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Photon from the annihilation process with CGC in the $pA$ collision

We discuss the photon production in the $pA$ collision in a framework of the color glass condensate (CGC) with expansion in terms of the proton color source $ρ_p$. We work in a regime where the color density $ρ_A$ of the nucleus is large enough to justify the CGC treatment, while soft gluons in the proton could be dominant over quark components but do not yet belong to the CGC regime, so that we can still expand the amplitude in powers of $ρ_p$. The zeroth-order contribution to the photon production is known to appear from the Bremsstrahlung process and the first-order corrections consist of the Bremsstrahlung diagrams with pair produced quarks and the annihilation diagrams of quarks involving a gluon sourced by $ρ_p$. Because the final states are different there is no interference between these two processes. In this work we elucidate calculation procedures in details focusing on the annihilation diagrams only. Using the McLerran-Venugopalan model for the color average we numerically calculate the photon production rate and discuss functional forms that fit the numerical results.