Prompt photon production in a bremsstrahlung in proton-proton collisions at $\sqrt{\mathbf{s}}$=10 GeV NICA energies

This paper investigates the kinematic dependencies and polarization effects on the differential cross-section and double spin asymmetry of prompt photon production via bremsstrahlung in proton-proton collisions at NICA energies of $\sqrt{s}=10$ GeV, noting that while this process constitutes a small fraction (0.03%) of the total prompt photon yield, it exhibits significant sensitivity to proton polarization at high transverse momenta.

The Gist

The Big Picture: A High-Speed Traffic Jam

Imagine two streams of cars (protons) speeding toward each other on a highway. Inside these cars are tiny passengers called quarks and gluons. When the cars crash, these passengers sometimes bounce off each other so hard that they spit out a flash of light—a photon.

In physics, we call these flashes "prompt photons" because they happen instantly during the crash, not later when the wreckage settles. Scientists want to understand exactly how often these flashes happen and what they tell us about the cars and passengers.

This paper focuses on a specific, somewhat rare type of crash called Bremsstrahlung (German for "braking radiation").

The Main Character: The "Braking" Photon

Usually, when two cars crash, the passengers might bounce off and hit a third car, or they might annihilate each other. But in Bremsstrahlung, two quarks crash, bounce off each other, and as they "brake" or change direction sharply, they emit a photon.

Think of it like a race car driver slamming on the brakes to avoid a wall. The sudden stop creates a loud screech (sound). In the quantum world, that "screech" is a flash of light (a photon).

The Paper's Main Finding:
The authors calculated that at the specific energy levels of the NICA facility (a particle accelerator in Russia, operating at 10 GeV), this "braking" type of photon is very rare. It accounts for only 0.03% of all the prompt photons produced. The other 99.97% come from two other, more common types of crashes (Compton scattering and annihilation).

The Experiment: Unpolarized vs. Polarized Cars

The researchers looked at two scenarios:

1. Unpolarized: The cars are driving normally, with their passengers spinning in random directions.
2. Polarized: The cars are driving with their passengers spinning in a specific, coordinated direction (like all the drivers holding their hands up).

The Surprising Discovery:
Even though the "braking" photons are rare, the direction the passengers are spinning (polarization) matters a lot when the crash is very hard (high transverse momentum).

• If the passengers spin in the same direction, the crash produces more braking photons.
• If they spin in opposite directions, the crash produces fewer braking photons.

It's like a dance: if two dancers spin the same way, they might create a bigger splash of water when they collide. If they spin opposite ways, the splash is smaller. The paper found that this "spin effect" gets stronger the harder the crash is.

The "Double Spin" Asymmetry

The paper also calculated something called "Double Spin Asymmetry." Imagine a scale that measures the difference between "same-spin crashes" and "opposite-spin crashes."

• The paper found that this scale swings wildly depending on the energy and angle of the crash.
• At certain specific speeds and angles, the scale hits zero. This means that at that exact moment, it doesn't matter which way the passengers are spinning; the result is the same. This is a "magic point" where the physics cancels itself out.

The Tools: Math vs. Simulation

To get these results, the authors used two different methods:

1. FeynCalc: A rigorous mathematical tool that calculates the "pure" physics of the crash, ignoring the messy details of what happens after the impact.
2. PYTHIA: A computer simulation that acts like a video game engine. It includes the "messy" stuff: how the particles shower, how they stick together, and how they turn into other particles (hadronization).

The Comparison:

• At low energies, the simulation (PYTHIA) showed fewer photons than the math (FeynCalc). This is because the simulation includes "soft" effects and noise that the pure math ignores.
• At high energies, the two methods agreed perfectly.

Why Does This Matter?

The NICA facility is unique because it operates at an energy level where the universe is transitioning from a "soup" of free particles (Quark-Gluon Plasma) back into solid matter (hadrons).

By studying these rare "braking" photons, especially when the protons are polarized (spinning in a specific way), scientists can:

• Better understand the internal structure of the proton.
• Test the rules of Quantum Chromodynamics (the theory of how quarks and gluons interact).
• Distinguish between different types of particle interactions in this specific energy range.

Summary in a Nutshell

• The Event: Two protons crash, and two quarks inside them "brake," creating a flash of light.
• The Rarity: This happens very rarely (0.03% of the time) compared to other crash types.
• The Twist: If the protons are "spinning" in a coordinated way, the number of flashes changes significantly, especially in hard crashes.
• The Result: The paper maps out exactly how often these flashes happen at different speeds and angles, confirming that while rare, this process is sensitive to the "spin" of the particles, offering a new way to probe the secrets of matter at the NICA facility.

Technical Summary

Technical Summary: Prompt Photon Production in Bremsstrahlung at NICA Energies

Problem Statement
This study investigates the production of prompt photons via the bremsstrahlung subprocess ($qq \to qq\gamma$) in proton-proton collisions at the Nuclotron-based Ion Collider Facility (NICA), specifically at a center-of-mass energy of $\sqrt{s} = 10$ GeV. While prompt photon production is a well-established tool for probing parton distribution functions (PDFs) and testing perturbative Quantum Chromodynamics (pQCD) at high energies (LHC, Tevatron), the dominant mechanisms there are Compton scattering ($qg \to q\gamma$) and quark-antiquark annihilation ($q\bar{q} \to g\gamma$). The bremsstrahlung channel is typically a minor contributor. However, at NICA energies, where the transition from partonic to hadronic degrees of freedom occurs and polarized proton beams are available, a detailed understanding of this specific subprocess is necessary to fully characterize the prompt photon spectrum and spin-dependent observables.

Methodology
The authors employed a theoretical framework combining analytical pQCD calculations with Monte Carlo simulations:

• Analytical Calculation: The process was modeled using 16 Feynman diagrams generated by FeynArts. Matrix elements were computed using FeynCalc. The study derived the squared matrix elements for both nonpolarized and longitudinally polarized proton collisions.
• Kinematic Framework: The differential cross-sections were calculated at the parton level and convoluted with Parton Distribution Functions (PDFs) to obtain hadron-level results. The analysis utilized modern PDF sets via LHAPDF (version 6.5.5) to account for scale dependence.
• Polarization: Longitudinal polarization effects were incorporated using polarized PDFs ($\Delta u, \Delta d, \Delta G$) and calculating the double spin asymmetry ($A_{LL}$).
• Simulation and Validation: Results were compared against simulations performed with PYTHIA 8.315. The PYTHIA setup included initial- and final-state parton showers (ISR/FSR), multiparton interactions (MPI), and hadronization (Lund string model), providing a benchmark that includes non-perturbative effects absent in the pure pQCD analytical approach.
• Variables: The study examined dependencies on the sum of collision energies ($\sqrt{s}$), transverse momentum ($p_T$), scattering angle cosine ($\cos\theta$), rapidity ($y$), and the variable $x_T$.

Key Contributions and Results

1. Magnitude of the Bremsstrahlung Channel: The authors determined that the bremsstrahlung subprocess ($qq \to qq\gamma$) accounts for approximately 0.03% of the total differential cross-section for prompt photon production at NICA energies. This is significantly smaller than the contributions from Compton scattering (>50%) and quark-antiquark annihilation (~43%).
2. Kinematic Dependencies (Nonpolarized):
   • Energy ($\sqrt{s}$): The cross-section rises rapidly at low energies ($\sqrt{s} < 3$ GeV), peaks in the intermediate range (3–7 GeV), and decreases at higher energies ($\sqrt{s} > 7$ GeV) due to the "dead cone" effect and coupling constant variations.
   • Transverse Momentum ($p_T$): The differential cross-section decreases rapidly as $p_T$ increases. The authors note that at low NICA energies, collisions are more likely to produce photons with large $p_T$ compared to high-energy regimes, a behavior distinct from Compton scattering.
   • Angular Distribution: The distribution is symmetric and bell-shaped relative to $\cos\theta$, with maximum production occurring near the collision axis ($\cos\theta \approx \pm 1$, corresponding to angles of ~16° and 164°) and a minimum at 90°.
3. Polarization Effects:
   • The polarization of protons significantly influences the differential cross-section, particularly at large values of transverse momentum ($p_T$).
   • Spin Alignment: When proton spins are parallel, the cross-section increases; when antiparallel, it decreases relative to the nonpolarized case.
   • Double Spin Asymmetry ($A_{LL}$): The study calculated $A_{LL}$ as a function of $p_T$ and $x_T$. A key finding is that the magnitude of the asymmetry $|A_{LL}|$ increases with $p_T$, indicating stronger spin dependence in "harder" collisions.
   • Zero-Asymmetry Point: The authors identified a non-trivial prediction where $A_{LL}$ vanishes at $x_T \approx 0.7\text{--}0.75$. At this specific kinematic point, the contributions of different Feynman diagrams cancel out, rendering the process insensitive to the initial spin configuration.
4. Comparison with PYTHIA:
   • The functional dependencies of the cross-sections on kinematic variables obtained via FeynCalc and PYTHIA were found to be identical in shape.
   • However, numerical values differed, particularly at low energies and low $p_T$. PYTHIA yielded lower cross-sections in these regions due to the inclusion of non-perturbative effects (soft radiation, hadronization, MPI) which are not present in the pure pQCD matrix element calculation.

Significance and Claims
The paper positions its work as a necessary component of the broader physics program at NICA, specifically for the Spin Physics Detector (SPD). The authors claim that while bremsstrahlung is a minor contributor to the total prompt photon yield, its specific kinematic behavior and spin dependence must be understood to isolate other signals and accurately interpret experimental data.

The study highlights that:

• Polarization effects in bremsstrahlung are most pronounced at large $p_T$, contrasting with Compton scattering and annihilation where polarization is more significant at small $p_T$.
• The double spin asymmetry for bremsstrahlung ($A_{LL} = \lambda_1 \lambda_2$) differs in sign from that of quark-antiquark annihilation ($A_{LL} = -\lambda_1 \lambda_2$), providing a distinct signature for experimental identification.
• The linear relationship between the double spin asymmetry and the product of beam polarizations ($P_1 P_2$) allows for direct interpretation of measured asymmetries in terms of beam polarization and underlying QCD dynamics.

The authors conclude that their analytical results, validated against PYTHIA simulations, provide a baseline for understanding the interplay between perturbative and non-perturbative QCD effects in prompt photon production at intermediate energies, aiding in the precise measurement of spin-dependent parton distributions.