content_image/412158.jpg content_image/412159.jpg content_image/412155.jpg content_image/412156.jpg content_image/412157.jpg content_image/412152.jpg content_image/412153.jpg content_image/412154.jpg content_image/412161.jpg
## Appendix A: The partial virtual-photon nucleon cross sections 

The longitudinal/transverse ( l/t ) separated photon- nucleon cross sections take the forms 

<img src='content_image/412158.jpg'>

<img src='content_image/412159.jpg'>

<img src='content_image/412155.jpg'>

<img src='content_image/412156.jpg'>

<img src='content_image/412157.jpg'>

where ( ... ) stands for the sum and average over the initial and final nucleon spins. The normalization factor reads 

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. 

## Appendix B: On the gauged electric amplitude 

The Lorentz tensor-vector decomposition of the nucleon-pole term in Eq. (9) is given by 

<img src='content_image/412152.jpg'>

<img src='content_image/412153.jpg'>

<img src='content_image/412154.jpg'>

<img src='content_image/412161.jpg'>

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- 

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