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Chapter 2. Relativistic Eikonal A(p, pN) Formalism 11The (A + 1)-body wave function in the final state readsΨ ⃗ k 1 ,m s1f , ⃗ k 2 ,m s2fA+1(⃗r 0 , ⃗r 1 , ⃗r 2 , ⃗r 3 )∣ ∣∣∣∣∣∣∣∣∣ φ ⃗k1 (⃗rm s1f 0 ) φ ⃗k2 (⃗rm s2f 0 ) φ α2 (⃗r 0 ) φ α3 (⃗r 0 )1 φ ⃗k1 (⃗rm= √ s1f 1 ) φ ⃗k2 (⃗rm s2f 1 ) φ α2 (⃗r 1 ) φ α3 (⃗r 1 )(A + 1)! φ ⃗k1 (⃗rm s1f 2 ) φ ⃗k2 (⃗rm s2f 2 ) φ α2 (⃗r 2 ) φ α3 (⃗r 2 ). (2.9)φ ⃗k1 (⃗rm s1f 3 ) φ ⃗k2 (⃗rm s2f 3 ) φ α2 (⃗r 3 ) φ α3 (⃗r 3 ) ∣Relative to the target nucleus ground state written in Eq. (2.7), the wave function of Eq. (2.9)refers to the situation whereby the struck proton resides in a state “α 1 ”, leaving the residualA − 1 nucleus as a hole state in that particular single-particle level. The outgoing protons arerepresented by relativistic plane waves.Since both the initial and the final wave functions are fully antisymmetrized, one can choosethe operator Ô(2) to act on two particular coordinates (⃗r 0 and ⃗r 1 ). Without any loss of generality,the A(p, 2p) matrix element of Eq. (2.6) can be written as∫ ∫ ∫ ∫M (p,2p) A(A + 1) 1fi= d⃗r 0 d⃗r 1 d⃗r 2 d⃗r 32 (A + 1)!∑∑ ∑×k,l∈{ ⃗ k 1 m s1f , ⃗ k 2 m s2f}m,n∈{α 2 ,α 3 }× ɛ klmn ɛ opqr φ † k (⃗r 0) φ † l (⃗r 1) φ † m (⃗r 2 ) φ † n (⃗r 3 )o,p∈{⃗p 1 m s1i ,α 1 }∑q,r∈{α 2 ,α 3 }× Ô (⃗r 0, ⃗r 1 ) φ o (⃗r 0 ) φ p (⃗r 1 ) φ q (⃗r 2 ) φ r (⃗r 3 ) , (2.10)with ɛ ijkl the Levi-Civita tensor. In the RPWIA,∫ ∫ ∫d⃗r 0 d⃗r 1 d⃗r 2 φ † k (⃗r 0) φ † l (⃗r 1) φ † m (⃗r 2 ) Ô (⃗r 0, ⃗r 1 ) φ o (⃗r 0 ) φ p (⃗r 1 ) φ q (⃗r 2 )∫ ∫ ∫= δ mq d⃗r 0 d⃗r 1 d⃗r 2 φ † k (⃗r 0) φ † l (⃗r 1)× Ô (⃗r 0, ⃗r 1 ) φ o (⃗r 0 ) φ p (⃗r 1 ) |φ q (⃗r 2 )| 2 . (2.11)Inserting this expression in Eq. (2.10), one obtains∫ ∫ ∫M (p,2p) A(A + 1) 1fi= d⃗r 0 d⃗r 12 (A + 1)!∑∑×k,l∈{ ⃗ k 1 m s1f , ⃗ k 2 m s2f}o,p∈{⃗p 1 m s1i ,α 1 }d⃗r 2∫d⃗r 3∑m,n∈{α 2 ,α 3 }× ɛ klmn ɛ opmn φ † k (⃗r 0) φ † l (⃗r 1) |φ m (⃗r 2 )| 2 |φ n (⃗r 3 )| 2× Ô (⃗r 0, ⃗r 1 ) φ o (⃗r 0 ) φ p (⃗r 1 ) . (2.12)There are (A − 1)! possible choices (permutations) for the indices m, n, . . . , all giving the samecontribution to the matrix element. Accordingly, the above expression can be rewritten asM (p,2p)fi= 1 ∫ ∫ ∫ ∫∑∑d⃗r 0 d⃗r 1 d⃗r 2 d⃗r 32k,l∈{ ⃗ k 1 m s1f , ⃗ k 2 m s2f}× ɛ klα2 α 3ɛ opα2 α 3φ † k (⃗r 0) φ † l (⃗r 1) |φ α2 (⃗r 2 )| 2 |φ α3 (⃗r 3 )| 2o,p∈{⃗p 1 m s1i ,α 1 }× Ô (⃗r 0, ⃗r 1 ) φ o (⃗r 0 ) φ p (⃗r 1 ) . (2.13)