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Chapter 2. Relativistic Eikonal A(p, pN) Formalism 33which reduces to the ROMEA result of Eq. (2.50) when terms of higher order than V opt /k areneglected. Accordingly, the function f( ⃗ b, z) should be of the form f = 1 + O(V opt /k 2 ). Substituting(2.88) into Eq. (2.87) and multiplying by e −i S(⃗b,z) on the right yields[ (f( ⃗ b, z) = 1 − 1 1 + b ∂ ) ]V c (2kv ∂b⃗ b, z) f( ⃗ b, z)+ z (1 + b ∂ ) ∫ zdz ′ ∂V c( ⃗ b, z ′ )f(2kvb ∂b −∞ ∂b⃗ b, z ′ )+ 12kv V so( ⃗ b, z) (⃗σ ·⃗b × ⃗ k − ikz) f( ⃗ b, z)+ 1 (1 + b ∂ ) ∫ [zdz ′ (z − z ′ ∂() V so (2kvb ∂b −∞∂b⃗ b, z ′ ) ⃗σ ·⃗b × ⃗ k) ] f( ⃗ b, z ′ )− i ∫ [z(dz ′ 2 + b ∂ ) ]V so (2v −∞ ∂b⃗ b, z) f( ⃗ b, z ′ ) . (2.89)In deriving this equation, we set e i S(⃗ b,z ′) e −i S(⃗ b,z) equal to 1, since higher-order terms are neglected.The difficulty in solving for f( ⃗ b, z) is that Eq. (2.89) is an integral equation. An expressionfor f( ⃗ b, z) can, however, be readily obtained by adding (1−f) terms, which introduce onlyhigher-order terms, to the right-hand side of Eq. (2.89). This is permitted since the solution isonly determined up to order V opt /k 2 . With this manipulation, the function f becomesf( ⃗ b, z) = 1 − 1 (1 + b ∂ )V c (2kv ∂b⃗ b, z) +z (1 + b ∂ ) ∫ zdz ′ ∂V c( ⃗ b, z ′ )2kvb ∂b −∞ ∂b+ 12kv V so( ⃗ b, z) (⃗σ ·⃗b × ⃗ k − ikz)+ 1 (1 + b ∂ ) ∫ zdz ′ (z − z ′ ) ∂ (V so (2kvb ∂b −∞ ∂b⃗ b, z ′ ) ⃗σ ·⃗b × ⃗ )k− i ∫ z(dz ′ 2 + b ∂ )V so (2v∂b⃗ b, z) . (2.90)−∞The eikonal factor of Eq. (2.88) whereby f is determined by (2.90) is a solution of the integralequation (2.83) to order V opt /k 2 and indeed reduces to the ROMEA result (2.50) when truncatedat order V opt /k. Furthermore, it can be easily verified that the derivative of η is of higher orderin V opt /k than η itself. Henceforth, results obtained with the eikonal factor given by Eqs. (2.88)and (2.90) are dubbed as the second-order relativistic optical model eikonal approximation(SOROMEA).