}\frac{S_{\gamma}(\underline{\alpha})}{-\overline{\alpha}_u p_1^2-\alpha_u p_2^2} \right.
{\nonumber}\\ 
& \phantom{\Big[}\left. -\frac{e_u}{2 \pi^2}\int_0^1\!\!du\;\int {\mathcal{D \underline{\alpha}} }\frac{(1-2u) T_{4}^{\gamma}(\underline{\alpha})}{{-\overline{\alpha}_up_1^2-\alpha_u p_2^2}} \right] p^{\alpha} p_{\beta} \sigma^{\beta \zeta} f_{\alpha \zeta} + \ldots \\ 
\label{diagramDProton}
T^{P}_{d}(p,q)=&\left[-\frac{e_d {\left\langle {\overline{q}}q\right\rangle}}{4 \pi^2} \int_0^1 \!\! du \; \int \!\! {\mathcal{D \underline{\alpha}} }\; \frac{\mathcal{S}(\underline{\alpha})+\widetilde{\mathcal{S}}(\underline{\alpha})}{-\overline{\alpha}_u p_1^2-\alpha_u p_2^2}  \right.
{\nonumber}\\
&\phantom{\Big[}\left. +\frac{e_d {\left\langle {\overline{q}}q\right\rangle}}{4 \pi^2} \int_0^1 \!\! du \; \int \!\! {\mathcal{D \underline{\alpha}} }\; (1-2u)\frac{T_2(\underline{\alpha}) - 2T_3(\underline{\alpha}) + T_4(\underline{\alpha})}{-\overline{\alpha}_u p_1^2-\alpha_u p_2^2} \right] 
{\nonumber}\\
&\phantom{ +\frac{e_d {\left\langle {\overline{q}}q\right\rangle}}{4 \pi^2} \int_0^1 \!\! du \; \int \!\! {\mathcal{D \underline{\alpha}} }\;} \times p^{\alpha} p_{\beta} \sigma^{\beta \zeta} f_{\alpha \zeta} + \ldots \mbox{ ,}\end{aligned}$$ where $\int Da = \int_0^1 d \alpha_q \;\int_0^1 d\alpha_{{\overline{q}}}\;\int_0^1d \alpha_{g} \; \delta(1-\alpha_q-\alpha_{{\overline{q}}}-\alpha_{g})$. The functions $\varphi$, which is of twist 2 and $T_i$, $\mathcal{S}$, $\widetilde{\mathcal{S}}$, $\mathcal{S}_{\gamma}$, $T_{4}^{\gamma}$, $\mathds{A}$ and $\mathds{B}$, which have twist 4, are defined in Appendix\[A\].

Borel transformation and continuum subtraction
----------------------------------------------

The sum rule for $F_2(0)$ can readily be obtained be equating the hadronic result, Eq., and the light-cone expansion, Eqs.(\[diagramBProton\], \[diagramCProton\], \[diagramDProton\]). As usual, a Borel transformation and a subsequent continuum subtraction are necessary to suppress the effects of excited states and extract the $p \to p \gamma$ ratio. The two independent momenta $p_1$ and $p_2$ allow a double Borel transformation, that can be performed using 