standard approach even if the latter includes potential correction terms. 4 So we aim to determine the domain of applicability of the Derbenev-Kondratenko formulas and the possibility in theory of polarization at the CEPC and FCC-ee energies. Of course both approaches focus on the equilibrium polarization and the polarization time. We use the name "Bloch" to reflect the analogy with equations for magnetization in condensed matter. 5 This paper concentrates on the Bloch approach. The cost of the numerical computations in the Bloch approach is considerable since the polarization density depends on six phase-space variables plus the time variable so that the numerical solution of the BE, the BE being a system of three PDEs in seven independent variables, is a nontrivial task which cannot be pursued with traditional approaches like the finite difference method. However we see at least five viable methods: 1. Approximating the BE by an effective BE via the Method of Averaging and solving the effective BE via spectral phase-space discretization, e.g., a collocation method, plus an implicit-explicit time discretization.
2. Solving the system of stochastic differential equations (SDEs), which underlies the BE, via Monte-Carlo spin tracking. See Ref. 6 for the system of SDEs underlying the BE.
3. Solving the Fokker-Planck equation, which underlies the BE, via the Gram-Charlier method.
4. Solving the BE via a deep learning method.
5. Solving the system of SDEs in a way that allows connections with the Derbenev-Kondratenko formulas to be established.
We will dwell on Method 1 in this paper. We plan to validate this method by one of the other four methods. More details on Method 1 can be found in Ref. 6. The method of averaging we use is discussed in Refs. 7-12. One hope tied to Method 1 is that the effective BE gives analytical insights into the spinresonance structure of the bunch. Note that Methods 1-4 are independent of the standard approach. In particular they do not rely on the invariant spin field. Note also that Methods 1-3 and 5 are based on knowing the system of SDEs, which underlies the BE. For details of this system of SDEs, see the invited ICAP18 paper of Ref. 6. Regarding Method 2 there is a large literature on the numerical solution of SDEs, see Refs. 13, 14 and references in Ref. 15.
By neglecting the spin-flip terms and the kinetic-polarization term in the BE one obtains an equation that we call the Reduced Bloch equation (RBE). The RBE approximation is sufficient for computing the radiative depolarization rate due to stochastic orbital effects and it shares the terms with the BE that are challenging to discretize. For details on our phase-space discretization and time discretization of the RBE, see Refs. 6,16,17 and 18.
We proceed as follows. In Section 2 we sketch the standard approach. In Section 3 we present, for the laboratory frame, the BE and its restriction, the RBE. In Section 4 we discuss the RBE in the beam frame and in Section 5 we show how, in the beam frame, the effective RBE is obtained via the method of averaging. In Section 6 we describe ongoing and future work.

Sketching the standard approach based on the Derbenev-Kondratenko formulas
We define the "time" θ = 2πs/C where s is the distance around the ring and C is the circumference. We denote by y a position in six-dimensional phase space of accelerator coordinates which we call beam-frame coordinates. In particular, following Ref. 19, y 6 is the relative deviation of the energy from the reference energy. Then if, f = f (θ, y) denotes the normalized 2π-periodic equilibrium phase-space density at θ and y and P loc = P loc (θ, y) denotes the local polarization vector of the bunch we have where P (θ) is the polarization vector of the bunch at θ. For a detailed discussion about P loc , see, e.g., Ref.
20. Here and in the following we use arrows on three-component column vectors.
Central to the standard approach is the invariant spin field (ISF)n =n(θ, y) defined as a normalized periodic solution of the Thomas-BMT-equation in phase space, i.e., such that 1. n(θ, y) = 1, 2.n(θ + 2π, y) =n(θ, y), and where L Liou denotes the Hamiltonian part of the Fokker-Planck operator L y FP , the latter being introduced in Section 3 below. For some of our work on the ISF see Refs. 21 and 22. The unit vector of the ISF on the closed orbit is denoted byn 0 (θ) and it is easily obtained as an eigenvector of the one-turn spin-transport map on the closed orbit. 19 There are many methods for computing the ISF but none are trivial (for a recent technique see Ref. 23). In fact the existence, in general, of the invariant spin field is a mathematical issue which is only partially resolved, see, e.g., Ref. 21. The standard approach assumes that a function P DK = P DK (θ) exists such that P loc (θ, y) ≈ P DK (θ)n(θ, y) . ( Thus, by (1) and (3), The approximation (3) leads to 2 where τ DK and P DK (∞) are given by the Derbenev-Kondratenko formulas with •b =b(θ, y) ≡ normalized magnetic field,β =β(θ, y) ≡ normalized velocity vector, γ 0 ≡ Lorentz factor of the reference particle, R(θ, y) ≡ radius of curvature in the external magnetic field, r e ≡ classical electron radius, m ≡ rest mass of electrons or positrons.
By (4) and for large θ where P DK (∞) is given by (6) and where the rhs of (9) is the approximate equilibrium polarization vector. Note that the latter is 2π-periodic in θ since f (θ, y) andn(θ, y) are 2π-periodic in θ. Defining we can write (7) as For details on (6), (7), (8), (10) and (11) see, e.g., Refs. 24 and 19. We now briefly characterize the various terms in the Derbenev-Kondratenko formulas. First, τ −1 dep is the radiative depolarization rate. Secondly, the term covers the kinetic polarization effect and the term in τ −1 DK which is proportional to 2/9 covers the Baier-Katkov correction.
We now sketch three approaches for computing P DK (∞) via the Derbenev-Kondratenko formulas. All three approaches use (6)  Approach (ii) is the most practiced while approach (i) is only feasible if one can compute f andn as accurately as needed (which is not easy!). Approach (iii), which was historically the first, is very simple and is often used for ballparking P DK (∞). Since the inception of the Derbenev-Kondratenko formulas correction terms to the rhs of (10) have been suspected. See Refs. 4, 28 as well as Z. Duan's contribution to this workshop. These correction terms, associated with so-called resonance crossing, in turn associated with large energy spread, are not as well understood as the rhs of (10), partly because of their peculiar form. Nevertheless, careful observation of spin motion during the Monte-Carlo tracking in approach (ii), might provide a way to investigate their existence and form.  27 This approach provides a useful first impression avoiding the computation of f andn. For more details on this approach see Ref. 19. Monte-Carlo tracking can also be extended beyond integrable orbital motion to include, as just one example, beam-beam forces. Note that Monte-Carlo tracking just gives an estimate of τ −1 dep but it does not provide an explanation. Nevertheless, insights into sources of depolarization can be obtained by switching off terms in the Thomas-BMT equation. In principal such diagnoses can also be applied in approach (i). Such investigations can the systematized under the heading of "spin matching". 19

The Bloch equation and the Reduced Bloch equation in the laboratory frame
In the previous section we used the beam frame and we will do so later. However the BE was first presented in Ref. 3 for the laboratory frame and in that frame it also has its simplest form. In this section we focus on the laboratory frame. In a semiclassical probabilistic description of an electron or positron bunch the spin-orbit dynamics is described by the spin-1/2 Wigner function ρ (also called the Stratonovich function) written as with z = ( r, p) where r and p are the position and momentum vectors of the phase space and t is the time, and where f lab is the phase-space density of particles normalized by dzf lab (t, z) = 1, η lab is the polarization density of the bunch and σ is the vector of the three Pauli matrices. As explained in Ref. 20, η lab is proportional to the spin angular momentum density. In fact it is given by η lab (t, z) = f lab (t, z) P loc,lab (t, z) where P loc,lab is the local polarization vector. Thus f lab = T r[ρ] and η lab = T r[ρ σ]. The polarization vector P lab (t) of the bunch is P lab (t) = dz η lab (t, z) = dzf lab (t, z) P loc,lab (t, z). Then, by neglecting collective effects and after several other approximations, the phase-space density evolves according to Ref. 3 via Using the units as in Ref. 3 the Fokker-Planck operator L lab F P is defined by where and with e being the electric charge of the electron or positron and E and B being the external electric and magnetic fields. The Fokker-Planck operator L lab F P whose explicit form is taken from Ref. 3 is a linear second-order partial differential operator and, with some additional approximations, is commonly used for electron synchrotrons and storage rings, see Ref. 29 and Section 2.5.4 in Ref. 19. As usual, since it is minuscule compared to all other forces, the Stern-Gerlach effect from the spin onto the orbit is neglected in (13). The polarization density η lab evolves via eq. 2 in Ref. 3, i.e., via that which we call the Bloch equation, namely where a(t, z) := e m 2 γ 2 ( p) ( p × B(t, r)) .
The BE was derived in Ref. 3 from the semiclassical approximation of quantum electrodynamics and it is a generalization, to the whole phase space, of the Baier-Katkov-Strakhovenko equation which just describes the evolution of polarization along a single deterministic trajectory. 30 Note also that, while the BE was new in 1975, the orbital Fokker-Planck equation (13) was already known thanks to research of the 1950s, e.g., Schwinger's paper on quantum corrections to synchrotron radiation. 31 The skew-symmetric matrix Ω lab (t, z) takes into account the Thomas-BMT spin-precession effect. Thus in the laboratory frame the Thomas-BMT-equation (2) reads as The quantum aspect of (13) and (20) is embodied in the factor in λ(t, z). For example Q rad is a quantum correction to the classical radiation reaction force F rad .
The terms −λ(t, z) 5 | a(t,z)| f lab (t, z) take into account spin flips due to synchrotron radiation and encapsulate the Sokolov-Ternov effect. The term λ(t, z) 5 √ 3 8 2 9m 2 γ 2 ( p) p p T η lab encapsulates the Baier-Katkov correction, and the term ∂ p · p λ(t, z) 1 mγ( p) p× a(t,z) | a(t,z)| f lab (t, z) encapsulates the kinetic-polarization effect. The only terms in (20) which couple the three components of η lab are the Thomas-BMT term and the Baier-Katkov correction term.
As mentioned above, there exists a system of SDEs underlying (20) (for details, see Ref. 6). In particular, f lab and η lab are related to a spin-orbit density P lab = P lab (t, z, s) via where P lab satisfies the Fokker-Planck equation corresponding to the system of SDEs in Ref. 6. These SDEs can be used as the basis for a Monte-Carlo spin tracking algorithm, i.e., for Method 2 mentioned in Section 1 above. This would extend the standard Monte-Carlo spin tracking algorithms, which we mentioned in Section 2 above, by taking into account all physical effects described by (20), i.e., the Sokolov-Ternov effect, the Baier-Katkov correction, the kinetic-polarization effect and, of course, spin diffusion. If we ignore the spin-flip terms and the kinetic-polarization term in the BE then (20) simplifies to the RBE The RBE models spin diffusion due to the effect of the stochastic orbital motion on the spin and thus contains those terms of the BE which are related to the radiative depolarization rate τ −1 dep . This effect is clearly seen in the SDEs (see, e.g., (28) and (29)).
Because the coefficients of L y FP are θ-dependent, the RBE (27) is numerically and analytically quite complex. So we first approximate it by treating the synchrotron radiation as a perturbation. Then, in order to solve it numerically to determine the long-time behavior that we need, we address the system of SDEs underlying (27) and apply the refined averaging technique presented in Ref. 32 (see also 7), for the orbital dynamics, and extend it to include spin. The averaged SDEs are then used to construct an approximate RBE which we call the effective RBE.
The system of SDEs underlying (27) reads as 5 where the orbital dynamics has been linearized in Y and Ω = Ω 0 + ǫ S C has been linearized in Y so that Also, A(θ) is a Hamiltonian matrix representing the nonradiative part of the orbital dynamics and Y has been scaled so that ǫ R is the size of the orbital effect of the synchrotron radiation. Thus ǫ R δA(θ) represents the orbital damping effects due to synchrotron radiation and the cavities, √ ǫ R ξ(θ) represents the associated quantum fluctuations, ξ is the white noise process and e 6 := (0, 0, 0, 0, 0, 1) T . In the spin equation (29), Ω 0 is the closed-orbit contribution to Ω so that ǫ S C(θ, Y ) is what remains and C(θ, Y ) is chosen O(1). Hence ǫ S estimates the size of Ω − Ω 0 . Both Ω 0 (θ) and C(θ, Y ) are, of course, skew-symmetric 3 × 3 matrices. We are interested in the situation where ǫ R and ǫ S are small in some appropriate sense.
Eqs. (28) and (29) can be obtained by transforming the system of SDEs in Ref. 6 from the laboratory frame to the beam frame. 33 However, since in this section we only deal with the RBE (not with the BE), (28) and (29) can also be found in older expositions on spin in high-energy electron storage rings, e.g., Ref. 34. Note that these expositions make approximations as for example with the linearity of (28) in Y and the linearity of C(θ, Y ) in Y .
With (28) and (29) the evolution equation for the spin-orbit joint probability density P = P(θ, y, s) is the following spin-orbit Fokker-Planck equation where L y FP is the orbital Fokker-Planck operator. The phase-space density f and the polarization density η corresponding to P are defined by which are the beam-frame analogs of (24) and (25). The local polarization vector P loc from Section 2 above is related to f and η by η(θ, y) = f (θ, y) P loc (θ, y) .

The Effective Reduced Bloch equation in the beam frame
The effective RBE is, by definition, an approximation of the RBE (27) obtained by approximating the system of SDEs (28) and (29) using the method of averaging, see Refs. 7-12. We call the system of SDEs underlying the effective RBE the effective system of SDEs. We now discuss first-order averaging in the case where ǫ := ǫ S = ǫ R is small. To apply the method of averaging to (28) and (29) we must transform them to a standard form for averaging, i.e., we must transform the variables Y, S to slowly varying variables. We do this by using a fundamental solution matrix X of the unperturbed ǫ = 0 part of (28), i.e., and a fundamental solution matrix Φ of the unperturbed ǫ = 0 part of (29), i.e., We thus transform Y and S into the slowly varying U and T via Hence (28) and (29) are transformed to where D and D are defined by Of course, (37) and (38) carry the same information as (28) and (29). Now, applying the method of averaging to (37) and (38), we obtain the following effective system of SDEs where the bar denotes θ-averaging, i.e., the operation lim T →∞ (1/T ) T 0 dθ · · · . Moreover ξ 1 , ..., ξ k are statistically independent versions of the white noise process and B is a 6 × k matrix which satisfies BB T =Ē with k = rank(Ē) and whereĒ is the θ-average of For physically reasonable A and Ω the fundamental matrices X and Φ are quasiperiodic functions whence D, D(·, U ) and E are quasiperiodic functions so that their θ averagesD,D(V ) andĒ exist. Our derivation of (41) from (37) is discussed in some detail in Ref. 6. We are close to showing that U = V + O(ǫ) on θ-intervals of length O(1/ǫ) and it seems likely that this error is valid for 0 ≤ θ < ∞, because of the radiation damping. This is a refinement of Ref. 32 and assumes a non-resonance condition. Since the sample paths of U are continuous and U is slowly varying it seems likely that T a is a good approximation to T and we are working on the error analysis. Spin-orbit resonances will be an important focus in the construction ofD(V ) from (40) which contains both the orbital frequencies in X and the spin precession frequency in Φ.
Since, by definition, the effective system of SDEs underly the effective RBE, the latter can be obtained from the former in the same way as we obtained (27) from (28) and (29) (recall the discussion after (32)). Thus the evolution equation for the spin-orbit probability density P V = P V (θ, v, t) is the following Fokker-Planck equation: where The polarization density η V corresponding to P V is defined by so that, by (44), the effective RBE is This then is the focus of our approach in Method 1. For more details on this section, see Refs. 6, 17 and 18.

Next steps
• Further development of Bloch-equation approach (numerical and theoretical), i.e., of Method 1 and with a realistic lattice.
• Development of validation methods, i.e., Methods 2-4. Note that Method 2 is an extension of the standard Monte-Carlo spin tracking algorithms and for that matter we will study Refs. 13, 14 and 15.
• Investigating the connection between the Bloch-equation approach and the standard approach based on the Derbenev-Kondratenko formulas, and studying the potential for correction terms 4 to τ −1 DK by using the RBE.