All order factorization for virtual Compton scattering at next-to-leading power

We discuss all-order factorization for the virtual Compton process at next-to-leading power (NLP) in the Λ_QCD/Q and \( \sqrt{-t} \)/Q expansion (twist-3), both in the double-deeply-virtual case and the single-deeply-virtual case. We use the soft-collinear effective theory (SCET) as the main theoretical tool. We conclude that collinear factorization holds in the double-deeply virtual case, where both photons are far off-shell. The agreement is found with the known results for the hard matching coefficients at leading order \( {\alpha}_s^0 \), and we can therefore connect the traditional approach with SCET. In the single-deeply-virtual case, commonly called deeply virtual Compton scattering (DVCS), the contribution of non-target collinear regions complicates the factorization. These include momentum modes collinear to the real photon and (ultra)soft interactions between the photon-collinear and target-collinear modes. However, such contributions appear only for the transversely polarized virtual photon at the NLP accuracy and in fact it is the only NLP ~ (Λ_QCD/Q)¹ ~ (\( \sqrt{-t} \)/Q)¹ contribution in that case. We therefore conclude that the DVCS amplitude for a longitudinally polarized virtual photon, where the leading power ~ (Λ_QCD/Q)⁰ ~ (\( \sqrt{-t} \)/Q)⁰ contribution vanishes, is free of non-target collinear contributions and the collinear factorization in terms of twist-3 GPDs holds in that case as well.