⚛️ phenomenology

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

This paper presents a comprehensive theoretical and numerical study of QCD and QED radiative corrections to prompt photon production via quark-antiquark annihilation in proton-proton collisions at NICA energies ( $\sqrt{s}=10$ GeV), utilizing FeynCalc and PYTHIA 8.316 to analyze differential cross-sections and spin asymmetries, revealing that these corrections are significant at low energies and small transverse momenta with excellent agreement between analytical calculations and simulations.

Original authors: Mohsun R. Alizada, Azar I. Ahmadov

Published 2026-10-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohsun R. Alizada, Azar I. 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

High-energy physics often feels like trying to understand a storm by studying a single raindrop. In the world of particle accelerators, scientists smash protons together at incredible speeds to break them apart and see what lies inside. Among the debris, photons—particles of light that carry no electric charge and no color charge—play a unique role. Because they do not interact with the strong nuclear force that binds protons together, they can escape the chaotic collision zone without being deflected or absorbed. They act as perfect messengers, flying straight out of the interaction point to tell researchers exactly what happened at the moment of impact. This makes them invaluable for mapping the internal structure of protons, specifically how the smaller particles inside, known as quarks and gluons, are distributed and how they move.

While much of this research happens at the world's most powerful machines, like the Large Hadron Collider, there is a different kind of physics happening at lower energies. At the Nuclotron-based Ion Collider Facility (NICA) in Russia, protons collide at a center-of-mass energy of 10 GeV. This is a much gentler collision compared to the trillions of electron volts seen at the Large Hadron Collider, but it offers a unique window into how matter transitions between different states, such as the formation of a quark-gluon plasma. In this lower-energy regime, the rules of the game change. The dominant processes that create light at the highest energies are not the same ones that matter here. To understand the data coming out of NICA, scientists must calculate exactly how these collisions produce photons, including the subtle, secondary effects that occur when particles emit extra radiation during the crash.

A team of researchers from Baku State University has undertaken a detailed study of one specific way photons are created at NICA: the annihilation of a quark and an antiquark. In this process, a particle of matter meets its antimatter counterpart, and they vanish, converting their mass into energy that manifests as a photon. However, the universe is rarely simple. When these particles collide, they often emit additional particles, such as extra gluons or photons, before or after the main event. These are called radiative corrections. The researchers wanted to know how much these extra emissions change the final result. They focused on three specific variations of this annihilation process: one where the quark and antiquark produce a photon and two gluons, another where they produce a photon and a quark-antiquark pair, and a third where they produce a photon and two photons.

To get a complete picture, the team performed two types of calculations. First, they used rigorous mathematical methods to derive exact formulas for the probability of these events happening. They calculated how the total amount of light produced changes depending on the energy of the collision, the angle at which the photon flies out, and how fast it moves sideways relative to the beam. They also looked at what happens when the protons are polarized, meaning their internal spins are aligned in a specific direction, rather than being randomly oriented. This allowed them to see how the spin of the protons influences the creation of these photons. Second, to ensure their math matched reality, they used a sophisticated computer simulation called PYTHIA. This program mimics the complex chain of events that happens in a real detector, including the cascading showers of particles that occur after the initial collision.

The results revealed a clear hierarchy in how these processes behave. The most common way for these extra particles to appear is through the emission of gluons, which are the carriers of the strong nuclear force. This channel dominates the total production, accounting for the vast majority of the events. The process involving two photons is extremely rare, suppressed by the nature of electromagnetic interactions. A key finding was that these extra emissions are most significant when the collision energy is low and the photons have a small amount of sideways momentum. As the energy increases, the relative importance of these corrections drops, and the behavior of the different processes begins to level off. The researchers also found that the polarization of the protons has a profound effect. When the protons are aligned, the probability of certain outcomes changes dramatically, particularly for the process involving a quark and an antiquark in the final state, which becomes the dominant channel at high sideways momenta.

The study confirmed that the mathematical formulas and the computer simulations agree very well in the central regions where most collisions occur. However, at the very edges of what is physically possible—when the photon carries away almost all the available energy—the two methods diverge. The strict mathematical formulas predict a sharp spike in probability that does not exist in the real world, a phenomenon known as a divergence. The computer simulation, which accounts for the natural "smearing" of energy and the emission of soft particles, smooths out these spikes, providing a more realistic picture. This difference highlights the necessity of using advanced simulation techniques to interpret data at the boundaries of the phase space.

Ultimately, this work provides a precise map of how quark-antiquark annihilation produces light at NICA energies. It shows that while the basic process is well understood, the inclusion of radiative corrections is essential for accurate predictions, especially at lower energies. The research demonstrates that at these specific energy levels, the annihilation channel is a competitive and vital source of photons, unlike at much higher energies where other processes take over. By clarifying the behavior of these corrections and the role of proton polarization, the study offers a solid foundation for future experiments at NICA, helping physicists distinguish between the signals of new physics and the expected background of known particle interactions.

Technical Summary: Radiative Corrections to Quark-Antiquark Annihilation in Prompt Photon Production at NICA Energies

Problem Statement
This work addresses the theoretical description of prompt photon production in proton-proton ($pp$) collisions at the moderate center-of-mass energy of $\sqrt{s} = 10$ GeV, corresponding to the NICA (Nuclotron-based Ion Collider Facility) energy regime. While high-energy colliders like the LHC are dominated by gluon-initiated Compton scattering ( $qg \to q\gamma$ ), the authors posit that at NICA energies, the annihilation channel of quark-antiquark pairs ( $q\bar{q} \to \gamma$ ) is highly competitive, contributing approximately 48% to the total cross-section. The study focuses specifically on the quantum-chromodynamic (QCD) and quantum-electrodynamic (QED) radiative corrections to this annihilation subprocess. The primary objective is to calculate the differential cross-sections and double spin asymmetries for three specific radiative correction subprocesses:

$q\bar{q} \to \gamma g$ (quark-antiquark annihilation to photon and gluon)
$q\bar{q} \to \gamma q\bar{q}$ (quark-antiquark annihilation to photon and quark-antiquark pair)
$q\bar{q} \to \gamma \gamma$ (quark-antiquark annihilation to two photons)

The study investigates these processes for both non-polarized and longitudinally polarized colliding protons to determine the impact of radiative corrections on kinematic distributions and spin observables.

Methodology
The research employs a dual approach combining analytical calculations and numerical simulations:

Analytical Calculation:
- The differential cross-sections for the three subprocesses were derived analytically using the FeynCalc package.
- Calculations were performed at the parton level and subsequently convoluted with Parton Distribution Functions (PDFs) to obtain hadron-level cross-sections.
- Both unpolarized and polarized PDFs were utilized to account for longitudinal proton polarization.
- The study utilized LHAPDF version 6.5.5 to access modern PDF sets with scale-dependent evolution, ensuring consistency in the evaluation of factorization and renormalization scales.
- Kinematic variables analyzed included the sum of collision energies ( $\sqrt{s}$ ), transverse momentum ( $p_T$ ), scattering angle cosine ( $\cos\theta$ ), rapidity ( $y$ ), and the scaling variable $\tau$ .
Numerical Simulation:
- Numerical modulation was performed using the PYTHIA 8.316 Monte Carlo event generator.
- The simulation was configured to isolate parton-level dynamics by disabling the hadronization stage (HadronLevel: all = off) to ensure a direct comparison with the analytical FeynCalc results.
- Initial State Radiation (ISR) and Final State Radiation (FSR) were enabled to account for parton showers, transverse recoil ( $k_T$ -kick), and multiparton final states.
- Events were classified topologically to separate the contributions of the three specific subprocesses based on the final state particle composition (e.g., selecting events with exactly one photon and specific gluon/quark counts).

Key Contributions and Results

Hierarchy of Cross-Sections: The study establishes a strict hierarchy among the radiative correction subprocesses. The gluon emission channel ( $q\bar{q} \to \gamma g$ ) consistently dominates the total integrated cross-section, accounting for approximately 80–100% of the total due to strong infrared and collinear enhancements associated with massless gluons. The two-photon channel ( $q\bar{q} \to \gamma \gamma$ ) is heavily suppressed. However, a radical change in dominance is observed as a function of transverse momentum ( $p_T$ ): while the gluon channel dominates at low and moderate $p_T$ , the quark channel ( $q\bar{q} \to \gamma q\bar{q}$ ) rapidly increases in share as $p_T$ increases, asymptotically approaching 100% at high momentum transfers. This is attributed to the "harder" spectrum of the fermion-line diagram compared to the steeper, rapidly decaying spectrum of the multi-gluon channel.
Kinematic Dependence:
- Energy ( $\sqrt{s}$ ): At low energies (near the threshold), radiative corrections are significant. As energy increases, the cross-sections exhibit a sharp rise followed by an asymptotic plateau, characteristic of hard cross-sections in the Bjorken scaling regime.
- Transverse Momentum ( $p_T$ ): The differential cross-sections show a monotonic decline with increasing $p_T$ . The contribution of radiative corrections is most significant at low $p_T$ .
- Angular and Rapidity Distributions: The angular distributions exhibit a characteristic "U-shape" (peaks at forward/backward scattering due to $t$ - and $u$ -channel poles) and a "two-humped" structure in rapidity.
Polarization Effects:
- The longitudinal polarization of colliding protons has a nearly identical effect on all three subprocesses.
- Double Spin Asymmetry ( $A_{LL}$ ): The study finds that spin correlations in multi-gluon/photon vertices are quickly smeared out or cancelled out. Conversely, the $q\bar{q} \to \gamma q\bar{q}$ channel efficiently conserves helicity in hard collisions, leading to macroscopic asymmetry values (approaching 100%) at large momentum transfers and transverse scattering angles, whereas the multi-gluon channel is suppressed over the entire range in the polarized case at high $p_T$ .
- The double spin asymmetry is maximal at low energies and vanishes at high energies where sea quarks and gluons (with low polarization) dominate.
Analytical vs. Simulation Comparison:
- Central Region: There is excellent agreement between the strict Leading Order (LO) analytical calculations (FeynCalc) and the PYTHIA 8.316 simulations in the central kinematic regions.
- Phase Space Boundaries: Significant discrepancies arise at kinematic boundaries (collinear limits, extreme rapidity, and $p_T \to \sqrt{s}/2$ ). The analytical LO calculations suffer from unphysical infrared and collinear divergences, whereas PYTHIA 8.316 regularizes these singularities via Sudakov form factors and parton showers. Consequently, the ratio of analytical to simulated cross-sections diverges near these boundaries.

Significance and Claims
The paper claims that the annihilation channel $q\bar{q} \to \gamma$ is a highly competitive mechanism at NICA energies, unlike at the LHC where gluon density suppresses this channel. The study demonstrates that radiative corrections are non-negligible, contributing approximately 7.8% to the main subprocess, with their impact being most pronounced at low collision energies and small transverse momenta.

The authors assert that the NICA energy regime provides a "clean environment" to test perturbative QCD (pQCD) calculations near the kinematic threshold without the overwhelming gluon backgrounds found at higher energies. The work validates the necessity of including higher-order radiative corrections and logarithmic resummation techniques (as modeled by parton showers in PYTHIA) to accurately describe spin observables and differential cross-sections, particularly near phase-space boundaries where fixed-order analytical calculations fail. The results confirm that the hierarchy of subprocesses (2 > 1 > 3 for total cross-sections) and the relations between their cross-sections hold true for both polarized and non-polarized collisions, while noting the specific dominance shift of the quark channel at high transverse momentum.

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