⚛️ phenomenology

The next-to-leading order of the differential cross-section of the subprocess of Compton scattering of quark-gluon of prompt photon production in proton-proton collisions at NICA energies

This paper presents a next-to-leading-order calculation of the differential cross-section for prompt photon production via quark-gluon Compton scattering in proton-proton collisions at NICA energies, demonstrating that higher-order corrections contribute approximately 15% to the cross-section and are more sensitive to proton polarization than leading-order results.

Original authors: Mohsun Rasim Alizada, Azar Inshalla Ahmadov

Published 2026-06-19

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CC BY 4.0

Original authors: Mohsun Rasim Alizada, Azar Inshalla Ahmadov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine two high-speed trains (protons) crashing into each other inside a giant, dark tunnel. Inside these trains, there are tiny, invisible passengers: quarks and gluons. When the trains collide, these passengers sometimes bump into each other and shoot out a flash of light—a "prompt photon." Because light doesn't get stopped by the messy debris of the crash, it acts like a perfect messenger, telling us exactly what happened inside the collision.

This paper is a detailed mathematical report on how to predict exactly how often these flashes of light happen when the trains crash at the specific speeds planned for a facility called NICA (located in Russia).

Here is the breakdown of their findings using simple analogies:

1. The "Recipe" for Accuracy: LO vs. NLO

The scientists are trying to write a recipe for predicting these light flashes.

LO (Leading Order): This is the "basic recipe." It's like baking a cake using only flour, eggs, and sugar. It gives you a cake, but it might not taste exactly like the real thing.
NLO (Next-to-Leading Order): This is the "advanced recipe." It adds the secret spices, the exact temperature of the oven, and the humidity of the room. It's much more complicated to calculate, but it's much closer to reality.

The Finding: The paper shows that at the high speeds of the NICA facility, the "advanced recipe" (NLO) is crucial. It adds about 15% more detail to the prediction compared to the basic recipe. If you only used the basic recipe, you'd be missing a significant chunk of the truth.

2. The "Traffic Jam" of Particles

The researchers looked at how the speed of the collision affects the number of light flashes.

The Analogy: Imagine the protons are like soft, round balls. When they move slowly, they are round and easy to hit. As they speed up, they get squashed flat, like a pancake or a disk, due to the laws of physics (Lorentz transformation).
The Result: The paper found that the number of light flashes increases as the trains speed up, but only up to a point (around 4.6 GeV). After that, because the protons have become so flat and "thin," the passengers inside are less likely to bump into each other. It's like trying to hit a flat sheet of paper with a needle; the chance of a hit drops because the target is so thin.

3. The "Direction" of the Flash

Where do these light flashes go?

The Analogy: Think of a sprinkler hose. Most of the water shoots out in a straight line, with very little spraying sideways.
The Result: The math shows that these light flashes prefer to shoot out almost straight ahead or almost straight backward (along the path of the trains), at angles of about 16 or 164 degrees. They rarely shoot out sideways.

4. The "Spin" of the Trains (Polarization)

This is the most unique part of the study. The scientists asked: "What if the trains are spinning in a specific direction when they crash?"

The Analogy: Imagine the trains are spinning tops. Sometimes they spin in the same direction (like two gears meshing), and sometimes they spin in opposite directions.
The Result: The paper found that the "advanced recipe" (NLO) is much more sensitive to this spinning than the "basic recipe" (LO).
- If the trains spin in opposite directions, the number of light flashes increases.
- If they spin in the same direction, the number decreases.
- Crucially, this effect is stronger when using the advanced NLO calculations. It's as if the "secret spices" in the advanced recipe react strongly to the direction of the spin, whereas the basic recipe barely notices it.

5. The "Speed Limit" of the Flash

The researchers also looked at how "hard" the flash hits (its transverse momentum).

The Analogy: Think of throwing a ball. It's easy to throw it gently (low speed), but it's very hard to throw it with extreme force (high speed).
The Result: The number of flashes drops off very quickly as you look for harder, faster flashes. The "advanced recipe" (NLO) is needed to accurately predict these rare, high-speed events, especially at the higher energy levels of the collision.

Summary

In short, this paper is a mathematical proof that to understand what happens when protons collide at NICA energies, you cannot just use the simple, old-school calculations. You must use the complex, "Next-to-Leading Order" math.

Why? Because at these speeds, the "advanced math" reveals that:

The collision shape changes from round to flat, changing the hit rate.
The direction of the light flashes is very specific (straight ahead/back).
The spin of the colliding protons has a much bigger impact on the results than previously thought, but only if you use the advanced math to see it.

The authors conclude that if scientists want to analyze data from the NICA facility correctly, they must include these advanced calculations, or their understanding of the proton's structure will be incomplete.

Technical Summary: Next-to-Leading Order Differential Cross-Section of Quark-Gluon Compton Scattering in Prompt Photon Production at NICA Energies

Problem Statement
The production of prompt photons in proton-proton collisions serves as a critical probe for the dynamic structure of the proton and the gluon distribution within it. While theoretical and experimental investigations have been conducted at high-energy facilities like the LHC and Tevatron, research at the lower energy range of the NICA (Nuclotron-based Ion Collider fAcility) facility (10–27 GeV) presents unique challenges. At these energies, the production of many elementary particles is restricted, making the accurate determination of differential cross-sections difficult. Previous studies by the authors addressed this process at Leading Order (LO), identifying mixed Compton scattering of quark-gluon ( $qg \to q\gamma$ ) and quark-antiquark annihilation ( $q\bar{q} \to g\gamma$ ) as dominant mechanisms. However, LO approximations are often insufficient for describing experimental data, particularly at large transverse momenta ( $p_T$ ), necessitating higher-order corrections. This paper addresses the need for Next-to-Leading Order (NLO) calculations of the differential cross-section for the quark-gluon Compton scattering subprocess, specifically investigating the impact of longitudinal polarization of colliding protons at NICA energies.

Methodology
The authors performed NLO calculations using perturbative Quantum Chromodynamics (pQCD) and the factorization theorem to separate short-range particle interactions from long-range hadronic effects.

Diagram Generation: Using FeynCalc, 773 one-loop Feynman diagrams were initially generated. A rigorous selection process excluded diagrams involving heavy particles not registered at NICA (H, G, Z, W, t, c, b, $\mu$ , $\tau$ , e), triangular loops vanishing by the Furry theorem, "seagull" diagrams, loops on external legs (excluded in the on-mass-shell renormalization scheme), and non-color-conserving photon-gluon transitions. This resulted in 11 contributing diagrams for the NLO calculation.
Regularization and Renormalization: Dimensional regularization was applied to handle ultraviolet and infrared divergences. The strong coupling constant ( $\alpha_s$ ) was evaluated at the factorization scale $\hat{s}$ .
Parton Distribution Functions (PDFs): To calculate hadron-level cross-sections, the authors utilized NLO-level PDFs ( $f(x, \mu)$ ) and polarized PDFs ( $\Delta f(x, \mu)$ ) via the LHAPDF library (version 6.5.5), ensuring consistency in scale variation.
Polarization: The study incorporated longitudinal polarization for both initial quarks and gluons. Polarized PDFs were used to account for the helicity of the initial particles, and the double-spin asymmetry ( $A_{LL}$ ) was calculated to quantify the influence of proton polarization.
Kinematic Variables: The analysis focused on dependencies regarding the center-of-mass energy ( $\sqrt{s}$ ), transverse momentum ( $p_T$ ), scattering angle ( $\theta$ ), rapidity ( $y$ ), and the variable $x_T = 2p_T/\sqrt{s}$ .

Key Results

Magnitude of NLO Corrections: The contribution of NLO corrections to the differential cross-section is significant at high collision energies. The NLO contribution constitutes approximately 15% of the LO calculation at NICA energies. This is notably lower than the 35% contribution observed at LHC and Tevatron energies, reflecting the energy dependence of radiative corrections.
Kinematic Dependencies:
- Energy ( $\sqrt{s}$ ): The total cross-section increases with energy up to a maximum at $\sqrt{s} = 4.6$ GeV, after which it decreases. The authors attribute this decrease to Lorentz contraction effects (transition from spherical to disk shape) reducing component collision probability, and a decrease in interaction time (strong coupling).
- Transverse Momentum ( $p_T$ ): The differential cross-section decreases rapidly as $p_T$ increases. The ratio of NLO to LO ( $R$ ) is highest at low $p_T$ (reaching ~6.5% at 1 GeV/c) and decreases as $p_T$ grows, indicating that higher-order corrections are more dominant in the soft gluon emission region.
- Scattering Angle: The cross-section is maximized when the photon scatters at angles approaching 16° or 164° (along the collision axis) and is minimized at 90°.
- Rapidity ( $y$ ): The distribution is symmetric around $y=0$ , with maxima at $y \approx \pm 1.95$ and a minimum at $y=0$ .
Polarization Effects:
- The influence of longitudinal polarization on NLO calculations is found to be more significant than on LO calculations.
- In the case of oppositely directed polarizations ( $P_1, P_2 = 0.9, -0.9$ ), the differential cross-section increases, while it decreases for aligned polarizations ( $P_1 = P_2 = \pm 0.9$ ).
- Double-spin asymmetry ( $A_{LL}$ ) is large at low $p_T$ and diminishes as $p_T$ increases, approaching zero at high energies where polarization effects become negligible.
Comparison with LO: While LO calculations generally yield larger absolute values for differential cross-sections, the relative difference between LO and NLO is small at both very low and very high $p_T$ . However, the NLO corrections become increasingly important for accuracy at high energies.

Significance and Conclusions
The paper establishes that NLO calculations are essential for the accurate theoretical description of prompt photon production in proton-proton collisions at NICA energies. The study demonstrates that while the absolute contribution of NLO terms (approx. 15%) is smaller at NICA energies compared to higher-energy colliders, the sensitivity of these corrections to proton polarization is enhanced in the NLO regime.

The authors conclude that the inclusion of NLO contributions is necessary for the simulation and analysis of experimental data at NICA and FAIR energies. The results highlight that new Feynman diagrams in NLO, involving virtual loops and additional kinematic configurations (such as gluon emission), introduce kinematic dependencies that are more sensitive to the spin orientation of the colliding protons than the constrained kinematics of LO. Consequently, neglecting these higher-order terms could lead to inaccuracies in determining parton distribution functions and testing perturbative QCD in the intermediate energy regime.