28 where ( ... ) stands for the sum and average over the initial and final nucleon spins. The normalization factor reads The longitudinal/transverse ( l/t ) separated photon- nucleon cross sections take the forms where q ∗ is a three momentum of the incoming virtual photon in the γ ∗ N center of mass frame. ϵ λ µ are the ba- sis vectors of circular polarization for the virtual photon with helicities λ = ± 1 , 0 quantized along the three mo- mentum ⃗ q , i.e. , and J µ is the nuclear transition axial-vector current de- scribing the pion production in the momentum space. The Lorentz tensor-vector decomposition of the nucleon-pole term in Eq. (9) is given by Appendix A: The partial virtual-photon nucleon cross sections N = α$_{e}$ 4 π 2 π M 2 N ( W 2 − M 2 $_{N}$) W q ∗ , (A6) $_{ϵ}$± µ = ∓ 1 √$_{2}$ (0 , 1 , ± i, 0) , (A7) ϵ 0 µ = 1 √ Q $_{2}$( √ ν 2 + Q $^{2}$, 0 , 0 , ν ) , (A8) 1 N dσ$_{L}$ dt = ( J $^{µ}$ϵ 0 $_{µ}$J $^{ν}$† $_{ϵ}$† 0 ν ) , (A1) 1 N dσ$_{T}$ dt = 1 2 ∑ λ $^{=}$±1 ( J $^{µ}$ϵλ $_{µ}$J $^{ν}$† $_{ϵ}$† λ ν ) , (A2) 1 N dσ$_{TT}$ dt = − 1 2 ∑ λ $^{=}$±1 ( J $^{µ}$ϵλ $_{µ}$J $^{ν}$† $_{ϵ}$†− λ ν ) , (A3) 1 N dσ$_{LT}$ dt = − 1 $_{2}$√$_{2}$ ∑ λ $^{=}$±1 λ [( J $^{µ}$ϵ 0 $_{µ}$J $^{ν}$† $_{ϵ}$† λ ν ) + ( J $^{µ}$ϵλ $_{µ}$J $^{ν}$† $_{ϵ}$† 0 ν )] , (A4) 1 N dσ$_{LT}$ ′ dt = − 1 $_{2}$√$_{2}$ ∑ λ $^{=}$±1 λ [( J $^{µ}$ϵ 0 $_{µ}$J $^{ν}$† $_{ϵ}$† λ ν ) − ( J $^{µ}$ϵλ $_{µ}$J $^{ν}$† $_{ϵ}$† 0 ν )] , (A5) Appendix B: On the gauged electric amplitude ¯ u$_{s}$ ′ ( $_{p}$′$_{)}$$_{γ}$$_{5}$ ( p + q )$_{σ}$ γ $^{σ}$γµ + M$_{p}$γ µ s − M 2 p + i 0 + u$_{s}$ ( p ) = ¯ u$_{s}$ ′ ( $_{p}$′$_{)}$$_{γ}$$_{5}$ [ iσ $^{µσ}$q$_{σ}$ s − M 2 p + i 0 + + ( p + p ′ + k ′ ) µ s − M 2 p + i 0 + ] u$_{s}$ ( p ) , (B1) pole term generates a large axial-tensor background to the meson-pole amplitude which has important conse- quences in photoproduction. For instance, since the in- terference of π with ρ (770) is trivially zero it is σ $^{µσ}$γ$_{5}$ in the nucleon-pole term which, by interference with ρ , is responsible for the π − /π + asymmetry. It is also this term which explains the forward peak in the forward π ± pro- duction and the rapid variation of the polarized photon asymmetry in the same region. where σ µσ = i $_{2}$[ γ $^{µ}$, γ$^{σ}$]. The axial-tensor term in the r.h.s. of Eq. (B1) is gauge invariant by itself. It turns out that only the orbital part proportional to p + p ′ + k ′ is needed to conserve the charge in the sum with the π -pole amplitude, the first term in Eq. (9). In photoproduction the orbital part has no physical significance because it ap- pears in the physical scattering amplitude multiplied by the polarization vector ϵ λ $_{µ}$. Since, q $^{µ}$ϵ ± µ = p $^{µ}$ϵ ± µ = 0 the product ( p + p ′ + k ′ ) ϵ ± µ = (2 p + q ) ϵ ± µ = 0. The nucleon- [10] C.J. Bebek et al. , Phys. Rev. D 13 , 25 (1976). [11] C.J. Bebek et al. , Phys. Rev. D 17 , 1693 (1978). [12] H. Ackermann et al. , Nucl. Phys. B 137 , 294 (1978). [13] P. Brauel et al. , Phys. Lett. B 65 , 181 (1976). [14] P. Brauel et al. , Phys. Lett. B 65 , 184 (1976). [15] P. Brauel et al. , Phys. Lett. B 69 , 253 (1977). [16] P. Brauel et al. , Z. Phys. C 3 , 101 (1979). [17] V. G. Neudatchin et al. , Nucl. Phys. A 739 , 124 (2004). [18] J. D. Sullivan, Phys. Lett. B 33 , 179 (1970). [19] J. C. Collins, L. Frankfurt and M. Strikman, Phys. Rev. [1] T. Horn et al. , Phys. Rev. Lett. 97 , 192001 (2006). [2] V. Tadevosyan et al. , Phys. Rev. C 75 , 055205 (2007). [3] T. Horn et al. , Phys. Rev. C 78 , 058201 (2008). [4] H. P. Blok et al. , Phys. Rev. C 78 , 045202 (2008). [5] G. M. Huber et al. , Phys. Rev. C 78 , 045203 (2008). [6] H. Avakian and L. Elouadrhiri [CLAS Collaboration], Phys. Part. Nucl. 35 , S114 (2004). [7] A. Airapetian et al. , Phys. Lett. B 659 , 486 (2008). [8] C.N. Brown et al. , Phys. Rev. D 8 , 92 (1973). [9] C.J. Bebek et al. , Phys. Rev. D 9 , 1229 (1974).