The next-to-leading order of the differential cross section of the subprocess of annihilation of quark-antiquark pair of prompt photon production in proton-proton collisions at NICA energies

Technical Summary: Next-to-Leading Order Differential Cross-Section for Quark-Antiquark Annihilation in Prompt Photon Production at NICA Energies

Problem Statement
Prompt photon production in proton-proton collisions serves as a critical probe for understanding the dynamic structure of the proton and the formation of quark-gluon plasma (QGP). While theoretical investigations have historically focused on high-energy regimes (LHC, Tevatron), the Nuclotron-based Ion Collider fAcility (NICA) operates at a lower center-of-mass energy ($\sqrt{s} = 10$ GeV). At these energies, the production of additional elementary particles is limited, which simplifies the determination of effective cross-sections but requires precise theoretical tools to distinguish between different production mechanisms. Previous work by the authors established Leading Order (LO) calculations for Compton scattering, annihilation, and bremsstrahlung subprocesses. However, LO approximations are often insufficient for describing experimental data, particularly at large transverse momenta, necessitating higher-order corrections. This study addresses the need for a complete Next-to-Leading Order (NLO) calculation specifically for the quark-antiquark annihilation subprocess ($q\bar{q} \to \gamma g$) in both nonpolarized and longitudinally polarized proton-proton collisions at NICA energies.

Methodology
The research employs a dual approach combining analytical fixed-order calculations with Monte Carlo (MC) event generation:

1. Analytical NLO Calculation:

   • Diagram Generation: Using FeynCalc, 773 one-loop diagrams were initially generated for the $q\bar{q} \to \gamma$ subprocess. After applying selection criteria—excluding heavy particles not registered at NICA, triangular loops vanishing by the Furry theorem, "seagull" diagrams, mass-shell renormalization exclusions, and non-color-conserving transitions—13 contributing diagrams were retained.
   • Regularization and Renormalization: Calculations were performed using dimensional regularization within the Minimal Supersymmetric Standard Model (MSSM) scheme. The strong coupling constant ($\alpha_s$) and renormalization scales were defined, with the factorization scale set equal to the partonic center-of-mass energy ($\hat{s}$).
   • Polarization: Longitudinal polarization was incorporated using polarized Parton Distribution Functions (PDFs) and helicity-dependent matrix elements. Double spin asymmetry ($A_{LL}$) was calculated based on cross-sections for parallel ($\uparrow\uparrow$) and anti-parallel ($\uparrow\downarrow$) proton polarizations.
   • PDFs: The LHAPDF 6.5.5 framework was utilized to integrate NLO PDFs (specifically CT14nlo) for hadron-level cross-section calculations.

2. Monte Carlo Simulation:

   • Framework: The PYTHIA 8.316 generator was interfaced with the POWHEG BOX (V2) to simulate prompt photon production. This NLO+PS (Parton Shower) approach matched NLO matrix elements with parton showering and hadronization.
   • Configuration: Simulations were run for $\sqrt{s} = 10$ GeV with a transverse momentum cut ($p_T^{min} = 0.5$ GeV/c) to regulate infrared divergences. The CT14nlo PDF set was used for initial-state hadron structure.
   • Event Selection: Prompt photons were identified via PDG code 22 with status code 23, applying fiducial cuts on $p_T$ and rapidity ($y$).

Key Contributions and Results
The paper presents the first complete NLO calculations for the $q\bar{q}$ annihilation subprocess in the context of NICA energies, comparing them against LO results and MC simulations.

• Magnitude of NLO Corrections: The study demonstrates that NLO contributions are significant at high collision energies. The NLO differential cross-section contributes approximately 15% relative to the LO calculation at NICA energies. This contrasts with higher-energy environments (LHC/Tevatron), where NLO contributions can reach 35%.
• Kinematic Dependence:
  • Energy ($\sqrt{s}$): The total cross-section peaks at $\sqrt{s} \approx 5.2$ GeV and decreases at higher energies due to the running of the QCD coupling constant and PDF behavior.
  • Transverse Momentum ($p_T$): The differential cross-section ($d\sigma/dp_T$) decreases sharply with increasing $p_T$. The ratio of NLO to LO corrections increases with $p_T$, reaching up to 30% at large transverse momenta.
  • Angular and Rapidity Distributions: The angular distribution ($d\sigma/d\cos\theta$) exhibits a symmetric peaked shape with a maximum at $\cos\theta = 0$ (transverse) and minima at $\cos\theta = \pm 1$, driven by $t$-channel virtual photon exchange. The rapidity distribution shows a maximum at $|y| \approx 1.95$.
• Polarization Effects:
  • Longitudinal polarization does not alter the shape of the kinematic distributions but significantly affects the magnitude of the cross-sections.
  • Asymmetry: The double spin asymmetry ($A_{LL}$) reaches a maximum positive value ($+1$) when quark and antiquark spins are parallel and a maximum negative value ($-1$) when anti-parallel. The study finds that the influence of longitudinal polarization is more significant for NLO calculations than for LO, particularly at high energies. This is attributed to the sensitivity of virtual loop corrections and higher-order parton scattering configurations to initial spin orientations.
• Analytical vs. Simulation Comparison:
  • The ratio of analytical (FeynCalc) to simulated (PYTHIA) cross-sections varies with kinematics. At low $p_T$, the analytical fixed-order calculation underestimates the cross-section compared to PYTHIA due to logarithmic divergences in the soft/collinear limit, which PYTHIA resolves via Sudakov resummation and Initial State Radiation (ISR).
  • At high $p_T$, the ratio approaches unity, indicating convergence between fixed-order theory and the NLO+PS simulation.

Significance and Claims
The authors assert that this work provides a necessary theoretical refinement for analyzing prompt photon production at NICA energies. The primary significance lies in the demonstration that NLO corrections are non-negligible (approx. 15%) even at these moderate energies, justifying their inclusion in future experimental analyses. Furthermore, the study highlights that longitudinal polarization effects are amplified in NLO calculations, suggesting that polarization measurements could serve as a sensitive probe for QCD dynamics in this energy regime. The successful reproduction of pQCD signatures by the PYTHIA 8.316 + POWHEG BOX simulation validates the use of NLO+PS tools for predicting observables in proton-proton collisions at NICA, offering a more realistic description of exclusive observables compared to fixed-order calculations alone.