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Chapter 2. Relativistic Eikonal A(p, pN) Formalism 31in elastic nuclear scattering by protons, antiprotons, and α particles [106, 107], heavy-ion scattering[108–111], and inclusive electron-nucleus scattering [112]. In this section, we develop asecond-order correction to the ROMEA framework of Section 2.2. Our formalism builds uponthe work of Baker [113], where an eikonal approximation for potential scattering was derivedto second order in 1/k. This approach is extended to account for the effect of the spin-orbitpotential.Like in the ROMEA approach of Section 2.2, the starting point is the Schrödinger-likeequation (2.42) for the upper component u (+)⃗ k,ms(⃗r). In the spin-orbit (V so (r) ⃗σ · ⃗L) and Darwin(V so (r) (−i⃗r · ˆ⃗p)) terms, as well as in the lower component (2.43), the momentum operator ˆ⃗p isreplaced by the asymptotic momentum ⃗ k, i.e., the EMA is adopted. For the upper component,one postulates a solution of the formu (+)⃗(⃗r) ≡ N η(⃗r) e i⃗k·⃗r χ 1 , (2.82)k,ms ms2i.e., a plane wave modulated by an eikonal factor η(⃗r). Here, N is a normalization factor.In the ROMEA approach, which adopts the first-order eikonal approximation, the Schrödinger-typeequation (2.42) was then linearized in ˆ⃗p leading to the solution (2.50). Despite the factthat it is retained as an exponential phase, this solution is, strictly speaking, only valid up tofirst order in V opt /k, with V opt ( ⃗ b, z) = V c ( ⃗ b, z) + V so ( ⃗ b, z) (⃗σ · ⃗b × ⃗ k − ikz). Mathematically,the exponential expression is not justified. However, it is commonly used because physicalintuition dictates that the effect of the scattering is to modulate the incoming plane wave by aphase change.In what follows, we will derive an expression for the eikonal factor η(⃗r) that is valid upto order V opt /k 2 . The momentum dependence in the spin-orbit and Darwin terms makes thatthese terms are retained up to order V so /k, while central terms are included up to order V c /k 2 .Note that the expansion is not expressed in terms of the Lorentz scalar and vector potentials V sand V v . Looking for a solution of the form (2.82) for the Schrödinger-like equation (2.42), Bakerarrived at the following equation for the eikonal factor (see Eq. (14) of Ref. [113]):∫ zη( ⃗ b, z) = 1 − i dz ′ V opt (v⃗ b, z ′ ) η( ⃗ b, z ′ ) + 1−∞2kv V opt( ⃗ b, z) η( ⃗ b, z)+ 1 ∫ z( 1dz ′ (z − z ′ )2kv −∞b + ∂ ) ∂(V opt (∂b ∂b⃗ b, z ′ ) η( ⃗ )b, z ′ ) , (2.83)where v = k/M N . Note that, apart from dropping contributions of order V opt /k 3 and higher,no additional assumptions were made when deriving Eq. (2.83). In Ref. [113], Eq. (2.83) wassubsequently solved for spherically symmetric potentials. The spin-orbit and Darwin terms,however, break the spherical symmetry and a novel method to solve Eq. (2.83) is required.To that purpose, we assume that the derivative of the function η is of higher order in 1/kthan η itself (as is true for the ROMEA solution (2.50)). This allows us to drop the ∂η/∂b contri-