Radiative corrections to the subprocess of annihilation of quark-antiquark pair of prompt photon production in proton-proton collisions at NICA energies $\sqrt{\mathbf{s}}$= 10 GeV

In this work, a comprehensive theoretical and numerical study of the quantum-chromodynamic and quantum-electrodynamic radiative corrections: 1. $q\bar{q}\to q\bar{q}\gamma$, 2. $q\bar{q}\to gg\gamma$, and 3. $q\bar{q}\to g\gamma \gamma$ to the annihilation of a quark-antiquark pair $q\bar{q}\to g\gamma$ of prompt photon production in proton-proton collisions at NICA energies $\sqrt{s}$ = 10 GeV was performed. The differential cross-sections and double spin asymmetries $A_{LL}$ of the radiative corrections were investigated analytically using the FeynCalc and modulated in PYTHIA 8.316. Radiative corrections are significant at low collision energies of protons $\sqrt{s}$ and small transverse momentum of photons $p_T$. It was shown that radiative corrections to the annihilation of a quark-antiquark pair process satisfy the following relations: $$2.{\sigma }_{\sqrt{s}}(q\bar{q}\to gg\gamma )>1.{\sigma }_{\sqrt{s}}(q\bar{q}\to q\bar{q}\gamma )>3.{\sigma }_{\sqrt{s}}(q\bar{q}\to g\gamma \gamma )$$ $$1.\frac{d\sigma (q\bar{q}\to q\bar{q}\gamma )}{d{p}_T^2}>3.\frac{d\sigma (q\bar{q}\to g\gamma \gamma )}{d{p}_T^2}>2.\frac{d\sigma (q\bar{q}\to gg\gamma )}{d{p}_T^2}$$ The polarization of colliding protons has almost the same effect on processes 1. $q\bar{q}\to q\bar{q}\gamma$, 2. $q\bar{q}\to gg\gamma$, and 3. $q\bar{q}\to g\gamma \gamma$ and in this case above indicated relation is also satisfied. A direct comparison between strict leading order analytical calculations in FeynCalc and PYTHIA 8.316 modulations revealed excellent agreement in the central kinematic regions.