3 is the ratio of longitudinal to transverse polarization of the virtual photon. The longitudinal/transverse ( l/t ) separated virtual-photon nucleon cross sections are given in Appendix A. The t -differential cross section for N ( γ ∗ , π ) N ′ integrated over φ is denoted here as and specify the notations and definitions of variables. The reaction (2) in the laboratory is shown in Figure 1 where the target nucleon is at rest, the z -axis is directed along the three momentum ⃗ q = (0 , 0 , √ ν 2 + Q $^{2}$) of the exchanged virtual photon γ ∗ with q = l − l ′ = ( ν, ⃗ q ), Q 2 = − q $^{2}$, ν = E$_{e}$ − E ′ e and l ( l ′ ) is the four momentum of incoming (deflected) electrons. In Figure 1 φ stands for the azimuthal angle between the electron scattering ( e, e ′ ) plane and γ ∗ N → πN ′ reaction plane. φ is zero when the pion is closest to the outgoing electron [32]. In exclusive reaction ( e, e ′ π ) we shall deal with an un- polarized target and, both unpolarized and polarized lep- ton beams. The differential cross section is given by The longitudinal beam single-spin asymmetry (SSA) in ( ⃗ e, e ′ π ) scattering is defined so that where dσ → refers to positive helicity h = +1 of the in- coming electron. The azimuthal moment associated with the beam SSA is given by [32] dσ$_{U}$ dt = dσ$_{T}$ dt + ε dσ$_{L}$ dt . (6) A$_{LU}$ ( φ ) ≡ dσ → ( φ ) − dσ ← ( φ ) dσ → ( φ ) + dσ ← ( φ ) , (7) A sin( φ ) LU = √ 2 ε (1 − ε ) dσ$_{LT}$ ′ dσ$_{T}$ + εdσ$_{L}$ . (8) dσ dQ $^{2}$dνdtdφ = Φ 2 π [ dσ$_{T}$ dt + ε dσ$_{L}$ dt + √ 2 ε (1 + ε ) dσ$_{LT}$ dt cos( φ ) + ε dσ$_{TT}$ dt cos(2 φ ) + h √ 2 ε (1 − ε ) dσ$_{LT}$ ′ dt sin( φ ) ] , (3) III. GAUGING THE PION EXCHANGE The diagrams describing the π + and/or π − electro- production amplitudes in exclusive reactions p ( e, e ′ π $^{+}$) n and n ( e, e ′ π − ) p are shown in Figure 2. At high energies the particles exchanged in the t -channel are understood as the Regge trajectories. In p ( e, e ′ π $^{+}$) n the s -channel nucleon-pole term ( ib diagram) is added to the t -channel π -pole exchange ( ia diagram) to conserve the charge of the system. Similarly, the diagram iia and the u -channel nucleon-pole diagram iib form a gauge invariant ampli- tude in n ( e, e ′ π − ) p . The last two diagrams ( iii and iv ) correspond to the exchange of vector V = ρ (770) and axial-vector A = a$_{1}$ (1260), b$_{1}$ (1235) Regge trajectories. where dσ$_{T}$ is the transverse cross section, dσ$_{L}$ is the longi- tudinal cross section, dσ$_{TT}$ is the cross section originating from the interference between the transverse components of the virtual photon, dσ$_{LT}$ is the cross section arising from the interference between the transverse and longi- tudinal polarizations of the virtual photon and dσ$_{LT}$ ′ is the beam-spin polarized cross section resulting from the interference between the transverse and longitudinal pho- tons and helicity h = ± 1 of the incoming electron. The virtual photon flux is conventionally defined as [33] where F $_{γππ}$( Q $^{2}$, t ) denotes the transition form factor of the pion and F s ( u $_{)}$( Q $^{2}$, s ( u ) , t ) stands for the proton s ( u )- channel transition form factor. In Eqs. (9) and (10) g$_{πNN}$ = 13 . 4 is the pseudoscalar πN coupling constant, t = k $^{2}$, s = W $^{2}$, k = k ′ − q = p − p ′ and other notations are obvious. − iJ µ s ( γ ∗ p → π $^{+}$n ) = √$_{2}$$_{g}$$_{πNN}$ ¯ u$_{s}$ ′ ( p ′ ) γ$_{5}$ [ F $_{γππ}$( Q $^{2}$, t ) ( k + k ′ ) µ t − m 2 π + i 0 + + F $_{s}$( Q $^{2}$, s, t ) ( p + q )$_{σ}$ γ $^{σ}$γµ + M$_{p}$γ µ s − M 2 p + i 0 + + [ F $_{γππ}$( Q $^{2}$, t ) − F $_{s}$( Q $^{2}$, s, t )] ( k − k ′ ) µ Q 2 ] u$_{s}$ ( p ) , (9) − iJ µ $_{u}$( γ ∗ n → π − p ) = − √$_{2}$$_{g}$$_{πNN}$ ¯ u$_{s}$ ′ ( p ′ ) [ F $_{γππ}$( Q $^{2}$, t ) ( k + k ′ ) µ t − m 2 π + i 0 + − F $_{u}$( Q $^{2}$, u, t ) γ $^{µ}$( p ′ − q )$_{σ}$ γ σ + M$_{p}$γ µ u M 2 p + i 0 + + [ F $_{γππ}$( Q $^{2}$, t ) − F $_{u}$( Q $^{2}$, u, t )] ( k − k ′ ) µ Q 2 ] γ$_{5}$u$_{s}$ ( p ) , (10) Φ = π E$_{e}$ ( E$_{e}$ − ν ) ( α$_{e}$ 2 π 2 E$_{e}$ − ν E$_{e}$ K Q 2 1 1 − ε ) , (4) with α$_{e}$ ≃ 1 / 137, K = ( W 2 − M 2 $_{N}$) / 2 M$_{N}$ and ε = 1 1 + 2 ν $^{2}$+ Q 2 4 E$_{e}$ ( $^{E$_{e}$}$−ν $^{)}$− Q 2 (5) The π -exchange currents describing the reactions p ( γ ∗ , π $^{+}$) n and n ( γ ∗ , π − ) p take the form [24]